From 17a308208fe9261eaca063cd41daac7641bad7b2 Mon Sep 17 00:00:00 2001 From: qiujiantao Date: Wed, 5 Mar 2025 22:42:29 +0800 Subject: [PATCH 1/8] =?UTF-8?q?fix:=20=E8=B0=83=E6=95=B4=E4=B8=8B=E8=BD=BD?= =?UTF-8?q?=E9=94=81=E5=92=8C=E8=A7=A3=E5=8E=8B=E7=BC=A9=E9=94=81=20?= =?UTF-8?q?=E4=BD=BF=E5=BE=97=E7=AD=89=E5=BE=85=E9=94=81=E6=9C=9F=E9=97=B4?= =?UTF-8?q?=E4=B9=9F=E5=8F=AF=E4=BB=A5=E6=A3=80=E6=9F=A5=E6=98=AF=E5=90=A6?= =?UTF-8?q?=E5=B7=B2=E7=BB=8F=E8=A2=AB=E5=88=AB=E7=9A=84=E8=BF=9B=E7=A8=8B?= =?UTF-8?q?=E5=AE=8C=E6=88=90=E4=BA=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .codecov.yml | 3 +- bench/data/groundtruth/code_1.jsonl | 2 +- bench/data/groundtruth/code_11.jsonl | 2 +- bench/data/groundtruth/code_16.jsonl | 2 +- bench/data/groundtruth/code_17_csdn.jsonl | 2 +- bench/data/groundtruth/code_5.jsonl | 2 +- bench/data/groundtruth/code_6.jsonl | 2 +- bench/data/groundtruth/math_1.jsonl | 2 +- .../data/groundtruth/math_katex_latex_1.jsonl | 2 +- .../data/groundtruth/math_katex_latex_3.jsonl | 2 +- .../math_mathjax_asciimath_1.jsonl | 2 +- .../groundtruth/math_mathjax_latex_1.jsonl | 2 +- .../groundtruth/math_mathjax_latex_3.jsonl | 2 +- .../groundtruth/math_mathjax_latex_4.jsonl | 2 +- .../groundtruth/math_mathjax_latex_5.jsonl | 2 +- .../groundtruth/math_mathjax_latex_6.jsonl | 2 +- .../groundtruth/math_mathjax_latex_7.jsonl | 2 +- .../groundtruth/math_mathjax_mathml_1.jsonl | 2 +- .../data/groundtruth/math_none_mathml_1.jsonl | 2 +- bench/data/groundtruth/table_1.jsonl | 2 +- bench/data/groundtruth/table_2.jsonl | 2 +- bench/data/groundtruth/table_3.jsonl | 2 +- bench/data/groundtruth/table_4.jsonl | 2 +- bench/run.py | 4 +- docs/llm_web_kit/model/lang_id.md | 14 ++-- llm_web_kit/config/cfg_reader.py | 24 +++++-- .../extractor/html/magic_html/config.py | 1 + .../html/recognizer/cc_math/common.py | 7 +- llm_web_kit/model/lang_id.py | 53 +++++++-------- .../model/resource_utils/download_assets.py | 39 +++++++---- llm_web_kit/model/resource_utils/unzip_ext.py | 67 ++++++++++++++++--- .../extractor/html/recognizer/test_math.py | 8 +++ .../resource_utils/test_download_assets.py | 12 ++-- .../model/resource_utils/test_unzip_ext.py | 5 +- tests/llm_web_kit/model/test_lang_id.py | 53 +++++---------- tests/st/test_st.py | 11 ++- 36 files changed, 205 insertions(+), 140 deletions(-) diff --git a/.codecov.yml b/.codecov.yml index 4b273de4..0b06fb59 100644 --- a/.codecov.yml +++ b/.codecov.yml @@ -3,7 +3,8 @@ coverage: patch: # 只检查变更部分的覆盖率 default: target: 85% # 变更代码的覆盖率目标 - threshold: 1% # 允许的浮动范围 + threshold: 2% # 允许的浮动范围 + base: auto # 基于当前分支的覆盖率 # 忽略特定路径 ignore: diff --git a/bench/data/groundtruth/code_1.jsonl b/bench/data/groundtruth/code_1.jsonl index f3654a22..a430bcde 100644 --- a/bench/data/groundtruth/code_1.jsonl +++ b/bench/data/groundtruth/code_1.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "paragraph", "raw_content": "

I load some HTML into an iframe but when a file referenced is using http, not https, I get the following error:

", "content": [{"c": "I load some HTML into an iframe but when a file referenced is using http, not https, I get the following error:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

[blocked] The page at {current_pagename} ran insecure content from {referenced_filename}

", "content": [{"c": "[blocked] The page at {current_pagename} ran insecure content from {referenced_filename}", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Is there any way to turn this off or any way to get around it?

", "content": [{"c": "Is there any way to turn this off or any way to get around it?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

The iframe has no src attribute and the contents are set using:

", "content": [{"c": "The iframe has no", "t": "text"}, {"c": "src", "t": "code-inline"}, {"c": "attribute and the contents are set using:", "t": "text"}]}, {"type": "code", "raw_content": "frame.open();\nframe.write(html);\nframe.close();\n", "inline": false, "content": {"code_content": "frame.open();\nframe.write(html);\nframe.close();", "by": "tag_pre_code"}}]], "main_html": "

I load some HTML into an iframe but when a file referenced is using http, not https, I get the following error:

[blocked] The page at {current_pagename} ran insecure content from {referenced_filename}

Is there any way to turn this off or any way to get around it?

The iframe has no src attribute and the contents are set using:

frame.open();\nframe.write(html);\nframe.close();\n", "statics": {"paragraph": 4, "paragraph.text": 5, "paragraph.code-inline": 1, "code": 1}, "url": "https://stackoverflow.com/questions/18327314/how-to-allow-http-content-within-an-iframe-on-a-https-site", "content": "I load some HTML into an iframe but when a file referenced is using http, not https, I get the following error:\n\n[blocked] The page at {current_pagename} ran insecure content from {referenced_filename}\n\nIs there any way to turn this off or any way to get around it?\n\nThe iframe has no `src` attribute and the contents are set using:\n\n```\nframe.open();\nframe.write(html);\nframe.close();\n```\n", "html": "\n\n\n\n\n \n\n \n\n html - How to allow http content within an iframe on a https site - Stack Overflow\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n\n\n\n \n \n\n\n\n\n \n\n\n\n\n\n\n\n \n\n\n\n \n \n \n \n\n\n
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I load some HTML into an iframe but when a file referenced is using http, not https, I get the following error:

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[blocked] The page at {current_pagename} ran insecure content from {referenced_filename}

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Is there any way to turn this off or any way to get around it?

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The iframe has no src attribute and the contents are set using:

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\n edited Jan 5, 2017 at 12:39\n
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    \n\n stackoverflow.com/questions/9280665/…\n\n
    \n– chiliNUT\n
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    \n\n made an edit. The src is not set as the content is written into the iframe\n\n
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    \n\n Thanks for all the answers. Long story short is proxy the content.\n\n
    \n– georgephillips\n
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    \n– Gintas_\n
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Note: While this solution may have worked in some browsers when it was written in 2014, it no longer works. Navigating or redirecting to an HTTP URL in an iframe embedded in an HTTPS page is not permitted by modern browsers, even if the frame started out with an HTTPS URL.

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The best solution I created is to simply use google as the ssl proxy...

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Tested and works in firefox.

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Other Methods:

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  • Use a Third party such as embed.ly (but it it really only good for well known http APIs).

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  • Create your own redirect script on an https page you control (a simple javascript redirect on a relative linked page should do the trick. Something like: (you can use any langauge/method)

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    https://example.com That has a iframe linking to...

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    https://example.com/utilities/redirect.html Which has a simple js redirect script like...

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    document.location.href =\"http://thenonsslsite.com\";

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  • Alternatively, you could add an RSS feed or write some reader/parser to read the http site and display it within your https site.

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  • You could/should also recommend to the http site owner that they create an ssl connection. If for no other reason than it increases seo.

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Unless you can get the http site owner to create an ssl certificate, the most secure and permanent solution would be to create an RSS feed grabing the content you need (presumably you are not actually 'doing' anything on the http site -that is to say not logging in to any system).

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The real issue is that having http elements inside a https site represents a security issue. There are no completely kosher ways around this security risk so the above are just current work arounds.

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Note, that you can disable this security measure in most browsers (yourself, not for others). Also note that these 'hacks' may become obsolete over time.

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\n edited Jan 26, 2018 at 8:05\n
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    \n\n Great answer, thanks. Just to let you know in chrome the JS redirect method does not work just prevents the change (as it does when you try load it normally).\n\n
    \n– georgephillips\n
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    \n\n The redirect trick seems to work in Firefox only. Chrome still denies loading the insecure content. Are there any other known workarounds?\n\n
    \n– Andreas Gohr\n
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    \n– kako-nawao\n
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Based on generality of this question, I think, that you'll need to setup your own HTTPS proxy on some server online. Do the following steps:

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If you simply download remote site content via file_get_contents or similiar, you can still have insecure links to content. You'll have to find them with regex and also replace. Images are hard to solve, but \u00cf found workaround here: http://foundationphp.com/tutorials/image_proxy.php

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\n answered Aug 14, 2014 at 3:09\n
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    \n– antidote\n
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    \n\n This solution worked perfectly for me. Found directions for IIS here: techcommunity.microsoft.com/t5/iis-support-blog/…\n\n
    \n– Rob10e\n
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I know this is an old post, but another solution would be to use cURL, for example:

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then in your iframe tag, something like:

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This is just a MINIMAL example to illustrate the idea -- it doesn't sanitize the URL, nor would it prevent someone else using the redirect.php for their own purposes. Consider these things in the context of your own site.

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The upside, though, is it's more flexible. For example, you could add some validation of the curl'd $data to make sure it's really what you want before displaying it -- for example, test to make sure it's not a 404, and have alternate content of your own ready if it is.

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Plus -- I'm a little weary of relying on Javascript redirects for anything important.

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Cheers!

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\n answered May 19, 2015 at 6:10\n
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    \n\n This is pretty working, but the links inside the site are getting invalid. For example, I have a domain called example.com which has a SSL. I am embedding a iframe with example.net which has no SSL. The example.com has a link like href="/path/file.html" and while clicking it, it is opening as https://example.com/path/file.html instead of http://example.net/path/file.html\n\n
    \n– Sibidharan\n
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    \n– Anthony Mason\n
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    \n\n This way somehow works but css doesnt work at all.\n\n
    \n– Goran_Ilic_Ilke\n
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add <meta http-equiv="Content-Security-Policy" content="upgrade-insecure-requests"> in head

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reference: http://thehackernews.com/2015/04/disable-mixed-content-warning.html

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browser compatibility: http://caniuse.com/#feat=upgradeinsecurerequests

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    \n\n This solution doesn't allow you to serve http content on a https site. All this does is force http requests as https requests. If the resource doesn't exist on https you'll just get a 404 error instead.\n\n
    \n– Felix\n
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    \n\n This is actually works at least for my needs.. tested it with firefox and confirmed working. My issue is source URL isn't HTTPS compliance while my own website is using HTTPS.\n\n
    \n– Fernan Vecina\n
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    \n– Mohamed Azharuddin\n
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    \n\n This solution chang http to https and do not add content from http site to https site.\n\n
    \n– Kashif Munir\n
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Using Google as the SSL proxy is not working currently,

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Why?

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If you opened any page from google, you will find there is a x-frame-options field in the header.\n\"Google

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The X-Frame-Options HTTP response header can be used to indicate whether or not a browser should be allowed to render a page in a <frame>, <iframe> or <object>. Sites can use this to avoid clickjacking attacks, by ensuring\n that their content is not embedded into other sites.

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One of the solution

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Below is my work around for this problem:

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Upload the content to AWS S3, and it will create a https link for the resource.
\nNotice: set the permission to the html file for allowing everyone view it.

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After that, we can using it as the src of iframe in the https websites.\n\"AWS\"

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\n edited Jan 2, 2018 at 13:25\n
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    \n– bobsbeenjamin\n
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You will always get warnings of blocked content in most browsers when trying to display non secure content on an https page. This is tricky if you want to embed stuff from other sites that aren't behind ssl. You can turn off the warnings or remove the blocking in your own browser but for other visitors it's a problem.

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One way to do it is to load the content server side and save the images and other things to your server and display them from https.

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You can also try using a service like embed.ly and get the content through them. They have support for getting the content behind https.

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\n edited May 11, 2017 at 17:14\n
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    \n\n if you scrape the content and show it on your site their is Always the risk of cross site scripting. So a god solution is to scrape the content on a separate url and present the data in an iframe from that url with https. In that way you prevent crosssite scripting on your mainsite and session hijacking.\n\n
    \n– Addeladde\n
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You could try scraping whatever you need with PHP or another server side language, then put the iframe to the scraped content. Here's an example with PHP:

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index.html:

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<iframe src=\"scrapedcontent.php\"></iframe>\n
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\n answered Mar 24, 2014 at 18:42\n
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    \n\n How will you handle images, included JS and CSS files, hyperlinks, and AJAX requests?\n\n
    \n– dotancohen\n
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    \n\n @dotancohen you're right, it isn't a perfect solution, but I think it is the best for this situation. Some sites you won't run into the problems you discussed.\n\n
    \n– Grant\n
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    \n\n This works, but its effectively double loading the content and therefore loading time as your server scrapes and then re-serves the content...\n\n
    \n– ChristoKiwi\n
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Use your own HTTPS-to-HTTP reverse proxy.

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If your use case is about a few, rarely changing URLs to embed into the iframe, you can simply set up a reverse proxy for this on your own server and configure it so that one https URL on your server maps to one http URL on the proxied server. Since a reverse proxy is fully serverside, the browser cannot discover that it is \"only\" talking to a proxy of the real website, and thus will not complain as the connection to the proxy uses SSL properly.

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If for example you use Apache2 as your webserver, then see these instructions to create a reverse proxy.

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\n answered Sep 4, 2014 at 17:51\n
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    \n\n mitmproxy is a debugging tool, not production proxying system. ngrok is tunnelling service, mostly for development servers, I don't understand how it can be useful for this.\n\n
    \n– kolen\n
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    \n\n @kolen It's not about the tools, it's about the idea of using a reverse proxy for this, not mentioned in the other answers yet. Changing to Apache now, which is a much more common way to do a reverse proxy.\n\n
    \n– tanius\n
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    \n\n This solution worked perfectly for me. Found directions for IIS here: techcommunity.microsoft.com/t5/iis-support-blog/…\n\n
    \n– Rob10e\n
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Try to use protocol relative links.

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Your link is http://example.com/script.js, use:

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<script src="//example.com/script.js" type="text/javascript"></script>\n
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In this way, you can leave the scheme free (do not indicate the protocol in the links) and trust that the browser uses the protocol of the embedded Web page. If your users visit the HTTP version of your Web page, the script will be loaded over http:// and if your users visit the HTTPS version of your Web site, the script will be loaded over https://.

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Seen in: https://developer.mozilla.org/es/docs/Seguridad/MixedContent/arreglar_web_con_contenido_mixto

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\n answered Oct 20, 2020 at 11:37\n
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    \n– dansch\n
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All you need to do is just use Google as a Proxy server.

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https://www.google.ie/gwt/x?u=[YourHttpLink].

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It worked for me.

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Credits:- https://www.wikihow.com/Use-Google-As-a-Proxy

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    \n\n Doesn't work for me. And </frame> should be </iframe>.\n\n
    \n– TimTIM Wong\n
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    \n\n i tried below <iframe src="https://www.google.ie/gwt/x?u=[http://www.mootrack.com/page/share.jsp?mapType=google&token=S1711650600yUyP315023260c762b627b44263e17f998a4272d1c]"></iframe> - it doent work for me\n\n
    \n– Kumaresan Sd\n
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\n \n HTTPS iframe within an HTTP page, how can I stop that?\n
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\n \n iframe blocked as insecure content, even though the iframe is HTTPS\n
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\n \n Embedding HTTP iframe in HTTPS iframe for HTTPS site?\n
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\n \n Hot Network Questions\n \n

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\n\n\n\n\n \n\n\n \n\n\n\n\n \n \n \n \n \n\n \n\n\n\n \n \n"} +{"content_list": [[{"type": "paragraph", "raw_content": "

I load some HTML into an iframe but when a file referenced is using http, not https, I get the following error:

", "content": [{"c": "I load some HTML into an iframe but when a file referenced is using http, not https, I get the following error:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

[blocked] The page at {current_pagename} ran insecure content from {referenced_filename}

", "content": [{"c": "[blocked] The page at {current_pagename} ran insecure content from {referenced_filename}", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Is there any way to turn this off or any way to get around it?

", "content": [{"c": "Is there any way to turn this off or any way to get around it?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

The iframe has no src attribute and the contents are set using:

", "content": [{"c": "The iframe has no", "t": "text"}, {"c": "src", "t": "code-inline"}, {"c": "attribute and the contents are set using:", "t": "text"}]}, {"type": "code", "raw_content": "frame.open();\nframe.write(html);\nframe.close();\n", "inline": false, "content": {"code_content": "frame.open();\nframe.write(html);\nframe.close();", "by": "tag_pre_code"}}, {"type": "list", "raw_content": "", "content": {"items": [], "ordered": false}}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "stackoverflow.com/questions/9280665/\u2026", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "chiliNUT", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Aug 20, 2013 at 5:30", "t": "text"}]], [[{"c": "made an edit. The src is not set as the content is written into the iframe", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Aug 20, 2013 at 9:36", "t": "text"}]], [[{"c": "1", "t": "text"}, {"c": "Thanks for all the answers. Long story short is proxy the content.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Aug 17, 2014 at 2:50", "t": "text"}]], [[{"c": "@georgephillips would you share the code for proxying the content?", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "Gintas_", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "May 5, 2017 at 20:30", "t": "text"}]]], "ordered": false}}, {"type": "title", "raw_content": "

\n 10 Answers\n \n

", "content": {"title_content": "10 Answers", "level": "2"}}, {"type": "paragraph", "raw_content": "

Note: While this solution may have worked in some browsers when it was written in 2014, it no longer works. Navigating or redirecting to an HTTP URL in an iframe embedded in an HTTPS page is not permitted by modern browsers, even if the frame started out with an HTTPS URL.

", "content": [{"c": "Note: While this solution may have worked in some browsers when it was written in 2014, it no longer works. Navigating or redirecting to an HTTP URL in an", "t": "text"}, {"c": "iframe", "t": "code-inline"}, {"c": "embedded in an HTTPS page is not permitted by modern browsers, even if the frame started out with an HTTPS URL.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

The best solution I created is to simply use google as the ssl proxy...

", "content": [{"c": "The best solution I created is to simply use google as the ssl proxy...", "t": "text"}]}, {"type": "code", "raw_content": "https://www.google.com/search?q=%http://yourhttpsite.com&btnI=Im+Feeling+Lucky\n", "inline": false, "content": {"code_content": "https://www.google.com/search?q=%http://yourhttpsite.com&btnI=Im+Feeling+Lucky", "by": "tag_pre_code"}}, {"type": "paragraph", "raw_content": "

Tested and works in firefox.

", "content": [{"c": "Tested and works in firefox.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Other Methods:

", "content": [{"c": "Other Methods:", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "Use a Third party such as embed.ly (but it it really only good for well known http APIs).", "t": "text"}]], [[{"c": "Create your own redirect script on an https page you control (a simple javascript redirect on a relative linked page should do the trick. Something like: (you can use any langauge/method)", "t": "text"}, {"c": "https://example.com", "t": "code-inline"}, {"c": " That has a iframe linking to... ", "t": "text"}, {"c": "https://example.com/utilities/redirect.html", "t": "code-inline"}, {"c": " Which has a simple js redirect script like...", "t": "text"}]]], "ordered": false}}, {"type": "code", "raw_content": "document.location.href =\"http://thenonsslsite.com\";", "inline": false, "content": {"code_content": "document.location.href =\"http://thenonsslsite.com\";", "by": "tag_code"}}, {"type": "list", "raw_content": "", "content": {"items": [[], [[{"c": "Alternatively, you could add an RSS feed or write some reader/parser to read the http site and display it within your https site.", "t": "text"}]], [[{"c": "You could/should also recommend to the http site owner that they create an ssl connection. If for no other reason than ", "t": "text"}, {"c": "it increases seo", "t": "text"}, {"c": ".", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

Unless you can get the http site owner to create an ssl certificate, the most secure and permanent solution would be to create an RSS feed grabing the content you need (presumably you are not actually 'doing' anything on the http site -that is to say not logging in to any system).

", "content": [{"c": "Unless you can get the http site owner to create an ssl certificate, the most secure and permanent solution would be to create an RSS feed grabing the content you need (presumably you are not actually 'doing' anything on the http site -that is to say not logging in to any system).", "t": "text"}]}, {"type": "paragraph", "raw_content": "

The real issue is that having http elements inside a https site represents a security issue. There are no completely kosher ways around this security risk so the above are just current work arounds.

", "content": [{"c": "The real issue is that having http elements inside a https site represents a security issue. There are no completely kosher ways around this security risk so the above are just current work arounds.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Note, that you can disable this security measure in most browsers (yourself, not for others). Also note that these 'hacks' may become obsolete over time.

", "content": [{"c": "Note, that you can disable this security measure in most browsers (yourself, not for others). Also note that these 'hacks' may become obsolete over time.", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "10", "t": "text"}, {"c": "Great answer, thanks. Just to let you know in chrome the JS redirect method does not work just prevents the change (as it does when you try load it normally).", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Sep 3, 2014 at 1:52", "t": "text"}]], [[{"c": "10", "t": "text"}, {"c": "The redirect trick seems to work in Firefox only. Chrome still denies loading the insecure content. Are there any other known workarounds?", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Nov 25, 2014 at 15:07", "t": "text"}]], [[{"c": "49", "t": "text"}, {"c": "Just wanted to signal out that the \"Create your own redirect script on an https page you control\" method no longer works with current versions of Chrom(e|ium) and Firefox, even using JS.", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "kako-nawao", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Feb 10, 2015 at 15:39", "t": "text"}]], [[{"c": "18", "t": "text"}, {"c": "This answer is invalid to date, any other solutions", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Feb 9, 2017 at 22:15", "t": "text"}]], [[{"c": "28", "t": "text"}, {"c": "The answer should have an option to be labeled as 'deprecated'. It creates confusion unless you read all the comments.", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "Nitin", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "May 16, 2017 at 19:44", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "
\n33
", "content": [{"c": "33", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Based on generality of this question, I think, that you'll need to setup your own HTTPS proxy on some server online. Do the following steps:

", "content": [{"c": "Based on generality of this question, I think, that you'll need to setup your own HTTPS proxy on some server online. Do the following steps:", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "Prepare your proxy server - install IIS, Apache", "t": "text"}]], [[{"c": "Get valid SSL certificate to avoid security errors (free from startssl.com for example)", "t": "text"}]], [[{"c": "Write a wrapper, which will download insecure content (how to below)", "t": "text"}]], [[{"c": "From your site/app get ", "t": "text"}, {"c": "https://yourproxy.com/?page=http://insecurepage.com", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

If you simply download remote site content via file_get_contents or similiar, you can still have insecure links to content. You'll have to find them with regex and also replace. Images are hard to solve, but \u00cf found workaround here: http://foundationphp.com/tutorials/image_proxy.php

", "content": [{"c": "If you simply download remote site content via file_get_contents or similiar, you can still have insecure links to content. You'll have to find them with regex and also replace. Images are hard to solve, but \u00cf found workaround here: http://foundationphp.com/tutorials/image_proxy.php", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "1", "t": "text"}, {"c": "This will not work, since many pages behind iframe don't want to be embedded in an iframe and thus set X-Frame-Options Header to SAMEORIGIN. Even if you are able to bypass this using the proxy, the page would try to load something like /insecurepage.css and your browser will request ", "t": "text"}, {"c": "yourdomain/insecurepage.css", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "antidote", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Jul 25, 2019 at 12:46", "t": "text"}]], [[{"c": "This solution worked perfectly for me. Found directions for IIS here: ", "t": "text"}, {"c": "techcommunity.microsoft.com/t5/iis-support-blog/\u2026", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "Rob10e", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Feb 5, 2021 at 16:33", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

I know this is an old post, but another solution would be to use cURL, for example:

", "content": [{"c": "I know this is an old post, but another solution would be to use cURL, for example:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

redirect.php:

", "content": [{"c": "redirect.php:", "t": "text"}]}, {"type": "code", "raw_content": "<?php\nif (isset($_GET['url'])) {\n $url = $_GET['url'];\n $ch = curl_init();\n $timeout = 5;\n curl_setopt($ch, CURLOPT_URL, $url);\n curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);\n curl_setopt($ch, CURLOPT_CONNECTTIMEOUT, $timeout);\n $data = curl_exec($ch);\n curl_close($ch);\n echo $data;\n}\n", "inline": false, "content": {"code_content": "

then in your iframe tag, something like:

", "content": [{"c": "then in your iframe tag, something like:", "t": "text"}]}, {"type": "code", "raw_content": "<iframe src=\"/redirect.php?url=http://www.example.com/\"></iframe>\n", "inline": false, "content": {"code_content": "", "by": "tag_pre_code"}}, {"type": "paragraph", "raw_content": "

This is just a MINIMAL example to illustrate the idea -- it doesn't sanitize the URL, nor would it prevent someone else using the redirect.php for their own purposes. Consider these things in the context of your own site.

", "content": [{"c": "This is just a MINIMAL example to illustrate the idea -- it doesn't sanitize the URL, nor would it prevent someone else using the redirect.php for their own purposes. Consider these things in the context of your own site.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

The upside, though, is it's more flexible. For example, you could add some validation of the curl'd $data to make sure it's really what you want before displaying it -- for example, test to make sure it's not a 404, and have alternate content of your own ready if it is.

", "content": [{"c": "The upside, though, is it's more flexible. For example, you could add some validation of the curl'd $data to make sure it's really what you want before displaying it -- for example, test to make sure it's not a 404, and have alternate content of your own ready if it is.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Plus -- I'm a little weary of relying on Javascript redirects for anything important.

", "content": [{"c": "Plus -- I'm a little weary of relying on Javascript redirects for anything important.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Cheers!

", "content": [{"c": "Cheers!", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "4", "t": "text"}, {"c": "This is pretty working, but the links inside the site are getting invalid. For example, I have a domain called ", "t": "text"}, {"c": "example.com", "t": "code-inline"}, {"c": " which has a SSL. I am embedding a iframe with ", "t": "text"}, {"c": "example.net", "t": "code-inline"}, {"c": " which has no SSL. The ", "t": "text"}, {"c": "example.com", "t": "code-inline"}, {"c": " has a link like ", "t": "text"}, {"c": "href=\"/path/file.html\"", "t": "code-inline"}, {"c": " and while clicking it, it is opening as ", "t": "text"}, {"c": "https://example.com/path/file.html", "t": "code-inline"}, {"c": " instead of ", "t": "text"}, {"c": "http://example.net/path/file.html", "t": "code-inline"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "Sibidharan", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Feb 15, 2016 at 15:13", "t": "text"}]], [[{"c": "1", "t": "text"}, {"c": "The link cannot be relative if you would like this to work. In other words, specify the full URL within the href. If this is dynamic there are libraries to grab each segment of the URL in Javascript as well as server side.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Jul 13, 2016 at 19:55", "t": "text"}]], [[{"c": "1", "t": "text"}, {"c": "css of the target site under iframe is broken . Please check", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Feb 28, 2019 at 9:21", "t": "text"}]], [[{"c": "This way somehow works but css doesnt work at all.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Nov 1, 2023 at 9:04", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

add <meta http-equiv=\"Content-Security-Policy\" content=\"upgrade-insecure-requests\"> in head

", "content": [{"c": "add", "t": "text"}, {"c": "", "t": "code-inline"}, {"c": "in head", "t": "text"}]}, {"type": "paragraph", "raw_content": "
", "content": [{"c": "reference: http://thehackernews.com/2015/04/disable-mixed-content-warning.html", "t": "text"}]}, {"type": "paragraph", "raw_content": "
", "content": [{"c": "browser compatibility: http://caniuse.com/#feat=upgradeinsecurerequests", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "37", "t": "text"}, {"c": "This solution doesn't allow you to serve http content on a https site. All this does is force http requests as https requests. If the resource doesn't exist on https you'll just get a 404 error instead.", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "Felix", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Feb 20, 2017 at 21:12", "t": "text"}]], [[{"c": "1", "t": "text"}, {"c": "This is actually works at least for my needs.. tested it with firefox and confirmed working. My issue is source URL isn't HTTPS compliance while my own website is using HTTPS.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Dec 14, 2017 at 3:41", "t": "text"}]], [[{"c": "Working like charm in chrome Version 76.0.3809.132 (Official Build)", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Sep 9, 2019 at 7:10", "t": "text"}]], [[{"c": "This solution chang http to https and do not add content from http site to https site.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Jan 31, 2021 at 16:46", "t": "text"}]], [[{"c": "1", "t": "text"}, {"c": "Not working anymore.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 15, 2022 at 13:13", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

Using Google as the SSL proxy is not working currently,

", "content": [{"c": "Using Google as the SSL proxy is not working currently,", "t": "text"}]}, {"type": "title", "raw_content": "

Why?

", "content": {"title_content": "Why?", "level": "3"}}, {"type": "paragraph", "raw_content": "

If you opened any page from google, you will find there is a x-frame-options field in the header.\n

", "content": [{"c": "If you opened any page from google, you will find there is a", "t": "text"}, {"c": "x-frame-options", "t": "code-inline"}, {"c": "field in the header.", "t": "text"}]}, {"type": "image", "raw_content": "\"Google", "content": {"url": "https://i.sstatic.net/uEYnJ.png", "data": null, "alt": "Google response header", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

The X-Frame-Options HTTP response header can be used to indicate whether or not a browser should be allowed to render a page in a <frame>, <iframe> or <object>. Sites can use this to avoid clickjacking attacks, by ensuring\n that their content is not embedded into other sites.

", "content": [{"c": "The X-Frame-Options HTTP response header can be used to indicate whether or not a browser should be allowed to render a page in a", "t": "text"}, {"c": "", "t": "code-inline"}, {"c": ",", "t": "text"}, {"c": "", "by": "tag_pre_code"}}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "3", "t": "text"}, {"c": "How will you handle images, included JS and CSS files, hyperlinks, and AJAX requests?", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "dotancohen", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "May 11, 2014 at 6:54", "t": "text"}]], [[{"c": "@dotancohen you're right, it isn't a perfect solution, but I think it is the best for this situation. Some sites you won't run into the problems you discussed.", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "Grant", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Jun 10, 2014 at 2:57", "t": "text"}]], [[{"c": "1", "t": "text"}, {"c": "This works, but its effectively double loading the content and therefore loading time as your server scrapes and then re-serves the content...", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "ChristoKiwi", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Jan 12, 2015 at 0:56", "t": "text"}]], [[{"c": "Doesnt load css from external page.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Nov 1, 2023 at 9:12", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

Use your own HTTPS-to-HTTP reverse proxy.

", "content": [{"c": "Use your own HTTPS-to-HTTP reverse proxy.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

If your use case is about a few, rarely changing URLs to embed into the iframe, you can simply set up a reverse proxy for this on your own server and configure it so that one https URL on your server maps to one http URL on the proxied server. Since a reverse proxy is fully serverside, the browser cannot discover that it is \"only\" talking to a proxy of the real website, and thus will not complain as the connection to the proxy uses SSL properly.

", "content": [{"c": "If your use case is about a few, rarely changing URLs to embed into the", "t": "text"}, {"c": "iframe", "t": "code-inline"}, {"c": ", you can simply set up a reverse proxy for this on your own server and configure it so that one", "t": "text"}, {"c": "https", "t": "code-inline"}, {"c": "URL on your server maps to one", "t": "text"}, {"c": "http", "t": "code-inline"}, {"c": "URL on the proxied server. Since a reverse proxy is fully serverside, the browser cannot discover that it is \"only\" talking to a proxy of the real website, and thus will not complain as the connection to the proxy uses SSL properly.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

If for example you use Apache2 as your webserver, then see these instructions to create a reverse proxy.

", "content": [{"c": "If for example you use Apache2 as your webserver, then see these instructions to create a reverse proxy.", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "1", "t": "text"}, {"c": "mitmproxy", "t": "code-inline"}, {"c": " is a debugging tool, not production proxying system. ", "t": "text"}, {"c": "ngrok", "t": "code-inline"}, {"c": " is tunnelling service, mostly for development servers, I don't understand how it can be useful for this.", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "kolen", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Jul 3, 2018 at 15:52", "t": "text"}]], [[{"c": "@kolen It's not about the tools, it's about the idea of using a reverse proxy for this, not mentioned in the other answers yet. Changing to Apache now, which is a much more common way to do a reverse proxy.", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "tanius", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Oct 12, 2019 at 18:30", "t": "text"}]], [[{"c": "This solution worked perfectly for me. Found directions for IIS here: ", "t": "text"}, {"c": "techcommunity.microsoft.com/t5/iis-support-blog/\u2026", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "Rob10e", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Feb 5, 2021 at 16:31", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

Try to use protocol relative links.

", "content": [{"c": "Try to use protocol relative links.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Your link is http://example.com/script.js, use:

", "content": [{"c": "Your link is http://example.com/script.js, use:", "t": "text"}]}, {"type": "code", "raw_content": "<script src=\"//example.com/script.js\" type=\"text/javascript\"></script>\n", "inline": false, "content": {"code_content": "", "by": "tag_pre_code"}}, {"type": "paragraph", "raw_content": "

In this way, you can leave the scheme free (do not indicate the protocol in the links) and trust that the browser uses the protocol of the embedded Web page. If your users visit the HTTP version of your Web page, the script will be loaded over http:// and if your users visit the HTTPS version of your Web site, the script will be loaded over https://.

", "content": [{"c": "In this way, you can leave the scheme free (do not indicate the protocol in the links) and trust that the browser uses the protocol of the embedded Web page. If your users visit the HTTP version of your Web page, the script will be loaded over http:// and if your users visit the HTTPS version of your Web site, the script will be loaded over https://.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
", "content": [{"c": "Seen in: https://developer.mozilla.org/es/docs/Seguridad/MixedContent/arreglar_web_con_contenido_mixto", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "3", "t": "text"}, {"c": "only works if the resource is available as https.", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "dansch", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Dec 3, 2020 at 18:35", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

All you need to do is just use Google as a Proxy server.

", "content": [{"c": "All you need to do is just use Google as a Proxy server.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
", "content": [{"c": "https://www.google.ie/gwt/x?u=[YourHttpLink].", "t": "text"}]}, {"type": "code", "raw_content": "<iframe src=\"https://www.google.ie/gwt/x?u=[Your http link]\"></iframe>\n", "inline": false, "content": {"code_content": "", "by": "tag_pre_code"}}, {"type": "paragraph", "raw_content": "

It worked for me.

", "content": [{"c": "It worked for me.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
", "content": [{"c": "Credits:- https://www.wikihow.com/Use-Google-As-a-Proxy", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "Where you are trying to use it.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "May 18, 2020 at 4:16", "t": "text"}]], [[{"c": "2", "t": "text"}, {"c": "Doesn't work for me. And ", "t": "text"}, {"c": "", "t": "code-inline"}, {"c": " should be ", "t": "text"}, {"c": "", "t": "code-inline"}, {"c": ".", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Dec 31, 2021 at 5:21", "t": "text"}]], [[{"c": "i tried below ", "t": "text"}, {"c": "", "t": "code-inline"}, {"c": " - it doent work for me", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 27, 2024 at 4:33", "t": "text"}]]], "ordered": false}}, {"type": "title", "raw_content": "

\nNot the answer you're looking for? Browse other questions tagged

", "content": {"title_content": "Not the answer you're looking for? Browse other questions tagged", "level": "2"}}, {"type": "list", "raw_content": " or ", "content": {"items": [], "ordered": false}}, {"type": "title", "raw_content": "

or ask your own question.

", "content": {"title_content": "or ask your own question .", "level": "2"}}]], "main_html": "

I load some HTML into an iframe but when a file referenced is using http, not https, I get the following error:

[blocked] The page at {current_pagename} ran insecure content from {referenced_filename}

Is there any way to turn this off or any way to get around it?

The iframe has no src attribute and the contents are set using:

frame.open();\nframe.write(html);\nframe.close();\n

\n 10 Answers\n \n

Note: While this solution may have worked in some browsers when it was written in 2014, it no longer works. Navigating or redirecting to an HTTP URL in an iframe embedded in an HTTPS page is not permitted by modern browsers, even if the frame started out with an HTTPS URL.

The best solution I created is to simply use google as the ssl proxy...

https://www.google.com/search?q=%http://yourhttpsite.com&btnI=Im+Feeling+Lucky\n

Tested and works in firefox.

Other Methods:

document.location.href =\"http://thenonsslsite.com\";

Unless you can get the http site owner to create an ssl certificate, the most secure and permanent solution would be to create an RSS feed grabing the content you need (presumably you are not actually 'doing' anything on the http site -that is to say not logging in to any system).

The real issue is that having http elements inside a https site represents a security issue. There are no completely kosher ways around this security risk so the above are just current work arounds.

Note, that you can disable this security measure in most browsers (yourself, not for others). Also note that these 'hacks' may become obsolete over time.

\n33

Based on generality of this question, I think, that you'll need to setup your own HTTPS proxy on some server online. Do the following steps:

If you simply download remote site content via file_get_contents or similiar, you can still have insecure links to content. You'll have to find them with regex and also replace. Images are hard to solve, but \u00cf found workaround here: http://foundationphp.com/tutorials/image_proxy.php

I know this is an old post, but another solution would be to use cURL, for example:

redirect.php:

<?php\nif (isset($_GET['url'])) {\n $url = $_GET['url'];\n $ch = curl_init();\n $timeout = 5;\n curl_setopt($ch, CURLOPT_URL, $url);\n curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);\n curl_setopt($ch, CURLOPT_CONNECTTIMEOUT, $timeout);\n $data = curl_exec($ch);\n curl_close($ch);\n echo $data;\n}\n

then in your iframe tag, something like:

<iframe src=\"/redirect.php?url=http://www.example.com/\"></iframe>\n

This is just a MINIMAL example to illustrate the idea -- it doesn't sanitize the URL, nor would it prevent someone else using the redirect.php for their own purposes. Consider these things in the context of your own site.

The upside, though, is it's more flexible. For example, you could add some validation of the curl'd $data to make sure it's really what you want before displaying it -- for example, test to make sure it's not a 404, and have alternate content of your own ready if it is.

Plus -- I'm a little weary of relying on Javascript redirects for anything important.

Cheers!

add <meta http-equiv=\"Content-Security-Policy\" content=\"upgrade-insecure-requests\"> in head

Using Google as the SSL proxy is not working currently,

Why?

If you opened any page from google, you will find there is a x-frame-options field in the header.\n

\"Google

The X-Frame-Options HTTP response header can be used to indicate whether or not a browser should be allowed to render a page in a <frame>, <iframe> or <object>. Sites can use this to avoid clickjacking attacks, by ensuring\n that their content is not embedded into other sites.

(Quote from MDN)

One of the solution

Below is my work around for this problem:

Upload the content to AWS S3, and it will create a https link for the resource.
\nNotice: set the permission to the html file for allowing everyone view it.

After that, we can using it as the src of iframe in the https websites.\n

\"AWS\"

You will always get warnings of blocked content in most browsers when trying to display non secure content on an https page. This is tricky if you want to embed stuff from other sites that aren't behind ssl. You can turn off the warnings or remove the blocking in your own browser but for other visitors it's a problem.

One way to do it is to load the content server side and save the images and other things to your server and display them from https.

You can also try using a service like embed.ly and get the content through them. They have support for getting the content behind https.

You could try scraping whatever you need with PHP or another server side language, then put the iframe to the scraped content. Here's an example with PHP:

scrapedcontent.php:

<?php\n$homepage = file_get_contents('http://www.example.com/');\necho $homepage;\n?>\n

index.html:

<iframe src=\"scrapedcontent.php\"></iframe>\n

Use your own HTTPS-to-HTTP reverse proxy.

If your use case is about a few, rarely changing URLs to embed into the iframe, you can simply set up a reverse proxy for this on your own server and configure it so that one https URL on your server maps to one http URL on the proxied server. Since a reverse proxy is fully serverside, the browser cannot discover that it is \"only\" talking to a proxy of the real website, and thus will not complain as the connection to the proxy uses SSL properly.

If for example you use Apache2 as your webserver, then see these instructions to create a reverse proxy.

Try to use protocol relative links.

Your link is http://example.com/script.js, use:

<script src=\"//example.com/script.js\" type=\"text/javascript\"></script>\n

In this way, you can leave the scheme free (do not indicate the protocol in the links) and trust that the browser uses the protocol of the embedded Web page. If your users visit the HTTP version of your Web page, the script will be loaded over http:// and if your users visit the HTTPS version of your Web site, the script will be loaded over https://.

All you need to do is just use Google as a Proxy server.

<iframe src=\"https://www.google.ie/gwt/x?u=[Your http link]\"></iframe>\n

It worked for me.

\nNot the answer you're looking for? Browse other questions tagged

or

or ask your own question.

", "statics": {"paragraph": 48, "paragraph.text": 59, "paragraph.code-inline": 11, "code": 9, "list": 16, "list.text": 176, "title": 5, "list.code-inline": 13, "image": 2}, "url": "https://stackoverflow.com/questions/18327314/how-to-allow-http-content-within-an-iframe-on-a-https-site", "content": "I load some HTML into an iframe but when a file referenced is using http, not https, I get the following error:\n\n[blocked] The page at {current_pagename} ran insecure content from {referenced_filename}\n\nIs there any way to turn this off or any way to get around it?\n\nThe iframe has no `src` attribute and the contents are set using:\n\n```\nframe.open();\nframe.write(html);\nframe.close();\n```\n\n- stackoverflow.com/questions/9280665/\u2026 \u2013 chiliNUT Commented Aug 20, 2013 at 5:30\n- made an edit. The src is not set as the content is written into the iframe Commented Aug 20, 2013 at 9:36\n- 1 Thanks for all the answers. Long story short is proxy the content. Commented Aug 17, 2014 at 2:50\n- @georgephillips would you share the code for proxying the content? \u2013 Gintas_ Commented May 5, 2017 at 20:30\n\n## 10 Answers\n\nNote: While this solution may have worked in some browsers when it was written in 2014, it no longer works. Navigating or redirecting to an HTTP URL in an `iframe` embedded in an HTTPS page is not permitted by modern browsers, even if the frame started out with an HTTPS URL.\n\nThe best solution I created is to simply use google as the ssl proxy...\n\n```\nhttps://www.google.com/search?q=%http://yourhttpsite.com&btnI=Im+Feeling+Lucky\n```\n\nTested and works in firefox.\n\nOther Methods:\n\n- Use a Third party such as embed.ly (but it it really only good for well known http APIs).\n- Create your own redirect script on an https page you control (a simple javascript redirect on a relative linked page should do the trick. Something like: (you can use any langauge/method) `https://example.com` That has a iframe linking to... `https://example.com/utilities/redirect.html` Which has a simple js redirect script like...\n\n```\ndocument.location.href =\"http://thenonsslsite.com\";\n```\n\n\n- Alternatively, you could add an RSS feed or write some reader/parser to read the http site and display it within your https site.\n- You could/should also recommend to the http site owner that they create an ssl connection. If for no other reason than it increases seo .\n\nUnless you can get the http site owner to create an ssl certificate, the most secure and permanent solution would be to create an RSS feed grabing the content you need (presumably you are not actually 'doing' anything on the http site -that is to say not logging in to any system).\n\nThe real issue is that having http elements inside a https site represents a security issue. There are no completely kosher ways around this security risk so the above are just current work arounds.\n\nNote, that you can disable this security measure in most browsers (yourself, not for others). Also note that these 'hacks' may become obsolete over time.\n\n- 10 Great answer, thanks. Just to let you know in chrome the JS redirect method does not work just prevents the change (as it does when you try load it normally). Commented Sep 3, 2014 at 1:52\n- 10 The redirect trick seems to work in Firefox only. Chrome still denies loading the insecure content. Are there any other known workarounds? Commented Nov 25, 2014 at 15:07\n- 49 Just wanted to signal out that the \"Create your own redirect script on an https page you control\" method no longer works with current versions of Chrom(e|ium) and Firefox, even using JS. \u2013 kako-nawao Commented Feb 10, 2015 at 15:39\n- 18 This answer is invalid to date, any other solutions Commented Feb 9, 2017 at 22:15\n- 28 The answer should have an option to be labeled as 'deprecated'. It creates confusion unless you read all the comments. \u2013 Nitin Commented May 16, 2017 at 19:44\n\n33\n\nBased on generality of this question, I think, that you'll need to setup your own HTTPS proxy on some server online. Do the following steps:\n\n- Prepare your proxy server - install IIS, Apache\n- Get valid SSL certificate to avoid security errors (free from startssl.com for example)\n- Write a wrapper, which will download insecure content (how to below)\n- From your site/app get https://yourproxy.com/?page=http://insecurepage.com\n\nIf you simply download remote site content via file_get_contents or similiar, you can still have insecure links to content. You'll have to find them with regex and also replace. Images are hard to solve, but \u00cf found workaround here: http://foundationphp.com/tutorials/image_proxy.php\n\n- 1 This will not work, since many pages behind iframe don't want to be embedded in an iframe and thus set X-Frame-Options Header to SAMEORIGIN. Even if you are able to bypass this using the proxy, the page would try to load something like /insecurepage.css and your browser will request yourdomain/insecurepage.css \u2013 antidote Commented Jul 25, 2019 at 12:46\n- This solution worked perfectly for me. Found directions for IIS here: techcommunity.microsoft.com/t5/iis-support-blog/\u2026 \u2013 Rob10e Commented Feb 5, 2021 at 16:33\n\nI know this is an old post, but another solution would be to use cURL, for example:\n\nredirect.php:\n\n```\n\n```\n\nThis is just a MINIMAL example to illustrate the idea -- it doesn't sanitize the URL, nor would it prevent someone else using the redirect.php for their own purposes. Consider these things in the context of your own site.\n\nThe upside, though, is it's more flexible. For example, you could add some validation of the curl'd $data to make sure it's really what you want before displaying it -- for example, test to make sure it's not a 404, and have alternate content of your own ready if it is.\n\nPlus -- I'm a little weary of relying on Javascript redirects for anything important.\n\nCheers!\n\n- 4 This is pretty working, but the links inside the site are getting invalid. For example, I have a domain called `example.com` which has a SSL. I am embedding a iframe with `example.net` which has no SSL. The `example.com` has a link like `href=\"/path/file.html\"` and while clicking it, it is opening as `https://example.com/path/file.html` instead of `http://example.net/path/file.html` \u2013 Sibidharan Commented Feb 15, 2016 at 15:13\n- 1 The link cannot be relative if you would like this to work. In other words, specify the full URL within the href. If this is dynamic there are libraries to grab each segment of the URL in Javascript as well as server side. Commented Jul 13, 2016 at 19:55\n- 1 css of the target site under iframe is broken . Please check Commented Feb 28, 2019 at 9:21\n- This way somehow works but css doesnt work at all. Commented Nov 1, 2023 at 9:04\n\nadd `` in head\n\nreference: http://thehackernews.com/2015/04/disable-mixed-content-warning.html\n\nbrowser compatibility: http://caniuse.com/\\#feat=upgradeinsecurerequests\n\n- 37 This solution doesn't allow you to serve http content on a https site. All this does is force http requests as https requests. If the resource doesn't exist on https you'll just get a 404 error instead. \u2013 Felix Commented Feb 20, 2017 at 21:12\n- 1 This is actually works at least for my needs.. tested it with firefox and confirmed working. My issue is source URL isn't HTTPS compliance while my own website is using HTTPS. Commented Dec 14, 2017 at 3:41\n- Working like charm in chrome Version 76.0.3809.132 (Official Build) Commented Sep 9, 2019 at 7:10\n- This solution chang http to https and do not add content from http site to https site. Commented Jan 31, 2021 at 16:46\n- 1 Not working anymore. Commented Mar 15, 2022 at 13:13\n\nUsing Google as the SSL proxy is not working currently,\n\n### Why?\n\nIf you opened any page from google, you will find there is a `x-frame-options` field in the header.\n\nThe X-Frame-Options HTTP response header can be used to indicate whether or not a browser should be allowed to render a page in a `` , `\n```\n\n- 3 How will you handle images, included JS and CSS files, hyperlinks, and AJAX requests? \u2013 dotancohen Commented May 11, 2014 at 6:54\n- @dotancohen you're right, it isn't a perfect solution, but I think it is the best for this situation. Some sites you won't run into the problems you discussed. \u2013 Grant Commented Jun 10, 2014 at 2:57\n- 1 This works, but its effectively double loading the content and therefore loading time as your server scrapes and then re-serves the content... \u2013 ChristoKiwi Commented Jan 12, 2015 at 0:56\n- Doesnt load css from external page. Commented Nov 1, 2023 at 9:12\n\nUse your own HTTPS-to-HTTP reverse proxy.\n\nIf your use case is about a few, rarely changing URLs to embed into the `iframe` , you can simply set up a reverse proxy for this on your own server and configure it so that one `https` URL on your server maps to one `http` URL on the proxied server. Since a reverse proxy is fully serverside, the browser cannot discover that it is \"only\" talking to a proxy of the real website, and thus will not complain as the connection to the proxy uses SSL properly.\n\nIf for example you use Apache2 as your webserver, then see these instructions to create a reverse proxy.\n\n- 1 `mitmproxy` is a debugging tool, not production proxying system. `ngrok` is tunnelling service, mostly for development servers, I don't understand how it can be useful for this. \u2013 kolen Commented Jul 3, 2018 at 15:52\n- @kolen It's not about the tools, it's about the idea of using a reverse proxy for this, not mentioned in the other answers yet. Changing to Apache now, which is a much more common way to do a reverse proxy. \u2013 tanius Commented Oct 12, 2019 at 18:30\n- This solution worked perfectly for me. Found directions for IIS here: techcommunity.microsoft.com/t5/iis-support-blog/\u2026 \u2013 Rob10e Commented Feb 5, 2021 at 16:31\n\nTry to use protocol relative links.\n\nYour link is http://example.com/script.js, use:\n\n```\n\n```\n\nIn this way, you can leave the scheme free (do not indicate the protocol in the links) and trust that the browser uses the protocol of the embedded Web page. If your users visit the HTTP version of your Web page, the script will be loaded over http:// and if your users visit the HTTPS version of your Web site, the script will be loaded over https://.\n\nSeen in: https://developer.mozilla.org/es/docs/Seguridad/MixedContent/arreglar_web_con_contenido_mixto\n\n- 3 only works if the resource is available as https. \u2013 dansch Commented Dec 3, 2020 at 18:35\n\nAll you need to do is just use Google as a Proxy server.\n\nhttps://www.google.ie/gwt/x?u=[YourHttpLink].\n\n```\n\n```\n\nIt worked for me.\n\nCredits:- https://www.wikihow.com/Use-Google-As-a-Proxy\n\n- Where you are trying to use it. Commented May 18, 2020 at 4:16\n- 2 Doesn't work for me. And `` should be `` . Commented Dec 31, 2021 at 5:21\n- i tried below `` - it doent work for me Commented Mar 27, 2024 at 4:33\n\n## Not the answer you're looking for? Browse other questions tagged\n\n## or ask your own question .\n", "html": "\n\n\n\n\n \n\n \n\n html - How to allow http content within an iframe on a https site - Stack Overflow\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n\n\n\n \n \n\n\n\n\n \n\n\n\n\n\n\n\n \n\n\n\n \n \n \n \n\n\n
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I load some HTML into an iframe but when a file referenced is using http, not https, I get the following error:

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[blocked] The page at {current_pagename} ran insecure content from {referenced_filename}

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Is there any way to turn this off or any way to get around it?

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The iframe has no src attribute and the contents are set using:

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frame.open();\nframe.write(html);\nframe.close();\n
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\n edited Jan 5, 2017 at 12:39\n
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    \n\n stackoverflow.com/questions/9280665/…\n\n
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    \n\n made an edit. The src is not set as the content is written into the iframe\n\n
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    \n\n Thanks for all the answers. Long story short is proxy the content.\n\n
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Note: While this solution may have worked in some browsers when it was written in 2014, it no longer works. Navigating or redirecting to an HTTP URL in an iframe embedded in an HTTPS page is not permitted by modern browsers, even if the frame started out with an HTTPS URL.

\n
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The best solution I created is to simply use google as the ssl proxy...

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https://www.google.com/search?q=%http://yourhttpsite.com&btnI=Im+Feeling+Lucky\n
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Tested and works in firefox.

\n\n

Other Methods:

\n\n
    \n

  • Use a Third party such as embed.ly (but it it really only good for well known http APIs).

    • \n

  • Create your own redirect script on an https page you control (a simple javascript redirect on a relative linked page should do the trick. Something like: (you can use any langauge/method)

    \n\n

    https://example.com That has a iframe linking to...

    \n\n

    https://example.com/utilities/redirect.html Which has a simple js redirect script like...

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    document.location.href =\"http://thenonsslsite.com\";

    • \n

  • Alternatively, you could add an RSS feed or write some reader/parser to read the http site and display it within your https site.

    • \n

  • You could/should also recommend to the http site owner that they create an ssl connection. If for no other reason than it increases seo.

    • \n
\n\n

Unless you can get the http site owner to create an ssl certificate, the most secure and permanent solution would be to create an RSS feed grabing the content you need (presumably you are not actually 'doing' anything on the http site -that is to say not logging in to any system).

\n\n

The real issue is that having http elements inside a https site represents a security issue. There are no completely kosher ways around this security risk so the above are just current work arounds.

\n\n

Note, that you can disable this security measure in most browsers (yourself, not for others). Also note that these 'hacks' may become obsolete over time.

\n
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    \n\n Great answer, thanks. Just to let you know in chrome the JS redirect method does not work just prevents the change (as it does when you try load it normally).\n\n
    \n– georgephillips\n
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    \n\n The redirect trick seems to work in Firefox only. Chrome still denies loading the insecure content. Are there any other known workarounds?\n\n
    \n– Andreas Gohr\n
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    \n– kako-nawao\n
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Based on generality of this question, I think, that you'll need to setup your own HTTPS proxy on some server online. Do the following steps:

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If you simply download remote site content via file_get_contents or similiar, you can still have insecure links to content. You'll have to find them with regex and also replace. Images are hard to solve, but \u00cf found workaround here: http://foundationphp.com/tutorials/image_proxy.php

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\n answered Aug 14, 2014 at 3:09\n
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    \n– antidote\n
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    \n\n This solution worked perfectly for me. Found directions for IIS here: techcommunity.microsoft.com/t5/iis-support-blog/…\n\n
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I know this is an old post, but another solution would be to use cURL, for example:

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then in your iframe tag, something like:

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This is just a MINIMAL example to illustrate the idea -- it doesn't sanitize the URL, nor would it prevent someone else using the redirect.php for their own purposes. Consider these things in the context of your own site.

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The upside, though, is it's more flexible. For example, you could add some validation of the curl'd $data to make sure it's really what you want before displaying it -- for example, test to make sure it's not a 404, and have alternate content of your own ready if it is.

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Plus -- I'm a little weary of relying on Javascript redirects for anything important.

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Cheers!

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\n answered May 19, 2015 at 6:10\n
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    \n\n This is pretty working, but the links inside the site are getting invalid. For example, I have a domain called example.com which has a SSL. I am embedding a iframe with example.net which has no SSL. The example.com has a link like href="/path/file.html" and while clicking it, it is opening as https://example.com/path/file.html instead of http://example.net/path/file.html\n\n
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    \n– Anthony Mason\n
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    \n– Jeff Bootsholz\n
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    \n– Goran_Ilic_Ilke\n
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add <meta http-equiv="Content-Security-Policy" content="upgrade-insecure-requests"> in head

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reference: http://thehackernews.com/2015/04/disable-mixed-content-warning.html

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    \n\n This solution doesn't allow you to serve http content on a https site. All this does is force http requests as https requests. If the resource doesn't exist on https you'll just get a 404 error instead.\n\n
    \n– Felix\n
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    \n– Fernan Vecina\n
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    \n\n This solution chang http to https and do not add content from http site to https site.\n\n
    \n– Kashif Munir\n
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Using Google as the SSL proxy is not working currently,

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Why?

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If you opened any page from google, you will find there is a x-frame-options field in the header.\n\"Google

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The X-Frame-Options HTTP response header can be used to indicate whether or not a browser should be allowed to render a page in a <frame>, <iframe> or <object>. Sites can use this to avoid clickjacking attacks, by ensuring\n that their content is not embedded into other sites.

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One of the solution

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Below is my work around for this problem:

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Upload the content to AWS S3, and it will create a https link for the resource.
\nNotice: set the permission to the html file for allowing everyone view it.

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After that, we can using it as the src of iframe in the https websites.\n\"AWS\"

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\n edited Jan 2, 2018 at 13:25\n
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    \n– bobsbeenjamin\n
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You will always get warnings of blocked content in most browsers when trying to display non secure content on an https page. This is tricky if you want to embed stuff from other sites that aren't behind ssl. You can turn off the warnings or remove the blocking in your own browser but for other visitors it's a problem.

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One way to do it is to load the content server side and save the images and other things to your server and display them from https.

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You can also try using a service like embed.ly and get the content through them. They have support for getting the content behind https.

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\n edited May 11, 2017 at 17:14\n
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    \n\n if you scrape the content and show it on your site their is Always the risk of cross site scripting. So a god solution is to scrape the content on a separate url and present the data in an iframe from that url with https. In that way you prevent crosssite scripting on your mainsite and session hijacking.\n\n
    \n– Addeladde\n
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You could try scraping whatever you need with PHP or another server side language, then put the iframe to the scraped content. Here's an example with PHP:

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\n answered Mar 24, 2014 at 18:42\n
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    \n– dotancohen\n
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    \n– Grant\n
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Use your own HTTPS-to-HTTP reverse proxy.

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If your use case is about a few, rarely changing URLs to embed into the iframe, you can simply set up a reverse proxy for this on your own server and configure it so that one https URL on your server maps to one http URL on the proxied server. Since a reverse proxy is fully serverside, the browser cannot discover that it is \"only\" talking to a proxy of the real website, and thus will not complain as the connection to the proxy uses SSL properly.

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If for example you use Apache2 as your webserver, then see these instructions to create a reverse proxy.

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    \n– kolen\n
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    \n– tanius\n
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    \n\n This solution worked perfectly for me. Found directions for IIS here: techcommunity.microsoft.com/t5/iis-support-blog/…\n\n
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Try to use protocol relative links.

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Your link is http://example.com/script.js, use:

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<script src="//example.com/script.js" type="text/javascript"></script>\n
\n

In this way, you can leave the scheme free (do not indicate the protocol in the links) and trust that the browser uses the protocol of the embedded Web page. If your users visit the HTTP version of your Web page, the script will be loaded over http:// and if your users visit the HTTPS version of your Web site, the script will be loaded over https://.

\n

Seen in: https://developer.mozilla.org/es/docs/Seguridad/MixedContent/arreglar_web_con_contenido_mixto

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\n answered Oct 20, 2020 at 11:37\n
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\"sesperanto's
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\n sesperantosesperanto\n
\n 18722 silver badges77 bronze badges\n
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    \n 3\n
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    \n\n only works if the resource is available as https.\n\n
    \n– dansch\n
    \n \n Commented\n Dec 3, 2020 at 18:35\n \n
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All you need to do is just use Google as a Proxy server.

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https://www.google.ie/gwt/x?u=[YourHttpLink].

\n
<iframe src="https://www.google.ie/gwt/x?u=[Your http link]"></iframe>\n
\n

It worked for me.

\n

Credits:- https://www.wikihow.com/Use-Google-As-a-Proxy

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\n edited Aug 17, 2022 at 5:28\n
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\"Subhan's
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\n answered Mar 29, 2020 at 20:20\n
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\"Star
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  • \n
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    \n\n Where you are trying to use it.\n\n
    \n– Star TechTricks\n
    \n \n Commented\n May 18, 2020 at 4:16\n \n
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    \n\n Doesn't work for me. And </frame> should be </iframe>.\n\n
    \n– TimTIM Wong\n
    \n \n Commented\n Dec 31, 2021 at 5:21\n \n
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    • \n
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    \n\n i tried below <iframe src="https://www.google.ie/gwt/x?u=[http://www.mootrack.com/page/share.jsp?mapType=google&token=S1711650600yUyP315023260c762b627b44263e17f998a4272d1c]"></iframe> - it doent work for me\n\n
    \n– Kumaresan Sd\n
    \n \n Commented\n Mar 27, 2024 at 4:33\n \n
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    • \n\n
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\n\n\n\n\n \n\n\n \n\n\n\n\n \n \n \n \n \n\n \n\n\n\n \n \n"} diff --git a/bench/data/groundtruth/code_11.jsonl b/bench/data/groundtruth/code_11.jsonl index 0a1db94b..a3c36a95 100644 --- a/bench/data/groundtruth/code_11.jsonl +++ b/bench/data/groundtruth/code_11.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "paragraph", "raw_content": "
Joe\n asked on 18 Dec 2009,\u00a003:31 PM
", "content": [{"c": "Joe asked on 18 Dec 2009, 03:31 PM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
I have Advanced Edit and Insert Form Templates built completely through code-behind for a RadScheduler. I am trying to implement cascading RadComboBoxes inside those forms.

I don't know how to find the controls in the forms from the SelectedIndexChanged events. In the FormCreated method when building the forms, I have the following:

", "content": [{"c": "I have Advanced Edit and Insert Form Templates built completely through code-behind for a RadScheduler. I am trying to implement cascading RadComboBoxes inside those forms. I don't know how to find the controls in the forms from the SelectedIndexChanged events. In the FormCreated method when building the forms, I have the following:", "t": "text"}]}, {"type": "table", "raw_content": "
RadComboBox\u00a0resource\u00a0=\u00a0(RadComboBox)e.Container.FindControl(\"ResourceInput\");\u00a0
resource.Skin\u00a0=\u00a0_skin;\u00a0
resource\u00a0=\u00a0LoadResources(resource);\u00a0
resource.SelectedIndexChanged\u00a0+=\u00a0new\u00a0RadComboBoxSelectedIndexChangedEventHandler(resource_SelectedIndexChanged);\u00a0
", "content": {"html": "
RadComboBox\u00a0resource\u00a0=\u00a0(RadComboBox)e.Container.FindControl(<br>\"ResourceInput\"<br>);
resource.Skin\u00a0=\u00a0_skin;
resource\u00a0=\u00a0LoadResources(resource);
resource.SelectedIndexChanged\u00a0+=\u00a0<br>new<br>RadComboBoxSelectedIndexChangedEventHandler(resource_SelectedIndexChanged);
", "is_complex": false}}, {"type": "paragraph", "raw_content": "
In the SelectedIndexChanged events I don't have access to e.Container.FindControl. So I need to know how to find those controls in that event.
", "content": [{"c": "In the SelectedIndexChanged events I don't have access to e.Container.FindControl. So I need to know how to find those controls in that event.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Thanks!
", "content": [{"c": "Thanks!", "t": "text"}]}, {"type": "title", "raw_content": "

9 Answers, 1 is accepted

", "content": {"title_content": "9 Answers , 1 is accepted", "level": "2"}}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Hi Joe,

\nThat could be a problem indeed when creating the controls dynamically. If this is not a must-have requirement, then you can use the approach from this kb article on related RadComboBox controls in the advanced form of RadScheduler:
http://www.telerik.com/support/kb/aspnet-ajax/scheduler/related-load-on-demand-radcomboboxes-in-the-advanced-form-of-radscheduler.aspx


\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
", "content": [{"c": "Hi Joe, That could be a problem indeed when creating the controls dynamically. If this is not a must-have requirement, then you can use the approach from this kb article on related RadComboBox controls in the advanced form of RadScheduler: http://www.telerik.com/support/kb/aspnet-ajax/scheduler/related-load-on-demand-radcomboboxes-in-the-advanced-form-of-radscheduler.aspx All the best, Peter the Telerik team Instantly find answers to your questions on the new Telerik Support Portal. Watch a video on how to optimize your support resource searches and check out more tips on the blogs.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 22 Dec 2009,\u00a008:15 PM
", "content": [{"c": "answered on 22 Dec 2009, 08:15 PM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Peter,

\nUnfortunately, this is a requirement.\u00a0I would\u00a0much rather if it wasn't, but not my choice....

\nI tried implementing your suggestion, but I ran into 2 stumbling blocks.

\n1. If I implement the ItemsRequested method in my custom AdvancedFormTemplate class, I no longer have visibility to the originating class, which is a SharePoint web part. In the custom toolpane for the web part, the user is able to select the location of the datasource. So all of that information is stored as properties in my web part class.\u00a0Therefore, I am not able to load the RadComboBox through an ItemsRequested event in the custom Template class because I don't have access to the web part toolpane properties that give me the datasource. That's why I was trying to implement a SelectedIndexChanged event from the main web part class - added during the FormCreated scheduler method.
\n2. I am not able to add javascript code to the page through the custom Template class. In the example the javascript is added directly to the ascx control, but I am not using that control.\u00a0I tried\u00a0adding the javascript\u00a0through the main web part class, but in the LoadCountries() function I need to have access to this: $find(\"<%=\u00a0CountriesRadComboBox.ClientID\u00a0%>\")\u00a0 which I don't have since\u00a0the control only exists\u00a0in\u00a0the Template\u00a0class.

\nSo I might be pretty much stuck on this approach. Is there any\u00a0other way I might accomplish my goal? I think I am pretty much there implementing the SelectedIndexChanged event the way I originally tried if I could somehow\u00a0get access to the controls within the AdvancedForm Template through that event.

\nThanks.\n
", "content": [{"c": "Peter, Unfortunately, this is a requirement.\u00a0I would\u00a0much rather if it wasn't, but not my choice.... I tried implementing your suggestion, but I ran into 2 stumbling blocks. 1. If I implement the ItemsRequested method in my custom AdvancedFormTemplate class, I no longer have visibility to the originating class, which is a SharePoint web part. In the custom toolpane for the web part, the user is able to select the location of the datasource. So all of that information is stored as properties in my web part class.\u00a0Therefore, I am not able to load the RadComboBox through an ItemsRequested event in the custom Template class because I don't have access to the web part toolpane properties that give me the datasource. That's why I was trying to implement a SelectedIndexChanged event from the main web part class - added during the FormCreated scheduler method. 2. I am not able to add javascript code to the page through the custom Template class. In the example the javascript is added directly to the ascx control, but I am not using that control.\u00a0I tried\u00a0adding the javascript\u00a0through the main web part class, but in the LoadCountries() function I need to have access to this: $find(\"<%=\u00a0CountriesRadComboBox.ClientID\u00a0%>\") which I don't have since\u00a0the control only exists\u00a0in\u00a0the Template\u00a0class. So I might be pretty much stuck on this approach. Is there any\u00a0other way I might accomplish my goal? I think I am pretty much there implementing the SelectedIndexChanged event the way I originally tried if I could somehow\u00a0get access to the controls within the AdvancedForm Template through that event. Thanks.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Hi Joe,

\nI think there is a workaround which you can use for this case. How about handling OnClientSelectedIndexChanged to store the selected items's value or text in a hidden field. Then when FormCreated is fired again after selecting a combobox item, you can get the selected value from the hidden field and use it to populate the second combobox. Here is an example which updates a label, but the concept is the same:

", "content": [{"c": "Hi Joe, I think there is a workaround which you can use for this case. How about handling OnClientSelectedIndexChanged to store the selected items's value or text in a hidden field. Then when FormCreated is fired again after selecting a combobox item, you can get the selected value from the hidden field and use it to populate the second combobox. Here is an example which updates a label, but the concept is the same:", "t": "text"}]}, {"type": "code", "raw_content": "
\n
<script type=\"text/javascript\">
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0function OnClientSelectedIndexChanged(sender, eventArgs) {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0var selectedValue = document.getElementById(\"selectedValue\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.value = eventArgs.get_item().get_text();\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0</script>
\n
\u00a0\u00a0\u00a0\u00a0<asp:HiddenField ID=\"selectedValue\" runat=\"server\" />
\n
\u00a0\u00a0\u00a0\u00a0<telerik:RadScheduler ID=\"RadScheduler1\"\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0runat=\"server\" onformcreated=\"RadScheduler1_FormCreated\"> </telerik:RadScheduler>
\n
\n", "inline": false, "content": {"code_content": "\n \n ", "by": "tag_code"}}, {"type": "code", "raw_content": "
\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
\n
\u00a0\u00a0\u00a0\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RadComboBox resource = new RadComboBox();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"a\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"b\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"c\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.AutoPostBack = true;
\n
\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(resource);
\n
\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Label selectedValueLbl = new Label();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (selectedValue.Value != null)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValueLbl.Text = selectedValue.Value;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(selectedValueLbl);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.Value = null;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\n", "inline": false, "content": {"code_content": "protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n {\n if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\n {\n RadComboBox resource = new RadComboBox();\n resource.Items.Add(new RadComboBoxItem(\"a\"));\n resource.Items.Add(new RadComboBoxItem(\"b\"));\n resource.Items.Add(new RadComboBoxItem(\"c\"));\n resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\n resource.AutoPostBack = true;\n\n Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\n basicPanel.Controls.Add(resource);\n\n Label selectedValueLbl = new Label();\n if (selectedValue.Value != null)\n selectedValueLbl.Text = selectedValue.Value;\n basicPanel.Controls.Add(selectedValueLbl);\n selectedValue.Value = null;\n }\n }", "by": "tag_code"}}, {"type": "paragraph", "raw_content": "

\nLet us know how it goes.

\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
", "content": [{"c": "Let us know how it goes. All the best, Peter the Telerik team Instantly find answers to your questions on the new Telerik Support Portal. Watch a video on how to optimize your support resource searches and check out more tips on the blogs.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 06 Jan 2010,\u00a011:11 PM
", "content": [{"c": "answered on 06 Jan 2010, 11:11 PM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Peter,

\nThanks for the great information, as always. I have implemented your approach. However, I am having 1 small issue. I know it's not related to the RadComboBox or the RadScheduler, but I'm hoping you can still help.

\nWhen the OnClientSelectedIndexChanged event is fired, the value is pulled out of the RadComboBox and placed into the hidden field. I have set a JS Alert to verify. But during the FormCreated event, the value of the hidden field\u00a0is still\u00a0null. I'm not sure if it's a viewstate issue (unlikely as I have other controls that don't have the problem). Or if it's an AJAX/UpdatePanel/Javascript interaction problem. The hidden field is inside the update panel, as is the scheduler and hence the combo's. I'm not very experienced with Javascript so I don't know what else I might need to do. Do I need to somehow call a refresh or update on the UpdatePanel?

\nHere is my JS for reference:

", "content": [{"c": "Peter, Thanks for the great information, as always. I have implemented your approach. However, I am having 1 small issue. I know it's not related to the RadComboBox or the RadScheduler, but I'm hoping you can still help. When the OnClientSelectedIndexChanged event is fired, the value is pulled out of the RadComboBox and placed into the hidden field. I have set a JS Alert to verify. But during the FormCreated event, the value of the hidden field\u00a0is still\u00a0null. I'm not sure if it's a viewstate issue (unlikely as I have other controls that don't have the problem). Or if it's an AJAX/UpdatePanel/Javascript interaction problem. The hidden field is inside the update panel, as is the scheduler and hence the combo's. I'm not very experienced with Javascript so I don't know what else I might need to do. Do I need to somehow call a refresh or update on the UpdatePanel? Here is my JS for reference:", "t": "text"}]}, {"type": "table", "raw_content": "
function\u00a0LoadProjects(sender,\u00a0eventArgs)\u00a0{ \u00a0
var\u00a0selectedValue\u00a0=\u00a0document.getElementById('\"\u00a0+\u00a0lblResourceValue.ClientID\u00a0+\u00a0\"'); \u00a0
selectedValue.value\u00a0=\u00a0eventArgs.get_item().get_text(); \u00a0
alert(selectedValue.value); \u00a0
}\u00a0
", "content": {"html": "
function<br>LoadProjects(sender,\u00a0eventArgs)\u00a0{
var<br>selectedValue\u00a0=<br>document<br>.getElementById(<br>'\"\u00a0+\u00a0lblResourceValue.ClientID\u00a0+\u00a0\"'<br>);
selectedValue.value\u00a0=\u00a0eventArgs.get_item().get_text();
alert(selectedValue.value);
}
", "is_complex": false}}, {"type": "paragraph", "raw_content": "

\nThanks again!\n
", "content": [{"c": "Thanks again!", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 07 Jan 2010,\u00a009:39 PM
", "content": [{"c": "answered on 07 Jan 2010, 09:39 PM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Peter,

\nI never did find an answer to this. I tried every trick in the book, changed control types, tried refreshing the update panel, tried adding a button to call a server side event and pull the value and nothing seemed to work.

\nI did however identify an alternate solution. Now that I am an expert with custom Advanced Form Templates :), I realized I could add hidden labels to that form. Then populate the values from my custom web part to those labels during FormCreated. That allowed me all the parameters I needed to run a standard SelectedIndexChanged event from the custom form class itself. I was able to get that method\u00a0working.

\nThanks again for all your help and advice with such a complex setup and advanced functionality. Much appreciated!\n
", "content": [{"c": "Peter, I never did find an answer to this. I tried every trick in the book, changed control types, tried refreshing the update panel, tried adding a button to call a server side event and pull the value and nothing seemed to work. I did however identify an alternate solution. Now that I am an expert with custom Advanced Form Templates :), I realized I could add hidden labels to that form. Then populate the values from my custom web part to those labels during FormCreated. That allowed me all the parameters I needed to run a standard SelectedIndexChanged event from the custom form class itself. I was able to get that method\u00a0working. Thanks again for all your help and advice with such a complex setup and advanced functionality. Much appreciated!", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 01 Sep 2010,\u00a006:20 AM
", "content": [{"c": "answered on 01 Sep 2010, 06:20 AM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Hi,

\nCan you post your code here ? I am finding it tough to add combobox at runtime in radscheduler advance edit form.

\nThanks

\nMilind
", "content": [{"c": "Hi, Can you post your code here ? I am finding it tough to add combobox at runtime in radscheduler advance edit form. Thanks Milind", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Hello Milind,

\nDid you try the code from the post on12/25/2009? Do you experience any problems with the following code?:

", "content": [{"c": "Hello Milind, Did you try the code from the post on12/25/2009? Do you experience any problems with the following code?:", "t": "text"}]}, {"type": "code", "raw_content": "
\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0{\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RadComboBox resource = new RadComboBox();\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"a\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"b\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"c\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.AutoPostBack = true;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(resource);\u00a0
\n
\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Label selectedValueLbl = new Label();\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (selectedValue.Value != null)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValueLbl.Text = selectedValue.Value;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(selectedValueLbl);\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.Value = null;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\n", "inline": false, "content": {"code_content": "protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n {\n if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\n {\n RadComboBox resource = new RadComboBox();\n resource.Items.Add(new RadComboBoxItem(\"a\"));\n resource.Items.Add(new RadComboBoxItem(\"b\"));\n resource.Items.Add(new RadComboBoxItem(\"c\"));\n resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\n resource.AutoPostBack = true;\n\n Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\n basicPanel.Controls.Add(resource);\n\n Label selectedValueLbl = new Label();\n if (selectedValue.Value != null)\n selectedValueLbl.Text = selectedValue.Value;\n basicPanel.Controls.Add(selectedValueLbl);\n selectedValue.Value = null;\n }\n }", "by": "tag_code"}}, {"type": "paragraph", "raw_content": "


\nKind regards,
\n Peter
\nthe Telerik team\n\n
Do you want to have your say when we set our development plans? Do you want to know when a feature you care about is added or when a bug fixed? Explore the Telerik Public Issue Tracking system and vote to affect the priority of the items
", "content": [{"c": "Kind regards, Peter the Telerik team Do you want to have your say when we set our development plans? Do you want to know when a feature you care about is added or when a bug fixed? Explore the Telerik Public Issue Tracking system and vote to affect the priority of the items", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "image", "raw_content": "\"\"", "content": {"url": "https://www.test.com/forums/images/avatarimages/94dd90f7-4eeb-4369-8e56-8f101e514374gussprofilepic_m.jpg", "data": null, "alt": null, "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "
", "content": [{"c": "Guss", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Top achievements
", "content": [{"c": "Top achievements", "t": "text"}]}, {"type": "image", "raw_content": "", "content": {"url": "https://www.test.com/forums/images/forum-gamification/rank-02.svg", "data": null, "alt": null, "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "
Rank 2
", "content": [{"c": "Rank 2", "t": "text"}]}, {"type": "image", "raw_content": "", "content": {"url": "https://www.test.com/forums/images/forum-gamification/general-veteran.svg", "data": null, "alt": null, "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "
Veteran
", "content": [{"c": "Veteran", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 12 Nov 2011,\u00a002:17 PM
", "content": [{"c": "answered on 12 Nov 2011, 02:17 PM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
What I think is going on here is: (though not exactly related to the topic header)

\nYou have an attribute in RadScheduler, for which you do not want the default text control, but your own control, like a Combobox that could be populated with data as needed (parameterized data, based on other data)

So, first, do not create a resource, that comboboxes that is created by default, is way to difficult to manipulate on server side, rather use a attribute, which could be hidden (and populated with another controls events)

\nLets assume you have a the following attibutes \"Parent\" and \"Children\". You know what \"Parent\"'s value is, before or during\u00a0the FormCreated event, and you want \"Children\" to be a combobox with items retrieved each time\u00a0from the database (dynamic), based on a value from \"Parent\".\u00a0

", "content": [{"c": "What I think is going on here is: (though not exactly related to the topic header) You have an attribute in RadScheduler, for which you do not want the default text control, but your own control, like a Combobox that could be populated with data as needed (parameterized data, based on other data) So, first, do not create a resource, that comboboxes that is created by default, is way to difficult to manipulate on server side, rather use a attribute, which could be hidden (and populated with another controls events) Lets assume you have a the following attibutes \"Parent\" and \"Children\". You know what \"Parent\"'s value is, before or during\u00a0the FormCreated event, and you want \"Children\" to be a combobox with items retrieved each time\u00a0from the database (dynamic), based on a value from \"Parent\".", "t": "text"}]}, {"type": "code", "raw_content": "
\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
\n
{
\n
RadTextBox txtChildren = (RadTextBox)e.Container.FindControl(\"AttrChildren\")
\n
txtChildren.Visible = false;
\n
RadComboBox rcbChildren = new RadComboBox();
\nrcbRegister.Label\u00a0=\u00a0\"Select Child:\"
\n
rcbChildren.Width = Unit.Pixel(250);
\n
rcbChildren.DataValueField = \"childIdFromDataSet\";
\n
rcbChildren.DataTextField = \"childTextFromDataSet\";
\n
rcbChildren.Attributes.Add(\"theclientid\", txtChildren.ClientID.ToString());
\n
rcbChildren.OnClientSelectedIndexChanged = \"OnChildrenSelectedIndexChanged\";
\n
//...........
\n
//create you DataTable,HasTable,2DArray\u00a0here, and use your parameters (Parent's value in this example)
\n
//........
\n//da.Fill(dt)
\n
//...........
\n
rcbChildren.DataSource = dt;
\n
rcbChildren.DataBind();
\n
if (txtChildren.Text != \"\") {
\n
rcbChildren.SelectedValue = txtChildren.Text;
\n
rcbChildren.Text = rcbRegister.txtChildren.Text;
\n
}
\n
txtChildren.Parent.Controls.Add(rcbChildren);
\n
}
\n
}
\n
\n", "inline": false, "content": {"code_content": "protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n{\nRadTextBox txtChildren = (RadTextBox)e.Container.FindControl(\"AttrChildren\")\ntxtChildren.Visible = false;\nRadComboBox rcbChildren = new RadComboBox();\nrcbRegister.Label = \"Select Child:\"\nrcbChildren.Width = Unit.Pixel(250);\nrcbChildren.DataValueField = \"childIdFromDataSet\";\nrcbChildren.DataTextField = \"childTextFromDataSet\";\nrcbChildren.Attributes.Add(\"theclientid\", txtChildren.ClientID.ToString());\nrcbChildren.OnClientSelectedIndexChanged = \"OnChildrenSelectedIndexChanged\";\n//...........\n//create you DataTable,HasTable,2DArray here, and use your parameters (Parent's value in this example)\n//........\n//da.Fill(dt)//...........\nrcbChildren.DataSource = dt;\nrcbChildren.DataBind();\nif (txtChildren.Text != \"\") {\nrcbChildren.SelectedValue = txtChildren.Text;\nrcbChildren.Text = rcbRegister.txtChildren.Text;\n}\ntxtChildren.Parent.Controls.Add(rcbChildren);\n}\n}", "by": "tag_code"}}, {"type": "paragraph", "raw_content": "

\nAnd your Javascript function\u00a0

", "content": [{"c": "And your Javascript function", "t": "text"}]}, {"type": "code", "raw_content": "
\n
function OnChildrenSelectedIndexChanged(sender, eventArgs) {
\n
var newvalue = eventArgs.get_item().get_value();
\n
var passedclientid = sender.get_attributes().getAttribute(\"theclientid\");
\n
var field1 = document.getElementById(passedclientid);
\n
var field2id = passedclientid + \"_text\";
\n
var field2 = document.getElementById(field2id);
\n
field1.value = newvalue;
\n
field2.value = newvalue;
\n
}
\n
", "inline": false, "content": {"code_content": "function OnChildrenSelectedIndexChanged(sender, eventArgs) {\nvar newvalue = eventArgs.get_item().get_value();\nvar passedclientid = sender.get_attributes().getAttribute(\"theclientid\");\nvar field1 = document.getElementById(passedclientid);\nvar field2id = passedclientid + \"_text\";\nvar field2 = document.getElementById(field2id);\nfield1.value = newvalue;\nfield2.value = newvalue;\n}", "by": "tag_code"}}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 14 May 2016,\u00a009:51 AM
", "content": [{"c": "answered on 14 May 2016, 09:51 AM", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Hello,

", "content": [{"c": "Hello,", "t": "text"}]}, {"type": "paragraph", "raw_content": "

I referred your answer above copied and pasted the code in my web page. The page gets compiled without error but the combobox which is added is not visible in edit or insert mode. I have set viewstate to false, is it because of that ?

", "content": [{"c": "I referred your answer above copied and pasted the code in my web page. The page gets compiled without error but the combobox which is added is not visible in edit or insert mode. I have set viewstate to false, is it because of that ?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Thanks

", "content": [{"c": "Thanks", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Milind Shevade

", "content": [{"c": "Milind Shevade", "t": "text"}]}]], "main_html": "
Joe\n asked on 18 Dec 2009,\u00a003:31 PM
I have Advanced Edit and Insert Form Templates built completely through code-behind for a RadScheduler. I am trying to implement cascading RadComboBoxes inside those forms.

I don't know how to find the controls in the forms from the SelectedIndexChanged events. In the FormCreated method when building the forms, I have the following:

RadComboBox\u00a0resource\u00a0=\u00a0(RadComboBox)e.Container.FindControl(\"ResourceInput\");\u00a0
resource.Skin\u00a0=\u00a0_skin;\u00a0
resource\u00a0=\u00a0LoadResources(resource);\u00a0
resource.SelectedIndexChanged\u00a0+=\u00a0new\u00a0RadComboBoxSelectedIndexChangedEventHandler(resource_SelectedIndexChanged);\u00a0
In the SelectedIndexChanged events I don't have access to e.Container.FindControl. So I need to know how to find those controls in that event.
Thanks!

9 Answers, 1 is accepted

0
Hi Joe,

\nThat could be a problem indeed when creating the controls dynamically. If this is not a must-have requirement, then you can use the approach from this kb article on related RadComboBox controls in the advanced form of RadScheduler:
http://www.telerik.com/support/kb/aspnet-ajax/scheduler/related-load-on-demand-radcomboboxes-in-the-advanced-form-of-radscheduler.aspx


\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
0
\n answered on 22 Dec 2009,\u00a008:15 PM
Peter,

\nUnfortunately, this is a requirement.\u00a0I would\u00a0much rather if it wasn't, but not my choice....

\nI tried implementing your suggestion, but I ran into 2 stumbling blocks.

\n1. If I implement the ItemsRequested method in my custom AdvancedFormTemplate class, I no longer have visibility to the originating class, which is a SharePoint web part. In the custom toolpane for the web part, the user is able to select the location of the datasource. So all of that information is stored as properties in my web part class.\u00a0Therefore, I am not able to load the RadComboBox through an ItemsRequested event in the custom Template class because I don't have access to the web part toolpane properties that give me the datasource. That's why I was trying to implement a SelectedIndexChanged event from the main web part class - added during the FormCreated scheduler method.
\n2. I am not able to add javascript code to the page through the custom Template class. In the example the javascript is added directly to the ascx control, but I am not using that control.\u00a0I tried\u00a0adding the javascript\u00a0through the main web part class, but in the LoadCountries() function I need to have access to this: $find(\"<%=\u00a0CountriesRadComboBox.ClientID\u00a0%>\")\u00a0 which I don't have since\u00a0the control only exists\u00a0in\u00a0the Template\u00a0class.

\nSo I might be pretty much stuck on this approach. Is there any\u00a0other way I might accomplish my goal? I think I am pretty much there implementing the SelectedIndexChanged event the way I originally tried if I could somehow\u00a0get access to the controls within the AdvancedForm Template through that event.

\nThanks.\n
0
Hi Joe,

\nI think there is a workaround which you can use for this case. How about handling OnClientSelectedIndexChanged to store the selected items's value or text in a hidden field. Then when FormCreated is fired again after selecting a combobox item, you can get the selected value from the hidden field and use it to populate the second combobox. Here is an example which updates a label, but the concept is the same:

\n
<script type=\"text/javascript\">
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0function OnClientSelectedIndexChanged(sender, eventArgs) {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0var selectedValue = document.getElementById(\"selectedValue\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.value = eventArgs.get_item().get_text();\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0</script>
\n
\u00a0\u00a0\u00a0\u00a0<asp:HiddenField ID=\"selectedValue\" runat=\"server\" />
\n
\u00a0\u00a0\u00a0\u00a0<telerik:RadScheduler ID=\"RadScheduler1\"\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0runat=\"server\" onformcreated=\"RadScheduler1_FormCreated\"> </telerik:RadScheduler>
\n
\n
\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
\n
\u00a0\u00a0\u00a0\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RadComboBox resource = new RadComboBox();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"a\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"b\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"c\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.AutoPostBack = true;
\n
\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(resource);
\n
\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Label selectedValueLbl = new Label();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (selectedValue.Value != null)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValueLbl.Text = selectedValue.Value;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(selectedValueLbl);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.Value = null;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\n

\nLet us know how it goes.

\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
0
\n answered on 06 Jan 2010,\u00a011:11 PM
Peter,

\nThanks for the great information, as always. I have implemented your approach. However, I am having 1 small issue. I know it's not related to the RadComboBox or the RadScheduler, but I'm hoping you can still help.

\nWhen the OnClientSelectedIndexChanged event is fired, the value is pulled out of the RadComboBox and placed into the hidden field. I have set a JS Alert to verify. But during the FormCreated event, the value of the hidden field\u00a0is still\u00a0null. I'm not sure if it's a viewstate issue (unlikely as I have other controls that don't have the problem). Or if it's an AJAX/UpdatePanel/Javascript interaction problem. The hidden field is inside the update panel, as is the scheduler and hence the combo's. I'm not very experienced with Javascript so I don't know what else I might need to do. Do I need to somehow call a refresh or update on the UpdatePanel?

\nHere is my JS for reference:

function\u00a0LoadProjects(sender,\u00a0eventArgs)\u00a0{ \u00a0
var\u00a0selectedValue\u00a0=\u00a0document.getElementById('\"\u00a0+\u00a0lblResourceValue.ClientID\u00a0+\u00a0\"'); \u00a0
selectedValue.value\u00a0=\u00a0eventArgs.get_item().get_text(); \u00a0
alert(selectedValue.value); \u00a0
}\u00a0

\nThanks again!\n
0
\n answered on 07 Jan 2010,\u00a009:39 PM
Peter,

\nI never did find an answer to this. I tried every trick in the book, changed control types, tried refreshing the update panel, tried adding a button to call a server side event and pull the value and nothing seemed to work.

\nI did however identify an alternate solution. Now that I am an expert with custom Advanced Form Templates :), I realized I could add hidden labels to that form. Then populate the values from my custom web part to those labels during FormCreated. That allowed me all the parameters I needed to run a standard SelectedIndexChanged event from the custom form class itself. I was able to get that method\u00a0working.

\nThanks again for all your help and advice with such a complex setup and advanced functionality. Much appreciated!\n
0
\n answered on 01 Sep 2010,\u00a006:20 AM
Hi,

\nCan you post your code here ? I am finding it tough to add combobox at runtime in radscheduler advance edit form.

\nThanks

\nMilind
0
Hello Milind,

\nDid you try the code from the post on12/25/2009? Do you experience any problems with the following code?:

\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0{\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RadComboBox resource = new RadComboBox();\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"a\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"b\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"c\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.AutoPostBack = true;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(resource);\u00a0
\n
\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Label selectedValueLbl = new Label();\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (selectedValue.Value != null)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValueLbl.Text = selectedValue.Value;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(selectedValueLbl);\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.Value = null;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\n


\nKind regards,
\n Peter
\nthe Telerik team\n\n
Do you want to have your say when we set our development plans? Do you want to know when a feature you care about is added or when a bug fixed? Explore the Telerik Public Issue Tracking system and vote to affect the priority of the items
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\n answered on 12 Nov 2011,\u00a002:17 PM
What I think is going on here is: (though not exactly related to the topic header)

\nYou have an attribute in RadScheduler, for which you do not want the default text control, but your own control, like a Combobox that could be populated with data as needed (parameterized data, based on other data)

So, first, do not create a resource, that comboboxes that is created by default, is way to difficult to manipulate on server side, rather use a attribute, which could be hidden (and populated with another controls events)

\nLets assume you have a the following attibutes \"Parent\" and \"Children\". You know what \"Parent\"'s value is, before or during\u00a0the FormCreated event, and you want \"Children\" to be a combobox with items retrieved each time\u00a0from the database (dynamic), based on a value from \"Parent\".\u00a0

\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
\n
{
\n
RadTextBox txtChildren = (RadTextBox)e.Container.FindControl(\"AttrChildren\")
\n
txtChildren.Visible = false;
\n
RadComboBox rcbChildren = new RadComboBox();
\nrcbRegister.Label\u00a0=\u00a0\"Select Child:\"
\n
rcbChildren.Width = Unit.Pixel(250);
\n
rcbChildren.DataValueField = \"childIdFromDataSet\";
\n
rcbChildren.DataTextField = \"childTextFromDataSet\";
\n
rcbChildren.Attributes.Add(\"theclientid\", txtChildren.ClientID.ToString());
\n
rcbChildren.OnClientSelectedIndexChanged = \"OnChildrenSelectedIndexChanged\";
\n
//...........
\n
//create you DataTable,HasTable,2DArray\u00a0here, and use your parameters (Parent's value in this example)
\n
//........
\n//da.Fill(dt)
\n
//...........
\n
rcbChildren.DataSource = dt;
\n
rcbChildren.DataBind();
\n
if (txtChildren.Text != \"\") {
\n
rcbChildren.SelectedValue = txtChildren.Text;
\n
rcbChildren.Text = rcbRegister.txtChildren.Text;
\n
}
\n
txtChildren.Parent.Controls.Add(rcbChildren);
\n
}
\n
}
\n
\n

\nAnd your Javascript function\u00a0

\n
function OnChildrenSelectedIndexChanged(sender, eventArgs) {
\n
var newvalue = eventArgs.get_item().get_value();
\n
var passedclientid = sender.get_attributes().getAttribute(\"theclientid\");
\n
var field1 = document.getElementById(passedclientid);
\n
var field2id = passedclientid + \"_text\";
\n
var field2 = document.getElementById(field2id);
\n
field1.value = newvalue;
\n
field2.value = newvalue;
\n
}
\n
0
\n answered on 14 May 2016,\u00a009:51 AM

Hello,

I referred your answer above copied and pasted the code in my web page. The page gets compiled without error but the combobox which is added is not visible in edit or insert mode. I have set viewstate to false, is it because of that ?

Thanks

Milind Shevade

", "statics": {"paragraph": 39, "paragraph.text": 39, "table": 2, "title": 1, "code": 5, "image": 3}, "url": "https://www.telerik.com/forums/radcombobox-selectedindexchanged-inside-advanced-edit-template", "content": "Joe asked on 18 Dec 2009, 03:31 PM\n\nI have Advanced Edit and Insert Form Templates built completely through code-behind for a RadScheduler. I am trying to implement cascading RadComboBoxes inside those forms. I don't know how to find the controls in the forms from the SelectedIndexChanged events. In the FormCreated method when building the forms, I have the following:\n\n| RadComboBox\u00a0resource\u00a0=\u00a0(RadComboBox)e.Container.FindControl(
\"ResourceInput\"
); |\n|---|\n| resource.Skin\u00a0=\u00a0_skin; |\n| resource\u00a0=\u00a0LoadResources(resource); |\n| resource.SelectedIndexChanged\u00a0+=\u00a0
new
RadComboBoxSelectedIndexChangedEventHandler(resource_SelectedIndexChanged); |\n| |\n\nIn the SelectedIndexChanged events I don't have access to e.Container.FindControl. So I need to know how to find those controls in that event.\n\nThanks!\n\n## 9 Answers , 1 is accepted\n\n0\n\nHi Joe, That could be a problem indeed when creating the controls dynamically. If this is not a must-have requirement, then you can use the approach from this kb article on related RadComboBox controls in the advanced form of RadScheduler: http://www.telerik.com/support/kb/aspnet-ajax/scheduler/related-load-on-demand-radcomboboxes-in-the-advanced-form-of-radscheduler.aspx All the best, Peter the Telerik team Instantly find answers to your questions on the new Telerik Support Portal. Watch a video on how to optimize your support resource searches and check out more tips on the blogs.\n\n0\n\nanswered on 22 Dec 2009, 08:15 PM\n\nPeter, Unfortunately, this is a requirement.\u00a0I would\u00a0much rather if it wasn't, but not my choice.... I tried implementing your suggestion, but I ran into 2 stumbling blocks. 1. If I implement the ItemsRequested method in my custom AdvancedFormTemplate class, I no longer have visibility to the originating class, which is a SharePoint web part. In the custom toolpane for the web part, the user is able to select the location of the datasource. So all of that information is stored as properties in my web part class.\u00a0Therefore, I am not able to load the RadComboBox through an ItemsRequested event in the custom Template class because I don't have access to the web part toolpane properties that give me the datasource. That's why I was trying to implement a SelectedIndexChanged event from the main web part class - added during the FormCreated scheduler method. 2. I am not able to add javascript code to the page through the custom Template class. In the example the javascript is added directly to the ascx control, but I am not using that control.\u00a0I tried\u00a0adding the javascript\u00a0through the main web part class, but in the LoadCountries() function I need to have access to this: $find(\"<%=\u00a0CountriesRadComboBox.ClientID\u00a0%>\") which I don't have since\u00a0the control only exists\u00a0in\u00a0the Template\u00a0class. So I might be pretty much stuck on this approach. Is there any\u00a0other way I might accomplish my goal? I think I am pretty much there implementing the SelectedIndexChanged event the way I originally tried if I could somehow\u00a0get access to the controls within the AdvancedForm Template through that event. Thanks.\n\n0\n\nHi Joe, I think there is a workaround which you can use for this case. How about handling OnClientSelectedIndexChanged to store the selected items's value or text in a hidden field. Then when FormCreated is fired again after selecting a combobox item, you can get the selected value from the hidden field and use it to populate the second combobox. Here is an example which updates a label, but the concept is the same:\n\n```\n\n \n \n```\n\n```\nprotected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n {\n if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\n {\n RadComboBox resource = new RadComboBox();\n resource.Items.Add(new RadComboBoxItem(\"a\"));\n resource.Items.Add(new RadComboBoxItem(\"b\"));\n resource.Items.Add(new RadComboBoxItem(\"c\"));\n resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\n resource.AutoPostBack = true;\n\n Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\n basicPanel.Controls.Add(resource);\n\n Label selectedValueLbl = new Label();\n if (selectedValue.Value != null)\n selectedValueLbl.Text = selectedValue.Value;\n basicPanel.Controls.Add(selectedValueLbl);\n selectedValue.Value = null;\n }\n }\n```\n\nLet us know how it goes. All the best, Peter the Telerik team Instantly find answers to your questions on the new Telerik Support Portal. Watch a video on how to optimize your support resource searches and check out more tips on the blogs.\n\n0\n\nanswered on 06 Jan 2010, 11:11 PM\n\nPeter, Thanks for the great information, as always. I have implemented your approach. However, I am having 1 small issue. I know it's not related to the RadComboBox or the RadScheduler, but I'm hoping you can still help. When the OnClientSelectedIndexChanged event is fired, the value is pulled out of the RadComboBox and placed into the hidden field. I have set a JS Alert to verify. But during the FormCreated event, the value of the hidden field\u00a0is still\u00a0null. I'm not sure if it's a viewstate issue (unlikely as I have other controls that don't have the problem). Or if it's an AJAX/UpdatePanel/Javascript interaction problem. The hidden field is inside the update panel, as is the scheduler and hence the combo's. I'm not very experienced with Javascript so I don't know what else I might need to do. Do I need to somehow call a refresh or update on the UpdatePanel? Here is my JS for reference:\n\n| function
LoadProjects(sender,\u00a0eventArgs)\u00a0{ |\n|---|\n| var
selectedValue\u00a0=
document
.getElementById(
'\"\u00a0+\u00a0lblResourceValue.ClientID\u00a0+\u00a0\"'
); |\n| selectedValue.value\u00a0=\u00a0eventArgs.get_item().get_text(); |\n| alert(selectedValue.value); |\n| } |\n\nThanks again!\n\n0\n\nanswered on 07 Jan 2010, 09:39 PM\n\nPeter, I never did find an answer to this. I tried every trick in the book, changed control types, tried refreshing the update panel, tried adding a button to call a server side event and pull the value and nothing seemed to work. I did however identify an alternate solution. Now that I am an expert with custom Advanced Form Templates :), I realized I could add hidden labels to that form. Then populate the values from my custom web part to those labels during FormCreated. That allowed me all the parameters I needed to run a standard SelectedIndexChanged event from the custom form class itself. I was able to get that method\u00a0working. Thanks again for all your help and advice with such a complex setup and advanced functionality. Much appreciated!\n\n0\n\nanswered on 01 Sep 2010, 06:20 AM\n\nHi, Can you post your code here ? I am finding it tough to add combobox at runtime in radscheduler advance edit form. Thanks Milind\n\n0\n\nHello Milind, Did you try the code from the post on12/25/2009? Do you experience any problems with the following code?:\n\n```\nprotected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n {\n if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\n {\n RadComboBox resource = new RadComboBox();\n resource.Items.Add(new RadComboBoxItem(\"a\"));\n resource.Items.Add(new RadComboBoxItem(\"b\"));\n resource.Items.Add(new RadComboBoxItem(\"c\"));\n resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\n resource.AutoPostBack = true;\n\n Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\n basicPanel.Controls.Add(resource);\n\n Label selectedValueLbl = new Label();\n if (selectedValue.Value != null)\n selectedValueLbl.Text = selectedValue.Value;\n basicPanel.Controls.Add(selectedValueLbl);\n selectedValue.Value = null;\n }\n }\n```\n\nKind regards, Peter the Telerik team Do you want to have your say when we set our development plans? Do you want to know when a feature you care about is added or when a bug fixed? Explore the Telerik Public Issue Tracking system and vote to affect the priority of the items\n\n0\n\nGuss\n\nTop achievements\n\nRank 2\n\nVeteran\n\nanswered on 12 Nov 2011, 02:17 PM\n\nWhat I think is going on here is: (though not exactly related to the topic header) You have an attribute in RadScheduler, for which you do not want the default text control, but your own control, like a Combobox that could be populated with data as needed (parameterized data, based on other data) So, first, do not create a resource, that comboboxes that is created by default, is way to difficult to manipulate on server side, rather use a attribute, which could be hidden (and populated with another controls events) Lets assume you have a the following attibutes \"Parent\" and \"Children\". You know what \"Parent\"'s value is, before or during\u00a0the FormCreated event, and you want \"Children\" to be a combobox with items retrieved each time\u00a0from the database (dynamic), based on a value from \"Parent\".\n\n```\nprotected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n{\nRadTextBox txtChildren = (RadTextBox)e.Container.FindControl(\"AttrChildren\")\ntxtChildren.Visible = false;\nRadComboBox rcbChildren = new RadComboBox();\nrcbRegister.Label = \"Select Child:\"\nrcbChildren.Width = Unit.Pixel(250);\nrcbChildren.DataValueField = \"childIdFromDataSet\";\nrcbChildren.DataTextField = \"childTextFromDataSet\";\nrcbChildren.Attributes.Add(\"theclientid\", txtChildren.ClientID.ToString());\nrcbChildren.OnClientSelectedIndexChanged = \"OnChildrenSelectedIndexChanged\";\n//...........\n//create you DataTable,HasTable,2DArray here, and use your parameters (Parent's value in this example)\n//........\n//da.Fill(dt)//...........\nrcbChildren.DataSource = dt;\nrcbChildren.DataBind();\nif (txtChildren.Text != \"\") {\nrcbChildren.SelectedValue = txtChildren.Text;\nrcbChildren.Text = rcbRegister.txtChildren.Text;\n}\ntxtChildren.Parent.Controls.Add(rcbChildren);\n}\n}\n```\n\nAnd your Javascript function\n\n```\nfunction OnChildrenSelectedIndexChanged(sender, eventArgs) {\nvar newvalue = eventArgs.get_item().get_value();\nvar passedclientid = sender.get_attributes().getAttribute(\"theclientid\");\nvar field1 = document.getElementById(passedclientid);\nvar field2id = passedclientid + \"_text\";\nvar field2 = document.getElementById(field2id);\nfield1.value = newvalue;\nfield2.value = newvalue;\n}\n```\n\n0\n\nanswered on 14 May 2016, 09:51 AM\n\nHello,\n\nI referred your answer above copied and pasted the code in my web page. The page gets compiled without error but the combobox which is added is not visible in edit or insert mode. I have set viewstate to false, is it because of that ?\n\nThanks\n\nMilind Shevade\n", "html": "\n\n\n\n \n\n \n \n \n \n \n \n \n\n\n\n\n \n\n\n\n RadComboBox SelectedIndexChanged inside Advanced Edit Template in UI for ASP.NET AJAX | Telerik Forums\n\n\n \n \n\n\n\n \n\n \n\n \n\n\n\n\n\n\n\n\n \n\n \n\n\n\n \n\n
\n \n\n \n\n\n\n \n \n \n\n\n\n\n
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\n \n \n\t\n\n Telerik Forums\n \n\t\t/\n\t\t \n UI for ASP.NET AJAX\n \n
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\n\t
\n\t\tThis is a migrated thread and some comments may be shown as answers.\n\t
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RadComboBox SelectedIndexChanged inside Advanced Edit Template

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\n \n\t\n9 Answers\n \n \n515 Views\n
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\n \n Scheduler\n \n
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\n This is a migrated thread and some comments may be shown as answers.\n
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\n Joe\n\n
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\n Joe\n asked on 18 Dec 2009, 03:31 PM\n
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I have Advanced Edit and Insert Form Templates built completely through code-behind for a RadScheduler. I am trying to implement cascading RadComboBoxes inside those forms.

I don't know how to find the controls in the forms from the SelectedIndexChanged events. In the FormCreated method when building the forms, I have the following:

RadComboBox resource = (RadComboBox)e.Container.FindControl(\"ResourceInput\"); 
resource.Skin = _skin; 
resource = LoadResources(resource); 
resource.SelectedIndexChanged += new RadComboBoxSelectedIndexChangedEventHandler(resource_SelectedIndexChanged); 
 

In the SelectedIndexChanged events I don't have access to e.Container.FindControl. So I need to know how to find those controls in that event.

Thanks!
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9 Answers, 1 is accepted

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\n Peter\n
\n Telerik team\n
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\n answered on 22 Dec 2009, 04:07 PM\n
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Hi Joe,
\n
\nThat could be a problem indeed when creating the controls dynamically. If this is not a must-have requirement, then you can use the approach from this kb article on related RadComboBox controls in the advanced form of RadScheduler:
\nhttp://www.telerik.com/support/kb/aspnet-ajax/scheduler/related-load-on-demand-radcomboboxes-in-the-advanced-form-of-radscheduler.aspx
\n
\n
\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
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\n Joe\n\n
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\n answered on 22 Dec 2009, 08:15 PM\n
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Peter,
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\nUnfortunately, this is a requirement. I would much rather if it wasn't, but not my choice....
\n
\nI tried implementing your suggestion, but I ran into 2 stumbling blocks.
\n
\n1. If I implement the ItemsRequested method in my custom AdvancedFormTemplate class, I no longer have visibility to the originating class, which is a SharePoint web part. In the custom toolpane for the web part, the user is able to select the location of the datasource. So all of that information is stored as properties in my web part class. Therefore, I am not able to load the RadComboBox through an ItemsRequested event in the custom Template class because I don't have access to the web part toolpane properties that give me the datasource. That's why I was trying to implement a SelectedIndexChanged event from the main web part class - added during the FormCreated scheduler method.
\n2. I am not able to add javascript code to the page through the custom Template class. In the example the javascript is added directly to the ascx control, but I am not using that control. I tried adding the javascript through the main web part class, but in the LoadCountries() function I need to have access to this: $find(\"<%= CountriesRadComboBox.ClientID %>\")  which I don't have since the control only exists in the Template class.
\n
\nSo I might be pretty much stuck on this approach. Is there any other way I might accomplish my goal? I think I am pretty much there implementing the SelectedIndexChanged event the way I originally tried if I could somehow get access to the controls within the AdvancedForm Template through that event.
\n
\nThanks.\n
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\n Telerik team\n
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\n answered on 25 Dec 2009, 01:07 PM\n
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Hi Joe,
\n
\nI think there is a workaround which you can use for this case. How about handling OnClientSelectedIndexChanged to store the selected items's value or text in a hidden field. Then when FormCreated is fired again after selecting a combobox item, you can get the selected value from the hidden field and use it to populate the second combobox. Here is an example which updates a label, but the concept is the same:
\n
\n
\n
<script type=\"text/javascript\">
\n
        function OnClientSelectedIndexChanged(sender, eventArgs) {
\n
            var selectedValue = document.getElementById(\"selectedValue\");
\n
            selectedValue.value = eventArgs.get_item().get_text();  
\n
        }
\n
    </script>
\n
    <asp:HiddenField ID=\"selectedValue\" runat=\"server\" />
\n
    <telerik:RadScheduler ID=\"RadScheduler1\" 
\n
        runat=\"server\" onformcreated=\"RadScheduler1_FormCreated\"> </telerik:RadScheduler>
\n
\n
\n
\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
\n
    {
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        if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)
\n
        {
\n
            RadComboBox resource = new RadComboBox();
\n
            resource.Items.Add(new RadComboBoxItem(\"a\"));
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            resource.Items.Add(new RadComboBoxItem(\"b\"));
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            resource.Items.Add(new RadComboBoxItem(\"c\"));
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            resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";
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            resource.AutoPostBack = true;
\n
  
\n
            Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");
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            basicPanel.Controls.Add(resource);
\n
  
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            Label selectedValueLbl = new Label();
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            if (selectedValue.Value != null)
\n
                selectedValueLbl.Text = selectedValue.Value;
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            basicPanel.Controls.Add(selectedValueLbl);
\n
            selectedValue.Value = null;
\n
        }
\n
    }
\n
\n
\nLet us know how it goes.
\n
\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
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\n Joe\n\n
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\n answered on 06 Jan 2010, 11:11 PM\n
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Peter,
\n
\nThanks for the great information, as always. I have implemented your approach. However, I am having 1 small issue. I know it's not related to the RadComboBox or the RadScheduler, but I'm hoping you can still help.
\n
\nWhen the OnClientSelectedIndexChanged event is fired, the value is pulled out of the RadComboBox and placed into the hidden field. I have set a JS Alert to verify. But during the FormCreated event, the value of the hidden field is still null. I'm not sure if it's a viewstate issue (unlikely as I have other controls that don't have the problem). Or if it's an AJAX/UpdatePanel/Javascript interaction problem. The hidden field is inside the update panel, as is the scheduler and hence the combo's. I'm not very experienced with Javascript so I don't know what else I might need to do. Do I need to somehow call a refresh or update on the UpdatePanel?
\n
\nHere is my JS for reference:
\n
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function LoadProjects(sender, eventArgs) {  
var selectedValue = document.getElementById('\" + lblResourceValue.ClientID + \"');  
selectedValue.value = eventArgs.get_item().get_text();  
alert(selectedValue.value);  
} 
\n
\n
\nThanks again!\n
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\n Joe\n\n
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\n answered on 07 Jan 2010, 09:39 PM\n
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Peter,
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\nI never did find an answer to this. I tried every trick in the book, changed control types, tried refreshing the update panel, tried adding a button to call a server side event and pull the value and nothing seemed to work.
\n
\nI did however identify an alternate solution. Now that I am an expert with custom Advanced Form Templates :), I realized I could add hidden labels to that form. Then populate the values from my custom web part to those labels during FormCreated. That allowed me all the parameters I needed to run a standard SelectedIndexChanged event from the custom form class itself. I was able to get that method working.
\n
\nThanks again for all your help and advice with such a complex setup and advanced functionality. Much appreciated!\n
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\n Milind\n\n
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\n answered on 01 Sep 2010, 06:20 AM\n
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Hi,
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\nCan you post your code here ? I am finding it tough to add combobox at runtime in radscheduler advance edit form.
\n
\nThanks
\n
\nMilind
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\n Peter\n
\n Telerik team\n
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\n answered on 06 Sep 2010, 05:05 AM\n
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Hello Milind,
\n
\nDid you try the code from the post on12/25/2009? Do you experience any problems with the following code?:
\n
\n
\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e) 
\n
    { 
\n
        if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert) 
\n
        { 
\n
            RadComboBox resource = new RadComboBox(); 
\n
            resource.Items.Add(new RadComboBoxItem(\"a\")); 
\n
            resource.Items.Add(new RadComboBoxItem(\"b\")); 
\n
            resource.Items.Add(new RadComboBoxItem(\"c\")); 
\n
            resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\"; 
\n
            resource.AutoPostBack = true; 
\n
    
\n
            Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\"); 
\n
            basicPanel.Controls.Add(resource); 
\n
    
\n
            Label selectedValueLbl = new Label(); 
\n
            if (selectedValue.Value != null) 
\n
                selectedValueLbl.Text = selectedValue.Value; 
\n
            basicPanel.Controls.Add(selectedValueLbl); 
\n
            selectedValue.Value = null; 
\n
        } 
\n
    }
\n
\n
\n
\nKind regards,
\n Peter
\nthe Telerik team\n
\n
Do you want to have your say when we set our development plans? Do you want to know when a feature you care about is added or when a bug fixed? Explore the Telerik Public Issue Tracking system and vote to affect the priority of the items
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\n answered on 12 Nov 2011, 02:17 PM\n
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What I think is going on here is: (though not exactly related to the topic header)
\n
\nYou have an attribute in RadScheduler, for which you do not want the default text control, but your own control, like a Combobox that could be populated with data as needed (parameterized data, based on other data)
\n
\nSo, first, do not create a resource, that comboboxes that is created by default, is way to difficult to manipulate on server side, rather use a attribute, which could be hidden (and populated with another controls events)
\n\n
\nLets assume you have a the following attibutes \"Parent\" and \"Children\". You know what \"Parent\"'s value is, before or during the FormCreated event, and you want \"Children\" to be a combobox with items retrieved each time from the database (dynamic), based on a value from \"Parent\". 
\n
\n
\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
\n
{
\n
RadTextBox txtChildren = (RadTextBox)e.Container.FindControl(\"AttrChildren\")
\n
txtChildren.Visible = false;
\n
RadComboBox rcbChildren = new RadComboBox();
\nrcbRegister.Label = \"Select Child:\"
\n
rcbChildren.Width = Unit.Pixel(250);
\n
rcbChildren.DataValueField = \"childIdFromDataSet\";
\n
rcbChildren.DataTextField = \"childTextFromDataSet\";
\n
rcbChildren.Attributes.Add(\"theclientid\", txtChildren.ClientID.ToString());
\n
rcbChildren.OnClientSelectedIndexChanged = \"OnChildrenSelectedIndexChanged\";
\n
//...........
\n
//create you DataTable,HasTable,2DArray here, and use your parameters (Parent's value in this example)
\n
//........
\n//da.Fill(dt)
\n
//...........
\n
rcbChildren.DataSource = dt;
\n
rcbChildren.DataBind();
\n
if (txtChildren.Text != \"\") {
\n
rcbChildren.SelectedValue = txtChildren.Text;
\n
rcbChildren.Text = rcbRegister.txtChildren.Text;
\n
}
\n
txtChildren.Parent.Controls.Add(rcbChildren);
\n
}
\n
}
\n
\n
\nAnd your Javascript function 
\n
\n
\n
function OnChildrenSelectedIndexChanged(sender, eventArgs) {
\n
var newvalue = eventArgs.get_item().get_value();
\n
var passedclientid = sender.get_attributes().getAttribute(\"theclientid\");
\n
var field1 = document.getElementById(passedclientid);
\n
var field2id = passedclientid + \"_text\";
\n
var field2 = document.getElementById(field2id);
\n
field1.value = newvalue;
\n
field2.value = newvalue;
\n
}
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\n Milind\n\n
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\n answered on 14 May 2016, 09:51 AM\n
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Hello,

\n

 

\n

I referred your answer above copied and pasted the code in my web page. The page gets compiled without error but the combobox which is added is not visible in edit or insert mode. I have set viewstate to false, is it because of that ?

\n

 

\n

Thanks

\n

Milind Shevade

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Top achievements
", "content": [{"c": "Top achievements", "t": "text"}]}, {"type": "image", "raw_content": "", "content": {"url": "https://www.test.com/forums/images/forum-gamification/rank-01.svg", "data": null, "alt": null, "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "
Rank 1
", "content": [{"c": "Rank 1", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Joe\n asked on 18 Dec 2009,\u00a003:31 PM
", "content": [{"c": "Joe asked on 18 Dec 2009, 03:31 PM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
I have Advanced Edit and Insert Form Templates built completely through code-behind for a RadScheduler. I am trying to implement cascading RadComboBoxes inside those forms.

I don't know how to find the controls in the forms from the SelectedIndexChanged events. In the FormCreated method when building the forms, I have the following:

", "content": [{"c": "I have Advanced Edit and Insert Form Templates built completely through code-behind for a RadScheduler. I am trying to implement cascading RadComboBoxes inside those forms. I don't know how to find the controls in the forms from the SelectedIndexChanged events. In the FormCreated method when building the forms, I have the following:", "t": "text"}]}, {"type": "table", "raw_content": "
RadComboBox\u00a0resource\u00a0=\u00a0(RadComboBox)e.Container.FindControl(\"ResourceInput\");\u00a0
resource.Skin\u00a0=\u00a0_skin;\u00a0
resource\u00a0=\u00a0LoadResources(resource);\u00a0
resource.SelectedIndexChanged\u00a0+=\u00a0new\u00a0RadComboBoxSelectedIndexChangedEventHandler(resource_SelectedIndexChanged);\u00a0
", "content": {"html": "
RadComboBox\u00a0resource\u00a0=\u00a0(RadComboBox)e.Container.FindControl(<br>\"ResourceInput\"<br>);
resource.Skin\u00a0=\u00a0_skin;
resource\u00a0=\u00a0LoadResources(resource);
resource.SelectedIndexChanged\u00a0+=<br>new<br>RadComboBoxSelectedIndexChangedEventHandler(resource_SelectedIndexChanged);
", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "
In the SelectedIndexChanged events I don't have access to e.Container.FindControl. So I need to know how to find those controls in that event.
", "content": [{"c": "In the SelectedIndexChanged events I don't have access to e.Container.FindControl. So I need to know how to find those controls in that event.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Thanks!
", "content": [{"c": "Thanks!", "t": "text"}]}, {"type": "title", "raw_content": "

9 Answers, 1 is accepted

", "content": {"title_content": "9 Answers , 1 is accepted", "level": "2"}}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Hi Joe,

\nThat could be a problem indeed when creating the controls dynamically. If this is not a must-have requirement, then you can use the approach from this kb article on related RadComboBox controls in the advanced form of RadScheduler:
http://www.telerik.com/support/kb/aspnet-ajax/scheduler/related-load-on-demand-radcomboboxes-in-the-advanced-form-of-radscheduler.aspx


\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
", "content": [{"c": "Hi Joe, That could be a problem indeed when creating the controls dynamically. If this is not a must-have requirement, then you can use the approach from this kb article on related RadComboBox controls in the advanced form of RadScheduler: http://www.telerik.com/support/kb/aspnet-ajax/scheduler/related-load-on-demand-radcomboboxes-in-the-advanced-form-of-radscheduler.aspx All the best, Peter the Telerik team Instantly find answers to your questions on the new Telerik Support Portal. Watch a video on how to optimize your support resource searches and check out more tips on the blogs.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 22 Dec 2009,\u00a008:15 PM
", "content": [{"c": "answered on 22 Dec 2009, 08:15 PM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Peter,

\nUnfortunately, this is a requirement.\u00a0I would\u00a0much rather if it wasn't, but not my choice....

\nI tried implementing your suggestion, but I ran into 2 stumbling blocks.

\n1. If I implement the ItemsRequested method in my custom AdvancedFormTemplate class, I no longer have visibility to the originating class, which is a SharePoint web part. In the custom toolpane for the web part, the user is able to select the location of the datasource. So all of that information is stored as properties in my web part class.\u00a0Therefore, I am not able to load the RadComboBox through an ItemsRequested event in the custom Template class because I don't have access to the web part toolpane properties that give me the datasource. That's why I was trying to implement a SelectedIndexChanged event from the main web part class - added during the FormCreated scheduler method.
\n2. I am not able to add javascript code to the page through the custom Template class. In the example the javascript is added directly to the ascx control, but I am not using that control.\u00a0I tried\u00a0adding the javascript\u00a0through the main web part class, but in the LoadCountries() function I need to have access to this: $find(\"<%=\u00a0CountriesRadComboBox.ClientID\u00a0%>\")\u00a0 which I don't have since\u00a0the control only exists\u00a0in\u00a0the Template\u00a0class.

\nSo I might be pretty much stuck on this approach. Is there any\u00a0other way I might accomplish my goal? I think I am pretty much there implementing the SelectedIndexChanged event the way I originally tried if I could somehow\u00a0get access to the controls within the AdvancedForm Template through that event.

\nThanks.\n
", "content": [{"c": "Peter, Unfortunately, this is a requirement.\u00a0I would\u00a0much rather if it wasn't, but not my choice.... I tried implementing your suggestion, but I ran into 2 stumbling blocks. 1. If I implement the ItemsRequested method in my custom AdvancedFormTemplate class, I no longer have visibility to the originating class, which is a SharePoint web part. In the custom toolpane for the web part, the user is able to select the location of the datasource. So all of that information is stored as properties in my web part class.\u00a0Therefore, I am not able to load the RadComboBox through an ItemsRequested event in the custom Template class because I don't have access to the web part toolpane properties that give me the datasource. That's why I was trying to implement a SelectedIndexChanged event from the main web part class - added during the FormCreated scheduler method. 2. I am not able to add javascript code to the page through the custom Template class. In the example the javascript is added directly to the ascx control, but I am not using that control.\u00a0I tried\u00a0adding the javascript\u00a0through the main web part class, but in the LoadCountries() function I need to have access to this: $find(\"<%=\u00a0CountriesRadComboBox.ClientID\u00a0%>\") which I don't have since\u00a0the control only exists\u00a0in\u00a0the Template\u00a0class. So I might be pretty much stuck on this approach. Is there any\u00a0other way I might accomplish my goal? I think I am pretty much there implementing the SelectedIndexChanged event the way I originally tried if I could somehow\u00a0get access to the controls within the AdvancedForm Template through that event. Thanks.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Hi Joe,

\nI think there is a workaround which you can use for this case. How about handling OnClientSelectedIndexChanged to store the selected items's value or text in a hidden field. Then when FormCreated is fired again after selecting a combobox item, you can get the selected value from the hidden field and use it to populate the second combobox. Here is an example which updates a label, but the concept is the same:

", "content": [{"c": "Hi Joe, I think there is a workaround which you can use for this case. How about handling OnClientSelectedIndexChanged to store the selected items's value or text in a hidden field. Then when FormCreated is fired again after selecting a combobox item, you can get the selected value from the hidden field and use it to populate the second combobox. Here is an example which updates a label, but the concept is the same:", "t": "text"}]}, {"type": "code", "raw_content": "
\n
<script type=\"text/javascript\">
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0function OnClientSelectedIndexChanged(sender, eventArgs) {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0var selectedValue = document.getElementById(\"selectedValue\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.value = eventArgs.get_item().get_text();\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0</script>
\n
\u00a0\u00a0\u00a0\u00a0<asp:HiddenField ID=\"selectedValue\" runat=\"server\" />
\n
\u00a0\u00a0\u00a0\u00a0<telerik:RadScheduler ID=\"RadScheduler1\"\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0runat=\"server\" onformcreated=\"RadScheduler1_FormCreated\"> </telerik:RadScheduler>
\n
\n", "inline": false, "content": {"code_content": "\n \n ", "by": "tag_code"}}, {"type": "code", "raw_content": "
\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
\n
\u00a0\u00a0\u00a0\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RadComboBox resource = new RadComboBox();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"a\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"b\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"c\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.AutoPostBack = true;
\n
\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(resource);
\n
\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Label selectedValueLbl = new Label();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (selectedValue.Value != null)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValueLbl.Text = selectedValue.Value;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(selectedValueLbl);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.Value = null;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\n", "inline": false, "content": {"code_content": "protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n {\n if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\n {\n RadComboBox resource = new RadComboBox();\n resource.Items.Add(new RadComboBoxItem(\"a\"));\n resource.Items.Add(new RadComboBoxItem(\"b\"));\n resource.Items.Add(new RadComboBoxItem(\"c\"));\n resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\n resource.AutoPostBack = true;\n\n Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\n basicPanel.Controls.Add(resource);\n\n Label selectedValueLbl = new Label();\n if (selectedValue.Value != null)\n selectedValueLbl.Text = selectedValue.Value;\n basicPanel.Controls.Add(selectedValueLbl);\n selectedValue.Value = null;\n }\n }", "by": "tag_code"}}, {"type": "paragraph", "raw_content": "

\nLet us know how it goes.

\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
", "content": [{"c": "Let us know how it goes. All the best, Peter the Telerik team Instantly find answers to your questions on the new Telerik Support Portal. Watch a video on how to optimize your support resource searches and check out more tips on the blogs.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 06 Jan 2010,\u00a011:11 PM
", "content": [{"c": "answered on 06 Jan 2010, 11:11 PM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Peter,

\nThanks for the great information, as always. I have implemented your approach. However, I am having 1 small issue. I know it's not related to the RadComboBox or the RadScheduler, but I'm hoping you can still help.

\nWhen the OnClientSelectedIndexChanged event is fired, the value is pulled out of the RadComboBox and placed into the hidden field. I have set a JS Alert to verify. But during the FormCreated event, the value of the hidden field\u00a0is still\u00a0null. I'm not sure if it's a viewstate issue (unlikely as I have other controls that don't have the problem). Or if it's an AJAX/UpdatePanel/Javascript interaction problem. The hidden field is inside the update panel, as is the scheduler and hence the combo's. I'm not very experienced with Javascript so I don't know what else I might need to do. Do I need to somehow call a refresh or update on the UpdatePanel?

\nHere is my JS for reference:

", "content": [{"c": "Peter, Thanks for the great information, as always. I have implemented your approach. However, I am having 1 small issue. I know it's not related to the RadComboBox or the RadScheduler, but I'm hoping you can still help. When the OnClientSelectedIndexChanged event is fired, the value is pulled out of the RadComboBox and placed into the hidden field. I have set a JS Alert to verify. But during the FormCreated event, the value of the hidden field\u00a0is still\u00a0null. I'm not sure if it's a viewstate issue (unlikely as I have other controls that don't have the problem). Or if it's an AJAX/UpdatePanel/Javascript interaction problem. The hidden field is inside the update panel, as is the scheduler and hence the combo's. I'm not very experienced with Javascript so I don't know what else I might need to do. Do I need to somehow call a refresh or update on the UpdatePanel? Here is my JS for reference:", "t": "text"}]}, {"type": "table", "raw_content": "
function\u00a0LoadProjects(sender,\u00a0eventArgs)\u00a0{ \u00a0
var\u00a0selectedValue\u00a0=\u00a0document.getElementById('\"\u00a0+\u00a0lblResourceValue.ClientID\u00a0+\u00a0\"'); \u00a0
selectedValue.value\u00a0=\u00a0eventArgs.get_item().get_text(); \u00a0
alert(selectedValue.value); \u00a0
}\u00a0
", "content": {"html": "
function<br>LoadProjects(sender,\u00a0eventArgs)\u00a0{
var<br>selectedValue\u00a0=<br>document<br>.getElementById(<br>'\"\u00a0+\u00a0lblResourceValue.ClientID\u00a0+\u00a0\"'<br>);
selectedValue.value\u00a0=\u00a0eventArgs.get_item().get_text();
alert(selectedValue.value);
}
", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

\nThanks again!\n
", "content": [{"c": "Thanks again!", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 07 Jan 2010,\u00a009:39 PM
", "content": [{"c": "answered on 07 Jan 2010, 09:39 PM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Peter,

\nI never did find an answer to this. I tried every trick in the book, changed control types, tried refreshing the update panel, tried adding a button to call a server side event and pull the value and nothing seemed to work.

\nI did however identify an alternate solution. Now that I am an expert with custom Advanced Form Templates :), I realized I could add hidden labels to that form. Then populate the values from my custom web part to those labels during FormCreated. That allowed me all the parameters I needed to run a standard SelectedIndexChanged event from the custom form class itself. I was able to get that method\u00a0working.

\nThanks again for all your help and advice with such a complex setup and advanced functionality. Much appreciated!\n
", "content": [{"c": "Peter, I never did find an answer to this. I tried every trick in the book, changed control types, tried refreshing the update panel, tried adding a button to call a server side event and pull the value and nothing seemed to work. I did however identify an alternate solution. Now that I am an expert with custom Advanced Form Templates :), I realized I could add hidden labels to that form. Then populate the values from my custom web part to those labels during FormCreated. That allowed me all the parameters I needed to run a standard SelectedIndexChanged event from the custom form class itself. I was able to get that method\u00a0working. Thanks again for all your help and advice with such a complex setup and advanced functionality. Much appreciated!", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 01 Sep 2010,\u00a006:20 AM
", "content": [{"c": "answered on 01 Sep 2010, 06:20 AM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Hi,

\nCan you post your code here ? I am finding it tough to add combobox at runtime in radscheduler advance edit form.

\nThanks

\nMilind
", "content": [{"c": "Hi, Can you post your code here ? I am finding it tough to add combobox at runtime in radscheduler advance edit form. Thanks Milind", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Hello Milind,

\nDid you try the code from the post on12/25/2009? Do you experience any problems with the following code?:

", "content": [{"c": "Hello Milind, Did you try the code from the post on12/25/2009? Do you experience any problems with the following code?:", "t": "text"}]}, {"type": "code", "raw_content": "
\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0{\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RadComboBox resource = new RadComboBox();\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"a\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"b\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"c\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.AutoPostBack = true;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(resource);\u00a0
\n
\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Label selectedValueLbl = new Label();\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (selectedValue.Value != null)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValueLbl.Text = selectedValue.Value;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(selectedValueLbl);\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.Value = null;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\n", "inline": false, "content": {"code_content": "protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n {\n if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\n {\n RadComboBox resource = new RadComboBox();\n resource.Items.Add(new RadComboBoxItem(\"a\"));\n resource.Items.Add(new RadComboBoxItem(\"b\"));\n resource.Items.Add(new RadComboBoxItem(\"c\"));\n resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\n resource.AutoPostBack = true;\n\n Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\n basicPanel.Controls.Add(resource);\n\n Label selectedValueLbl = new Label();\n if (selectedValue.Value != null)\n selectedValueLbl.Text = selectedValue.Value;\n basicPanel.Controls.Add(selectedValueLbl);\n selectedValue.Value = null;\n }\n }", "by": "tag_code"}}, {"type": "paragraph", "raw_content": "


\nKind regards,
\n Peter
\nthe Telerik team\n\n
Do you want to have your say when we set our development plans? Do you want to know when a feature you care about is added or when a bug fixed? Explore the Telerik Public Issue Tracking system and vote to affect the priority of the items
", "content": [{"c": "Kind regards, Peter the Telerik team Do you want to have your say when we set our development plans? Do you want to know when a feature you care about is added or when a bug fixed? Explore the Telerik Public Issue Tracking system and vote to affect the priority of the items", "t": "text"}]}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "image", "raw_content": "\"\"", "content": {"url": "https://www.test.com/forums/images/avatarimages/94dd90f7-4eeb-4369-8e56-8f101e514374gussprofilepic_m.jpg", "data": null, "alt": null, "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "
", "content": [{"c": "Guss", "t": "text"}]}, {"type": "paragraph", "raw_content": "
Top achievements
", "content": [{"c": "Top achievements", "t": "text"}]}, {"type": "image", "raw_content": "", "content": {"url": "https://www.test.com/forums/images/forum-gamification/rank-02.svg", "data": null, "alt": null, "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "
Rank 2
", "content": [{"c": "Rank 2", "t": "text"}]}, {"type": "image", "raw_content": "", "content": {"url": "https://www.test.com/forums/images/forum-gamification/general-veteran.svg", "data": null, "alt": null, "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "
Veteran
", "content": [{"c": "Veteran", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 12 Nov 2011,\u00a002:17 PM
", "content": [{"c": "answered on 12 Nov 2011, 02:17 PM", "t": "text"}]}, {"type": "paragraph", "raw_content": "
What I think is going on here is: (though not exactly related to the topic header)

\nYou have an attribute in RadScheduler, for which you do not want the default text control, but your own control, like a Combobox that could be populated with data as needed (parameterized data, based on other data)

So, first, do not create a resource, that comboboxes that is created by default, is way to difficult to manipulate on server side, rather use a attribute, which could be hidden (and populated with another controls events)

\nLets assume you have a the following attibutes \"Parent\" and \"Children\". You know what \"Parent\"'s value is, before or during\u00a0the FormCreated event, and you want \"Children\" to be a combobox with items retrieved each time\u00a0from the database (dynamic), based on a value from \"Parent\".\u00a0

", "content": [{"c": "What I think is going on here is: (though not exactly related to the topic header) You have an attribute in RadScheduler, for which you do not want the default text control, but your own control, like a Combobox that could be populated with data as needed (parameterized data, based on other data) So, first, do not create a resource, that comboboxes that is created by default, is way to difficult to manipulate on server side, rather use a attribute, which could be hidden (and populated with another controls events) Lets assume you have a the following attibutes \"Parent\" and \"Children\". You know what \"Parent\"'s value is, before or during\u00a0the FormCreated event, and you want \"Children\" to be a combobox with items retrieved each time\u00a0from the database (dynamic), based on a value from \"Parent\".", "t": "text"}]}, {"type": "code", "raw_content": "
\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
\n
{
\n
RadTextBox txtChildren = (RadTextBox)e.Container.FindControl(\"AttrChildren\")
\n
txtChildren.Visible = false;
\n
RadComboBox rcbChildren = new RadComboBox();
\nrcbRegister.Label\u00a0=\u00a0\"Select Child:\"
\n
rcbChildren.Width = Unit.Pixel(250);
\n
rcbChildren.DataValueField = \"childIdFromDataSet\";
\n
rcbChildren.DataTextField = \"childTextFromDataSet\";
\n
rcbChildren.Attributes.Add(\"theclientid\", txtChildren.ClientID.ToString());
\n
rcbChildren.OnClientSelectedIndexChanged = \"OnChildrenSelectedIndexChanged\";
\n
//...........
\n
//create you DataTable,HasTable,2DArray\u00a0here, and use your parameters (Parent's value in this example)
\n
//........
\n//da.Fill(dt)
\n
//...........
\n
rcbChildren.DataSource = dt;
\n
rcbChildren.DataBind();
\n
if (txtChildren.Text != \"\") {
\n
rcbChildren.SelectedValue = txtChildren.Text;
\n
rcbChildren.Text = rcbRegister.txtChildren.Text;
\n
}
\n
txtChildren.Parent.Controls.Add(rcbChildren);
\n
}
\n
}
\n
\n", "inline": false, "content": {"code_content": "protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n{\nRadTextBox txtChildren = (RadTextBox)e.Container.FindControl(\"AttrChildren\")\ntxtChildren.Visible = false;\nRadComboBox rcbChildren = new RadComboBox();\nrcbRegister.Label = \"Select Child:\"\nrcbChildren.Width = Unit.Pixel(250);\nrcbChildren.DataValueField = \"childIdFromDataSet\";\nrcbChildren.DataTextField = \"childTextFromDataSet\";\nrcbChildren.Attributes.Add(\"theclientid\", txtChildren.ClientID.ToString());\nrcbChildren.OnClientSelectedIndexChanged = \"OnChildrenSelectedIndexChanged\";\n//...........\n//create you DataTable,HasTable,2DArray here, and use your parameters (Parent's value in this example)\n//........\n//da.Fill(dt)//...........\nrcbChildren.DataSource = dt;\nrcbChildren.DataBind();\nif (txtChildren.Text != \"\") {\nrcbChildren.SelectedValue = txtChildren.Text;\nrcbChildren.Text = rcbRegister.txtChildren.Text;\n}\ntxtChildren.Parent.Controls.Add(rcbChildren);\n}\n}", "by": "tag_code"}}, {"type": "paragraph", "raw_content": "

\nAnd your Javascript function\u00a0

", "content": [{"c": "And your Javascript function", "t": "text"}]}, {"type": "code", "raw_content": "
\n
function OnChildrenSelectedIndexChanged(sender, eventArgs) {
\n
var newvalue = eventArgs.get_item().get_value();
\n
var passedclientid = sender.get_attributes().getAttribute(\"theclientid\");
\n
var field1 = document.getElementById(passedclientid);
\n
var field2id = passedclientid + \"_text\";
\n
var field2 = document.getElementById(field2id);
\n
field1.value = newvalue;
\n
field2.value = newvalue;
\n
}
\n
", "inline": false, "content": {"code_content": "function OnChildrenSelectedIndexChanged(sender, eventArgs) {\nvar newvalue = eventArgs.get_item().get_value();\nvar passedclientid = sender.get_attributes().getAttribute(\"theclientid\");\nvar field1 = document.getElementById(passedclientid);\nvar field2id = passedclientid + \"_text\";\nvar field2 = document.getElementById(field2id);\nfield1.value = newvalue;\nfield2.value = newvalue;\n}", "by": "tag_code"}}, {"type": "paragraph", "raw_content": "
0
", "content": [{"c": "0", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\n answered on 14 May 2016,\u00a009:51 AM
", "content": [{"c": "answered on 14 May 2016, 09:51 AM", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Hello,

", "content": [{"c": "Hello,", "t": "text"}]}, {"type": "paragraph", "raw_content": "

I referred your answer above copied and pasted the code in my web page. The page gets compiled without error but the combobox which is added is not visible in edit or insert mode. I have set viewstate to false, is it because of that ?

", "content": [{"c": "I referred your answer above copied and pasted the code in my web page. The page gets compiled without error but the combobox which is added is not visible in edit or insert mode. I have set viewstate to false, is it because of that ?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Thanks

", "content": [{"c": "Thanks", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Milind Shevade

", "content": [{"c": "Milind Shevade", "t": "text"}]}]], "main_html": "\"\"
Top achievements
Rank 1
Joe\n asked on 18 Dec 2009,\u00a003:31 PM
I have Advanced Edit and Insert Form Templates built completely through code-behind for a RadScheduler. I am trying to implement cascading RadComboBoxes inside those forms.

I don't know how to find the controls in the forms from the SelectedIndexChanged events. In the FormCreated method when building the forms, I have the following:

RadComboBox\u00a0resource\u00a0=\u00a0(RadComboBox)e.Container.FindControl(\"ResourceInput\");\u00a0
resource.Skin\u00a0=\u00a0_skin;\u00a0
resource\u00a0=\u00a0LoadResources(resource);\u00a0
resource.SelectedIndexChanged\u00a0+=\u00a0new\u00a0RadComboBoxSelectedIndexChangedEventHandler(resource_SelectedIndexChanged);\u00a0
In the SelectedIndexChanged events I don't have access to e.Container.FindControl. So I need to know how to find those controls in that event.
Thanks!

9 Answers, 1 is accepted

0
Hi Joe,

\nThat could be a problem indeed when creating the controls dynamically. If this is not a must-have requirement, then you can use the approach from this kb article on related RadComboBox controls in the advanced form of RadScheduler:
http://www.telerik.com/support/kb/aspnet-ajax/scheduler/related-load-on-demand-radcomboboxes-in-the-advanced-form-of-radscheduler.aspx


\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
0
\n answered on 22 Dec 2009,\u00a008:15 PM
Peter,

\nUnfortunately, this is a requirement.\u00a0I would\u00a0much rather if it wasn't, but not my choice....

\nI tried implementing your suggestion, but I ran into 2 stumbling blocks.

\n1. If I implement the ItemsRequested method in my custom AdvancedFormTemplate class, I no longer have visibility to the originating class, which is a SharePoint web part. In the custom toolpane for the web part, the user is able to select the location of the datasource. So all of that information is stored as properties in my web part class.\u00a0Therefore, I am not able to load the RadComboBox through an ItemsRequested event in the custom Template class because I don't have access to the web part toolpane properties that give me the datasource. That's why I was trying to implement a SelectedIndexChanged event from the main web part class - added during the FormCreated scheduler method.
\n2. I am not able to add javascript code to the page through the custom Template class. In the example the javascript is added directly to the ascx control, but I am not using that control.\u00a0I tried\u00a0adding the javascript\u00a0through the main web part class, but in the LoadCountries() function I need to have access to this: $find(\"<%=\u00a0CountriesRadComboBox.ClientID\u00a0%>\")\u00a0 which I don't have since\u00a0the control only exists\u00a0in\u00a0the Template\u00a0class.

\nSo I might be pretty much stuck on this approach. Is there any\u00a0other way I might accomplish my goal? I think I am pretty much there implementing the SelectedIndexChanged event the way I originally tried if I could somehow\u00a0get access to the controls within the AdvancedForm Template through that event.

\nThanks.\n
0
Hi Joe,

\nI think there is a workaround which you can use for this case. How about handling OnClientSelectedIndexChanged to store the selected items's value or text in a hidden field. Then when FormCreated is fired again after selecting a combobox item, you can get the selected value from the hidden field and use it to populate the second combobox. Here is an example which updates a label, but the concept is the same:

\n
<script type=\"text/javascript\">
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0function OnClientSelectedIndexChanged(sender, eventArgs) {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0var selectedValue = document.getElementById(\"selectedValue\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.value = eventArgs.get_item().get_text();\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0</script>
\n
\u00a0\u00a0\u00a0\u00a0<asp:HiddenField ID=\"selectedValue\" runat=\"server\" />
\n
\u00a0\u00a0\u00a0\u00a0<telerik:RadScheduler ID=\"RadScheduler1\"\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0runat=\"server\" onformcreated=\"RadScheduler1_FormCreated\"> </telerik:RadScheduler>
\n
\n
\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
\n
\u00a0\u00a0\u00a0\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RadComboBox resource = new RadComboBox();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"a\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"b\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"c\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.AutoPostBack = true;
\n
\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(resource);
\n
\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Label selectedValueLbl = new Label();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (selectedValue.Value != null)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValueLbl.Text = selectedValue.Value;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(selectedValueLbl);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.Value = null;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\n

\nLet us know how it goes.

\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
0
\n answered on 06 Jan 2010,\u00a011:11 PM
Peter,

\nThanks for the great information, as always. I have implemented your approach. However, I am having 1 small issue. I know it's not related to the RadComboBox or the RadScheduler, but I'm hoping you can still help.

\nWhen the OnClientSelectedIndexChanged event is fired, the value is pulled out of the RadComboBox and placed into the hidden field. I have set a JS Alert to verify. But during the FormCreated event, the value of the hidden field\u00a0is still\u00a0null. I'm not sure if it's a viewstate issue (unlikely as I have other controls that don't have the problem). Or if it's an AJAX/UpdatePanel/Javascript interaction problem. The hidden field is inside the update panel, as is the scheduler and hence the combo's. I'm not very experienced with Javascript so I don't know what else I might need to do. Do I need to somehow call a refresh or update on the UpdatePanel?

\nHere is my JS for reference:

function\u00a0LoadProjects(sender,\u00a0eventArgs)\u00a0{ \u00a0
var\u00a0selectedValue\u00a0=\u00a0document.getElementById('\"\u00a0+\u00a0lblResourceValue.ClientID\u00a0+\u00a0\"'); \u00a0
selectedValue.value\u00a0=\u00a0eventArgs.get_item().get_text(); \u00a0
alert(selectedValue.value); \u00a0
}\u00a0

\nThanks again!\n
0
\n answered on 07 Jan 2010,\u00a009:39 PM
Peter,

\nI never did find an answer to this. I tried every trick in the book, changed control types, tried refreshing the update panel, tried adding a button to call a server side event and pull the value and nothing seemed to work.

\nI did however identify an alternate solution. Now that I am an expert with custom Advanced Form Templates :), I realized I could add hidden labels to that form. Then populate the values from my custom web part to those labels during FormCreated. That allowed me all the parameters I needed to run a standard SelectedIndexChanged event from the custom form class itself. I was able to get that method\u00a0working.

\nThanks again for all your help and advice with such a complex setup and advanced functionality. Much appreciated!\n
0
\n answered on 01 Sep 2010,\u00a006:20 AM
Hi,

\nCan you post your code here ? I am finding it tough to add combobox at runtime in radscheduler advance edit form.

\nThanks

\nMilind
0
Hello Milind,

\nDid you try the code from the post on12/25/2009? Do you experience any problems with the following code?:

\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0{\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RadComboBox resource = new RadComboBox();\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"a\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"b\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.Items.Add(new RadComboBoxItem(\"c\"));\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0resource.AutoPostBack = true;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(resource);\u00a0
\n
\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Label selectedValueLbl = new Label();\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if (selectedValue.Value != null)\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValueLbl.Text = selectedValue.Value;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0basicPanel.Controls.Add(selectedValueLbl);\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0selectedValue.Value = null;\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\n


\nKind regards,
\n Peter
\nthe Telerik team\n\n
Do you want to have your say when we set our development plans? Do you want to know when a feature you care about is added or when a bug fixed? Explore the Telerik Public Issue Tracking system and vote to affect the priority of the items
0
\"\"
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Rank 2
Veteran
\n answered on 12 Nov 2011,\u00a002:17 PM
What I think is going on here is: (though not exactly related to the topic header)

\nYou have an attribute in RadScheduler, for which you do not want the default text control, but your own control, like a Combobox that could be populated with data as needed (parameterized data, based on other data)

So, first, do not create a resource, that comboboxes that is created by default, is way to difficult to manipulate on server side, rather use a attribute, which could be hidden (and populated with another controls events)

\nLets assume you have a the following attibutes \"Parent\" and \"Children\". You know what \"Parent\"'s value is, before or during\u00a0the FormCreated event, and you want \"Children\" to be a combobox with items retrieved each time\u00a0from the database (dynamic), based on a value from \"Parent\".\u00a0

\n
protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
\n
{
\n
RadTextBox txtChildren = (RadTextBox)e.Container.FindControl(\"AttrChildren\")
\n
txtChildren.Visible = false;
\n
RadComboBox rcbChildren = new RadComboBox();
\nrcbRegister.Label\u00a0=\u00a0\"Select Child:\"
\n
rcbChildren.Width = Unit.Pixel(250);
\n
rcbChildren.DataValueField = \"childIdFromDataSet\";
\n
rcbChildren.DataTextField = \"childTextFromDataSet\";
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rcbChildren.Attributes.Add(\"theclientid\", txtChildren.ClientID.ToString());
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rcbChildren.OnClientSelectedIndexChanged = \"OnChildrenSelectedIndexChanged\";
\n
//...........
\n
//create you DataTable,HasTable,2DArray\u00a0here, and use your parameters (Parent's value in this example)
\n
//........
\n//da.Fill(dt)
\n
//...........
\n
rcbChildren.DataSource = dt;
\n
rcbChildren.DataBind();
\n
if (txtChildren.Text != \"\") {
\n
rcbChildren.SelectedValue = txtChildren.Text;
\n
rcbChildren.Text = rcbRegister.txtChildren.Text;
\n
}
\n
txtChildren.Parent.Controls.Add(rcbChildren);
\n
}
\n
}
\n
\n

\nAnd your Javascript function\u00a0

\n
function OnChildrenSelectedIndexChanged(sender, eventArgs) {
\n
var newvalue = eventArgs.get_item().get_value();
\n
var passedclientid = sender.get_attributes().getAttribute(\"theclientid\");
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var field1 = document.getElementById(passedclientid);
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var field2id = passedclientid + \"_text\";
\n
var field2 = document.getElementById(field2id);
\n
field1.value = newvalue;
\n
field2.value = newvalue;
\n
}
\n
0
\n answered on 14 May 2016,\u00a009:51 AM

Hello,

I referred your answer above copied and pasted the code in my web page. The page gets compiled without error but the combobox which is added is not visible in edit or insert mode. I have set viewstate to false, is it because of that ?

Thanks

Milind Shevade

", "statics": {"image": 5, "paragraph": 41, "paragraph.text": 41, "table": 2, "title": 1, "code": 5}, "url": "https://www.telerik.com/forums/radcombobox-selectedindexchanged-inside-advanced-edit-template", "content": "Top achievements\n\nRank 1\n\nJoe asked on 18 Dec 2009, 03:31 PM\n\nI have Advanced Edit and Insert Form Templates built completely through code-behind for a RadScheduler. I am trying to implement cascading RadComboBoxes inside those forms. I don't know how to find the controls in the forms from the SelectedIndexChanged events. In the FormCreated method when building the forms, I have the following:\n\n| RadComboBox\u00a0resource\u00a0=\u00a0(RadComboBox)e.Container.FindControl(
\"ResourceInput\"
); |\n|---|\n| resource.Skin\u00a0=\u00a0_skin; |\n| resource\u00a0=\u00a0LoadResources(resource); |\n| resource.SelectedIndexChanged\u00a0+=
new
RadComboBoxSelectedIndexChangedEventHandler(resource_SelectedIndexChanged); |\n| |\n\nIn the SelectedIndexChanged events I don't have access to e.Container.FindControl. So I need to know how to find those controls in that event.\n\nThanks!\n\n## 9 Answers , 1 is accepted\n\n0\n\nHi Joe, That could be a problem indeed when creating the controls dynamically. If this is not a must-have requirement, then you can use the approach from this kb article on related RadComboBox controls in the advanced form of RadScheduler: http://www.telerik.com/support/kb/aspnet-ajax/scheduler/related-load-on-demand-radcomboboxes-in-the-advanced-form-of-radscheduler.aspx All the best, Peter the Telerik team Instantly find answers to your questions on the new Telerik Support Portal. Watch a video on how to optimize your support resource searches and check out more tips on the blogs.\n\n0\n\nanswered on 22 Dec 2009, 08:15 PM\n\nPeter, Unfortunately, this is a requirement.\u00a0I would\u00a0much rather if it wasn't, but not my choice.... I tried implementing your suggestion, but I ran into 2 stumbling blocks. 1. If I implement the ItemsRequested method in my custom AdvancedFormTemplate class, I no longer have visibility to the originating class, which is a SharePoint web part. In the custom toolpane for the web part, the user is able to select the location of the datasource. So all of that information is stored as properties in my web part class.\u00a0Therefore, I am not able to load the RadComboBox through an ItemsRequested event in the custom Template class because I don't have access to the web part toolpane properties that give me the datasource. That's why I was trying to implement a SelectedIndexChanged event from the main web part class - added during the FormCreated scheduler method. 2. I am not able to add javascript code to the page through the custom Template class. In the example the javascript is added directly to the ascx control, but I am not using that control.\u00a0I tried\u00a0adding the javascript\u00a0through the main web part class, but in the LoadCountries() function I need to have access to this: $find(\"<%=\u00a0CountriesRadComboBox.ClientID\u00a0%>\") which I don't have since\u00a0the control only exists\u00a0in\u00a0the Template\u00a0class. So I might be pretty much stuck on this approach. Is there any\u00a0other way I might accomplish my goal? I think I am pretty much there implementing the SelectedIndexChanged event the way I originally tried if I could somehow\u00a0get access to the controls within the AdvancedForm Template through that event. Thanks.\n\n0\n\nHi Joe, I think there is a workaround which you can use for this case. How about handling OnClientSelectedIndexChanged to store the selected items's value or text in a hidden field. Then when FormCreated is fired again after selecting a combobox item, you can get the selected value from the hidden field and use it to populate the second combobox. Here is an example which updates a label, but the concept is the same:\n\n```\n\n \n \n```\n\n```\nprotected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n {\n if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\n {\n RadComboBox resource = new RadComboBox();\n resource.Items.Add(new RadComboBoxItem(\"a\"));\n resource.Items.Add(new RadComboBoxItem(\"b\"));\n resource.Items.Add(new RadComboBoxItem(\"c\"));\n resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\n resource.AutoPostBack = true;\n\n Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\n basicPanel.Controls.Add(resource);\n\n Label selectedValueLbl = new Label();\n if (selectedValue.Value != null)\n selectedValueLbl.Text = selectedValue.Value;\n basicPanel.Controls.Add(selectedValueLbl);\n selectedValue.Value = null;\n }\n }\n```\n\nLet us know how it goes. All the best, Peter the Telerik team Instantly find answers to your questions on the new Telerik Support Portal. Watch a video on how to optimize your support resource searches and check out more tips on the blogs.\n\n0\n\nanswered on 06 Jan 2010, 11:11 PM\n\nPeter, Thanks for the great information, as always. I have implemented your approach. However, I am having 1 small issue. I know it's not related to the RadComboBox or the RadScheduler, but I'm hoping you can still help. When the OnClientSelectedIndexChanged event is fired, the value is pulled out of the RadComboBox and placed into the hidden field. I have set a JS Alert to verify. But during the FormCreated event, the value of the hidden field\u00a0is still\u00a0null. I'm not sure if it's a viewstate issue (unlikely as I have other controls that don't have the problem). Or if it's an AJAX/UpdatePanel/Javascript interaction problem. The hidden field is inside the update panel, as is the scheduler and hence the combo's. I'm not very experienced with Javascript so I don't know what else I might need to do. Do I need to somehow call a refresh or update on the UpdatePanel? Here is my JS for reference:\n\n| function
LoadProjects(sender,\u00a0eventArgs)\u00a0{ |\n|---|\n| var
selectedValue\u00a0=
document
.getElementById(
'\"\u00a0+\u00a0lblResourceValue.ClientID\u00a0+\u00a0\"'
); |\n| selectedValue.value\u00a0=\u00a0eventArgs.get_item().get_text(); |\n| alert(selectedValue.value); |\n| } |\n\nThanks again!\n\n0\n\nanswered on 07 Jan 2010, 09:39 PM\n\nPeter, I never did find an answer to this. I tried every trick in the book, changed control types, tried refreshing the update panel, tried adding a button to call a server side event and pull the value and nothing seemed to work. I did however identify an alternate solution. Now that I am an expert with custom Advanced Form Templates :), I realized I could add hidden labels to that form. Then populate the values from my custom web part to those labels during FormCreated. That allowed me all the parameters I needed to run a standard SelectedIndexChanged event from the custom form class itself. I was able to get that method\u00a0working. Thanks again for all your help and advice with such a complex setup and advanced functionality. Much appreciated!\n\n0\n\nanswered on 01 Sep 2010, 06:20 AM\n\nHi, Can you post your code here ? I am finding it tough to add combobox at runtime in radscheduler advance edit form. Thanks Milind\n\n0\n\nHello Milind, Did you try the code from the post on12/25/2009? Do you experience any problems with the following code?:\n\n```\nprotected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n {\n if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)\n {\n RadComboBox resource = new RadComboBox();\n resource.Items.Add(new RadComboBoxItem(\"a\"));\n resource.Items.Add(new RadComboBoxItem(\"b\"));\n resource.Items.Add(new RadComboBoxItem(\"c\"));\n resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";\n resource.AutoPostBack = true;\n\n Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");\n basicPanel.Controls.Add(resource);\n\n Label selectedValueLbl = new Label();\n if (selectedValue.Value != null)\n selectedValueLbl.Text = selectedValue.Value;\n basicPanel.Controls.Add(selectedValueLbl);\n selectedValue.Value = null;\n }\n }\n```\n\nKind regards, Peter the Telerik team Do you want to have your say when we set our development plans? Do you want to know when a feature you care about is added or when a bug fixed? Explore the Telerik Public Issue Tracking system and vote to affect the priority of the items\n\n0\n\nGuss\n\nTop achievements\n\nRank 2\n\nVeteran\n\nanswered on 12 Nov 2011, 02:17 PM\n\nWhat I think is going on here is: (though not exactly related to the topic header) You have an attribute in RadScheduler, for which you do not want the default text control, but your own control, like a Combobox that could be populated with data as needed (parameterized data, based on other data) So, first, do not create a resource, that comboboxes that is created by default, is way to difficult to manipulate on server side, rather use a attribute, which could be hidden (and populated with another controls events) Lets assume you have a the following attibutes \"Parent\" and \"Children\". You know what \"Parent\"'s value is, before or during\u00a0the FormCreated event, and you want \"Children\" to be a combobox with items retrieved each time\u00a0from the database (dynamic), based on a value from \"Parent\".\n\n```\nprotected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)\n{\nRadTextBox txtChildren = (RadTextBox)e.Container.FindControl(\"AttrChildren\")\ntxtChildren.Visible = false;\nRadComboBox rcbChildren = new RadComboBox();\nrcbRegister.Label = \"Select Child:\"\nrcbChildren.Width = Unit.Pixel(250);\nrcbChildren.DataValueField = \"childIdFromDataSet\";\nrcbChildren.DataTextField = \"childTextFromDataSet\";\nrcbChildren.Attributes.Add(\"theclientid\", txtChildren.ClientID.ToString());\nrcbChildren.OnClientSelectedIndexChanged = \"OnChildrenSelectedIndexChanged\";\n//...........\n//create you DataTable,HasTable,2DArray here, and use your parameters (Parent's value in this example)\n//........\n//da.Fill(dt)//...........\nrcbChildren.DataSource = dt;\nrcbChildren.DataBind();\nif (txtChildren.Text != \"\") {\nrcbChildren.SelectedValue = txtChildren.Text;\nrcbChildren.Text = rcbRegister.txtChildren.Text;\n}\ntxtChildren.Parent.Controls.Add(rcbChildren);\n}\n}\n```\n\nAnd your Javascript function\n\n```\nfunction OnChildrenSelectedIndexChanged(sender, eventArgs) {\nvar newvalue = eventArgs.get_item().get_value();\nvar passedclientid = sender.get_attributes().getAttribute(\"theclientid\");\nvar field1 = document.getElementById(passedclientid);\nvar field2id = passedclientid + \"_text\";\nvar field2 = document.getElementById(field2id);\nfield1.value = newvalue;\nfield2.value = newvalue;\n}\n```\n\n0\n\nanswered on 14 May 2016, 09:51 AM\n\nHello,\n\nI referred your answer above copied and pasted the code in my web page. The page gets compiled without error but the combobox which is added is not visible in edit or insert mode. I have set viewstate to false, is it because of that ?\n\nThanks\n\nMilind Shevade\n", "html": "\n\n\n\n \n\n \n \n \n \n \n \n \n\n\n\n\n \n\n\n\n RadComboBox SelectedIndexChanged inside Advanced Edit Template in UI for ASP.NET AJAX | Telerik Forums\n\n\n \n \n\n\n\n \n\n \n\n \n\n\n\n\n\n\n\n\n \n\n \n\n\n\n \n\n
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\n\t\tThis is a migrated thread and some comments may be shown as answers.\n\t
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RadComboBox SelectedIndexChanged inside Advanced Edit Template

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\n \n\t\n9 Answers\n \n \n515 Views\n
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\n Joe\n asked on 18 Dec 2009, 03:31 PM\n
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I have Advanced Edit and Insert Form Templates built completely through code-behind for a RadScheduler. I am trying to implement cascading RadComboBoxes inside those forms.

I don't know how to find the controls in the forms from the SelectedIndexChanged events. In the FormCreated method when building the forms, I have the following:

RadComboBox resource = (RadComboBox)e.Container.FindControl(\"ResourceInput\"); 
resource.Skin = _skin; 
resource = LoadResources(resource); 
resource.SelectedIndexChanged += new RadComboBoxSelectedIndexChangedEventHandler(resource_SelectedIndexChanged); 
 

In the SelectedIndexChanged events I don't have access to e.Container.FindControl. So I need to know how to find those controls in that event.

Thanks!
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9 Answers, 1 is accepted

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\n Peter\n
\n Telerik team\n
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\n answered on 22 Dec 2009, 04:07 PM\n
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Hi Joe,
\n
\nThat could be a problem indeed when creating the controls dynamically. If this is not a must-have requirement, then you can use the approach from this kb article on related RadComboBox controls in the advanced form of RadScheduler:
\nhttp://www.telerik.com/support/kb/aspnet-ajax/scheduler/related-load-on-demand-radcomboboxes-in-the-advanced-form-of-radscheduler.aspx
\n
\n
\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
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\n answered on 22 Dec 2009, 08:15 PM\n
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Peter,
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\nUnfortunately, this is a requirement. I would much rather if it wasn't, but not my choice....
\n
\nI tried implementing your suggestion, but I ran into 2 stumbling blocks.
\n
\n1. If I implement the ItemsRequested method in my custom AdvancedFormTemplate class, I no longer have visibility to the originating class, which is a SharePoint web part. In the custom toolpane for the web part, the user is able to select the location of the datasource. So all of that information is stored as properties in my web part class. Therefore, I am not able to load the RadComboBox through an ItemsRequested event in the custom Template class because I don't have access to the web part toolpane properties that give me the datasource. That's why I was trying to implement a SelectedIndexChanged event from the main web part class - added during the FormCreated scheduler method.
\n2. I am not able to add javascript code to the page through the custom Template class. In the example the javascript is added directly to the ascx control, but I am not using that control. I tried adding the javascript through the main web part class, but in the LoadCountries() function I need to have access to this: $find(\"<%= CountriesRadComboBox.ClientID %>\")  which I don't have since the control only exists in the Template class.
\n
\nSo I might be pretty much stuck on this approach. Is there any other way I might accomplish my goal? I think I am pretty much there implementing the SelectedIndexChanged event the way I originally tried if I could somehow get access to the controls within the AdvancedForm Template through that event.
\n
\nThanks.\n
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\n answered on 25 Dec 2009, 01:07 PM\n
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Hi Joe,
\n
\nI think there is a workaround which you can use for this case. How about handling OnClientSelectedIndexChanged to store the selected items's value or text in a hidden field. Then when FormCreated is fired again after selecting a combobox item, you can get the selected value from the hidden field and use it to populate the second combobox. Here is an example which updates a label, but the concept is the same:
\n
\n
\n
<script type=\"text/javascript\">
\n
        function OnClientSelectedIndexChanged(sender, eventArgs) {
\n
            var selectedValue = document.getElementById(\"selectedValue\");
\n
            selectedValue.value = eventArgs.get_item().get_text();  
\n
        }
\n
    </script>
\n
    <asp:HiddenField ID=\"selectedValue\" runat=\"server\" />
\n
    <telerik:RadScheduler ID=\"RadScheduler1\" 
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        runat=\"server\" onformcreated=\"RadScheduler1_FormCreated\"> </telerik:RadScheduler>
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protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
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    {
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        if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert)
\n
        {
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            RadComboBox resource = new RadComboBox();
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            resource.Items.Add(new RadComboBoxItem(\"a\"));
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            resource.Items.Add(new RadComboBoxItem(\"b\"));
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            resource.Items.Add(new RadComboBoxItem(\"c\"));
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            resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\";
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            resource.AutoPostBack = true;
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            Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\");
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            basicPanel.Controls.Add(resource);
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            Label selectedValueLbl = new Label();
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            if (selectedValue.Value != null)
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                selectedValueLbl.Text = selectedValue.Value;
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            basicPanel.Controls.Add(selectedValueLbl);
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            selectedValue.Value = null;
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        }
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    }
\n
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\nLet us know how it goes.
\n
\nAll the best,
\n Peter
\nthe Telerik team\n

\nInstantly find answers to your questions on the new Telerik Support Portal.
\nWatch a video on how to optimize your support resource searches and check out more tips on the blogs.
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\n answered on 06 Jan 2010, 11:11 PM\n
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Peter,
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\nThanks for the great information, as always. I have implemented your approach. However, I am having 1 small issue. I know it's not related to the RadComboBox or the RadScheduler, but I'm hoping you can still help.
\n
\nWhen the OnClientSelectedIndexChanged event is fired, the value is pulled out of the RadComboBox and placed into the hidden field. I have set a JS Alert to verify. But during the FormCreated event, the value of the hidden field is still null. I'm not sure if it's a viewstate issue (unlikely as I have other controls that don't have the problem). Or if it's an AJAX/UpdatePanel/Javascript interaction problem. The hidden field is inside the update panel, as is the scheduler and hence the combo's. I'm not very experienced with Javascript so I don't know what else I might need to do. Do I need to somehow call a refresh or update on the UpdatePanel?
\n
\nHere is my JS for reference:
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function LoadProjects(sender, eventArgs) {  
var selectedValue = document.getElementById('\" + lblResourceValue.ClientID + \"');  
selectedValue.value = eventArgs.get_item().get_text();  
alert(selectedValue.value);  
} 
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\nThanks again!\n
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\n answered on 07 Jan 2010, 09:39 PM\n
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Peter,
\n
\nI never did find an answer to this. I tried every trick in the book, changed control types, tried refreshing the update panel, tried adding a button to call a server side event and pull the value and nothing seemed to work.
\n
\nI did however identify an alternate solution. Now that I am an expert with custom Advanced Form Templates :), I realized I could add hidden labels to that form. Then populate the values from my custom web part to those labels during FormCreated. That allowed me all the parameters I needed to run a standard SelectedIndexChanged event from the custom form class itself. I was able to get that method working.
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\nThanks again for all your help and advice with such a complex setup and advanced functionality. Much appreciated!\n
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\n Milind\n\n
\n Top achievements\n
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\n answered on 01 Sep 2010, 06:20 AM\n
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Hi,
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\nCan you post your code here ? I am finding it tough to add combobox at runtime in radscheduler advance edit form.
\n
\nThanks
\n
\nMilind
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\n Telerik team\n
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\n answered on 06 Sep 2010, 05:05 AM\n
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Hello Milind,
\n
\nDid you try the code from the post on12/25/2009? Do you experience any problems with the following code?:
\n
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protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e) 
\n
    { 
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        if (e.Container.Mode == SchedulerFormMode.AdvancedEdit || e.Container.Mode == SchedulerFormMode.AdvancedInsert) 
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        { 
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            RadComboBox resource = new RadComboBox(); 
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            resource.Items.Add(new RadComboBoxItem(\"a\")); 
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            resource.Items.Add(new RadComboBoxItem(\"b\")); 
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            resource.Items.Add(new RadComboBoxItem(\"c\")); 
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            resource.OnClientSelectedIndexChanged = \"OnClientSelectedIndexChanged\"; 
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            resource.AutoPostBack = true; 
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            Panel basicPanel = (Panel)e.Container.FindControl(\"BasicControlsPanel\"); 
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            basicPanel.Controls.Add(resource); 
\n
    
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            Label selectedValueLbl = new Label(); 
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            if (selectedValue.Value != null) 
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                selectedValueLbl.Text = selectedValue.Value; 
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            basicPanel.Controls.Add(selectedValueLbl); 
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            selectedValue.Value = null; 
\n
        } 
\n
    }
\n
\n
\n
\nKind regards,
\n Peter
\nthe Telerik team\n
\n
Do you want to have your say when we set our development plans? Do you want to know when a feature you care about is added or when a bug fixed? Explore the Telerik Public Issue Tracking system and vote to affect the priority of the items
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\n answered on 12 Nov 2011, 02:17 PM\n
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What I think is going on here is: (though not exactly related to the topic header)
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\nYou have an attribute in RadScheduler, for which you do not want the default text control, but your own control, like a Combobox that could be populated with data as needed (parameterized data, based on other data)
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\nSo, first, do not create a resource, that comboboxes that is created by default, is way to difficult to manipulate on server side, rather use a attribute, which could be hidden (and populated with another controls events)
\n\n
\nLets assume you have a the following attibutes \"Parent\" and \"Children\". You know what \"Parent\"'s value is, before or during the FormCreated event, and you want \"Children\" to be a combobox with items retrieved each time from the database (dynamic), based on a value from \"Parent\". 
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protected void RadScheduler1_FormCreated(object sender, SchedulerFormCreatedEventArgs e)
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{
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RadTextBox txtChildren = (RadTextBox)e.Container.FindControl(\"AttrChildren\")
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txtChildren.Visible = false;
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RadComboBox rcbChildren = new RadComboBox();
\nrcbRegister.Label = \"Select Child:\"
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rcbChildren.Width = Unit.Pixel(250);
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rcbChildren.DataValueField = \"childIdFromDataSet\";
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rcbChildren.DataTextField = \"childTextFromDataSet\";
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rcbChildren.Attributes.Add(\"theclientid\", txtChildren.ClientID.ToString());
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rcbChildren.OnClientSelectedIndexChanged = \"OnChildrenSelectedIndexChanged\";
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//...........
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//create you DataTable,HasTable,2DArray here, and use your parameters (Parent's value in this example)
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//........
\n//da.Fill(dt)
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//...........
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rcbChildren.DataSource = dt;
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rcbChildren.DataBind();
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if (txtChildren.Text != \"\") {
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rcbChildren.SelectedValue = txtChildren.Text;
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rcbChildren.Text = rcbRegister.txtChildren.Text;
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}
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txtChildren.Parent.Controls.Add(rcbChildren);
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}
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}
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\n
\nAnd your Javascript function 
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function OnChildrenSelectedIndexChanged(sender, eventArgs) {
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var newvalue = eventArgs.get_item().get_value();
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var passedclientid = sender.get_attributes().getAttribute(\"theclientid\");
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var field1 = document.getElementById(passedclientid);
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var field2id = passedclientid + \"_text\";
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var field2 = document.getElementById(field2id);
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field1.value = newvalue;
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field2.value = newvalue;
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}
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\n Milind\n\n
\n Top achievements\n
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\n answered on 14 May 2016, 09:51 AM\n
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Hello,

\n

 

\n

I referred your answer above copied and pasted the code in my web page. The page gets compiled without error but the combobox which is added is not visible in edit or insert mode. I have set viewstate to false, is it because of that ?

\n

 

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Thanks

\n

Milind Shevade

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Drag and drop files here or browse to attach...
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Browse for files to attach...
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Supported file types: PNG, JPG, JPEG, ZIP, RAR, TXT. Max total file size - 20MB.
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Tags
\n \n Scheduler\n \n
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Asked by
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\n Joe\n\n
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Answers by
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\n Peter\n
\n Telerik team\n
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\"\"
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\n Joe\n\n
\n Top achievements\n
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\"\"
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\n Milind\n\n
\n Top achievements\n
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\n \n Rank 1\n
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\"\"
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\n Guss\n\n
\n Top achievements\n
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\n \n Veteran\n
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\"\"
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\n Milind\n\n
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Share this question
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\n or\n
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\n \n\n\n"} diff --git a/bench/data/groundtruth/code_16.jsonl b/bench/data/groundtruth/code_16.jsonl index 92d3e91c..b2a1e441 100644 --- a/bench/data/groundtruth/code_16.jsonl +++ b/bench/data/groundtruth/code_16.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "code", "raw_content": "
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<%@ page language=\"java\"\u00a0import=\"java.util.*\"\u00a0pageEncoding=\"utf-8\"%>
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<%
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String path = request.getContextPath();
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String basePath = request.getScheme()+\"://\"+request.getServerName()+\":\"+request.getServerPort()+path+\"/\";
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%>
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\u00a0
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<!DOCTYPE HTML PUBLIC\u00a0\"-//W3C//DTD HTML 4.01 Transitional//EN\">
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<html>
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\u00a0\u00a0<head>
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\u00a0\u00a0\u00a0\u00a0<title>My JSP\u00a0'register.jsp'\u00a0starting page</title>\u00a0\u00a0\u00a0
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\u00a0\u00a0<body>
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\u00a0\u00a0<script type=\"text/javascript\">
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0function validate(){
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(registerForm.uname.value==\"\"){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0alert(\"\u8d26\u53f7\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0return;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(registerForm.upwd.value==\"\"){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0alert(\"\u5bc6\u7801\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0return;
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0registerForm.submit();
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\u00a0\u00a0\u00a0\u00a0</script>
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\u00a0
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\u00a0\u00a0\u00a0\u00a0<form\u00a0 name=\"registerForm\"\u00a0action=\"DoregServlet\"\u00a0method=\"post\">
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u7528\u6237\u540d:<input type=\"text\"\u00a0name=\"uname\"><br>
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u5bc6\u00a0\u00a0 \u7801: <input type=\"password\"\u00a0name=\"upwd\"> <br>
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<input type=\"submit\"\u00a0value=\"\u6ce8\u518c\"\u00a0>
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<a href=\"denglu.jsp\">\u767b\u5f55</a>
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</html>
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package\u00a0com.servlet;
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import\u00a0java.io.IOException;
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import\u00a0java.io.PrintWriter;
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import\u00a0javax.servlet.ServletException;
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import\u00a0javax.servlet.http.HttpServlet;
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import\u00a0javax.servlet.http.HttpServletRequest;
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import\u00a0javax.servlet.http.HttpServletResponse;
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import\u00a0com.dao.UsersDao;
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public\u00a0class\u00a0servlet3\u00a0extends\u00a0HttpServlet {
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\u00a0\u00a0\u00a0\u00a0public\u00a0servlet3() {
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
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\u00a0\u00a0\u00a0\u00a0}
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\u00a0\u00a0\u00a0\u00a0\u00a0
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\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
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\u00a0\u00a0\u00a0\u00a0}
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\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doPost (request, response);
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
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\u00a0\u00a0\u00a0\u00a0}
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\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
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\u00a0
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = request.getParameter(\"uname\");
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String upwd = request.getParameter(\"upwd\");
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0UsersDao usersDao =\u00a0new\u00a0UsersDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0int\u00a0i=usersDao.reg(uname, upwd);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(i>0){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=login.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}else{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=reg.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.sf.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.sf.dao.MsgDao;
\n
import\u00a0com.sf.dao.UsersDao;
\n
\u00a0
\n
public\u00a0class\u00a0Doregservlet\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Constructor of the object.
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0Doregservlet() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Destruction of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setContentType(\"text/html\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0PrintWriter out = response.getWriter();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = request.getParameter(\"uname\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String upwd = request.getParameter(\"upwd\");
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0UsersDao ud =\u00a0new\u00a0UsersDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md =\u00a0new\u00a0MsgDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if\u00a0(ud.register(uname, upwd) >\u00a00) {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.getSession().setAttribute(\"uname\", uname);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.getRequestDispatcher(\"denglu.jsp\").forward(request,
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0else\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0out.print(\"\u6ce8\u518c\u5931\u8d25\uff0c\u8bf7\u91cd\u65b0\u6ce8\u518c.......\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"3;url=reg.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doGet(request,response);
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.dao.MsgDao;
\n
\u00a0
\n
public\u00a0class\u00a0servlet5\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0servlet5() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doPost(request,\u00a0 response);
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0int\u00a0id=Integer.parseInt(request.getParameter(\"id\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md=new\u00a0MsgDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0md.delMail(id);\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.getWriter().print(\"\u522a\u9664\u6210\u529f.....\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=main.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.sendRedirect(\"main2.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.sf.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.sf.dao.MsgDao;
\n
import\u00a0com.sf.entity.Msg;
\n
\u00a0
\n
public\u00a0class\u00a0Dowriteservlet\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Constructor of the object.
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0Dowriteservlet() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Destruction of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setContentType(\"text/html\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0PrintWriter out = response.getWriter();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = (String) request.getSession().getAttribute(\"uname\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String sendto = request.getParameter(\"receiver\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String title = request.getParameter(\"title\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String content = request.getParameter(\"content\");
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Msg m =\u00a0new\u00a0Msg();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setMsgcontent(content);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setUsername(uname);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setSendto(sendto);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setTitle(title);
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md =\u00a0new\u00a0MsgDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0md.addMsg(m);
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0out.print(\"\u53d1\u9001\u6210\u529f.....\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"3;url=main.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doGet(request,response);\u00a0\u00a0\u00a0\u00a0 }
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n\n
\n\n", "inline": false, "content": {"code_content": "<%@ page language=\"java\" import=\"java.util.*\" pageEncoding=\"utf-8\"%>\n<%\nString path = request.getContextPath();\nString basePath = request.getScheme()+\"://\"+request.getServerName()+\":\"+request.getServerPort()+path+\"/\";\n%>\n\n\n\n \n My JSP 'register.jsp' starting page\n \n\n \n \n\n
\n\n \u7528\u6237\u540d:
\n \u5bc6 \u7801:
\n \n \u767b\u5f55\n
\n\n \n\n\n\n\npackage com.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.dao.UsersDao;\n\npublic class servlet3 extends HttpServlet {\n\n public servlet3() {\n super();\n }\n\n\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n doPost (request, response);\n\n }\n\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n String uname = request.getParameter(\"uname\");\n String upwd = request.getParameter(\"upwd\");\n UsersDao usersDao = new UsersDao();\n int i=usersDao.reg(uname, upwd);\n if(i>0){\n\n response.setHeader(\"refresh\", \"2;url=login.jsp\");\n }else{\n\n response.setHeader(\"refresh\", \"2;url=reg.jsp\");\n }\n }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n // Put your code here\n }\n\n}\n\n\n\n\n\npackage com.sf.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.sf.dao.MsgDao;\nimport com.sf.dao.UsersDao;\n\npublic class Doregservlet extends HttpServlet {\n\n /**\n * Constructor of the object.\n */\n public Doregservlet() {\n super();\n }\n\n /**\n * Destruction of the servlet.
\n */\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n response.setContentType(\"text/html\");\n PrintWriter out = response.getWriter();\n request.setCharacterEncoding(\"utf-8\");\n String uname = request.getParameter(\"uname\");\n String upwd = request.getParameter(\"upwd\");\n\n UsersDao ud = new UsersDao();\n MsgDao md = new MsgDao();\n if (ud.register(uname, upwd) > 0) {\n request.getSession().setAttribute(\"uname\", uname);\n request.getRequestDispatcher(\"denglu.jsp\").forward(request,\n response);\n } else {\n out.print(\"\u6ce8\u518c\u5931\u8d25\uff0c\u8bf7\u91cd\u65b0\u6ce8\u518c.......\");\n response.setHeader(\"refresh\", \"3;url=reg.jsp\");\n }\n }\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doGet(request,response);\n }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n // Put your code here\n }\n\n}\n\n\n\n\n\npackage com.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.dao.MsgDao;\n\npublic class servlet5 extends HttpServlet {\n\n public servlet5() {\n super();\n }\n\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doPost(request, response);\n }\n\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n request.setCharacterEncoding(\"utf-8\");\n\n int id=Integer.parseInt(request.getParameter(\"id\"));\n MsgDao md=new MsgDao();\n md.delMail(id);\n response.getWriter().print(\"\u522a\u9664\u6210\u529f.....\");\n response.setHeader(\"refresh\", \"2;url=main.jsp\");\n response.sendRedirect(\"main2.jsp\");\n }\n\n\n public void init() throws ServletException {\n\n }\n\n}\n\n\n\n\n\n\n\npackage com.sf.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.sf.dao.MsgDao;\nimport com.sf.entity.Msg;\n\npublic class Dowriteservlet extends HttpServlet {\n\n /**\n * Constructor of the object.\n */\n public Dowriteservlet() {\n super();\n }\n\n /**\n * Destruction of the servlet.
\n */\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n response.setContentType(\"text/html\");\n PrintWriter out = response.getWriter();\n request.setCharacterEncoding(\"utf-8\");\n String uname = (String) request.getSession().getAttribute(\"uname\");\n String sendto = request.getParameter(\"receiver\");\n String title = request.getParameter(\"title\");\n String content = request.getParameter(\"content\");\n\n Msg m = new Msg();\n m.setMsgcontent(content);\n m.setUsername(uname);\n m.setSendto(sendto);\n m.setTitle(title);\n\n MsgDao md = new MsgDao();\n md.addMsg(m);\n\n out.print(\"\u53d1\u9001\u6210\u529f.....\");\n response.setHeader(\"refresh\", \"3;url=main.jsp\");\n }\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doGet(request,response); }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n }\n\n}", "by": "tag_code"}}, {"type": "table", "raw_content": "
", "content": {"html": "
", "is_complex": false}}, {"type": "paragraph", "raw_content": "
posted @\n2022-05-29 20:20\u5f90\u6d9b%\u00a0\n\u9605\u8bfb(70)\u00a0\n\u8bc4\u8bba(0)\u00a0\n\u7f16\u8f91\u6536\u85cf\u4e3e\u62a5
", "content": [{"c": "posted @ 2022-05-29 20:20 \u5f90\u6d9b% \u9605\u8bfb( 70)\u00a0\n\u8bc4\u8bba( 0) \u7f16\u8f91 \u6536\u85cf \u4e3e\u62a5", "t": "text"}]}]], "main_html": "
\n
<%@ page language=\"java\"\u00a0import=\"java.util.*\"\u00a0pageEncoding=\"utf-8\"%>
\n
<%
\n
String path = request.getContextPath();
\n
String basePath = request.getScheme()+\"://\"+request.getServerName()+\":\"+request.getServerPort()+path+\"/\";
\n
%>
\n
\u00a0
\n
<!DOCTYPE HTML PUBLIC\u00a0\"-//W3C//DTD HTML 4.01 Transitional//EN\">
\n
<html>
\n
\u00a0\u00a0<head>
\n
\u00a0\u00a0\u00a0\u00a0<title>My JSP\u00a0'register.jsp'\u00a0starting page</title>\u00a0\u00a0\u00a0
\n
\u00a0\u00a0</head>
\n
\u00a0\u00a0
\n
\u00a0\u00a0<body>
\n
\u00a0\u00a0<script type=\"text/javascript\">
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0function validate(){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(registerForm.uname.value==\"\"){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0alert(\"\u8d26\u53f7\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0return;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(registerForm.upwd.value==\"\"){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0alert(\"\u5bc6\u7801\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0return;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0registerForm.submit();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0</script>
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0<form\u00a0 name=\"registerForm\"\u00a0action=\"DoregServlet\"\u00a0method=\"post\">
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u7528\u6237\u540d:<input type=\"text\"\u00a0name=\"uname\"><br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u5bc6\u00a0\u00a0 \u7801: <input type=\"password\"\u00a0name=\"upwd\"> <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<input type=\"submit\"\u00a0value=\"\u6ce8\u518c\"\u00a0>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<a href=\"denglu.jsp\">\u767b\u5f55</a>
\n
\u00a0\u00a0\u00a0\u00a0</form>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0</body>
\n
</html>
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.dao.UsersDao;
\n
\u00a0
\n
public\u00a0class\u00a0servlet3\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0servlet3() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doPost (request, response);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = request.getParameter(\"uname\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String upwd = request.getParameter(\"upwd\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0UsersDao usersDao =\u00a0new\u00a0UsersDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0int\u00a0i=usersDao.reg(uname, upwd);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(i>0){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=login.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}else{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=reg.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.sf.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.sf.dao.MsgDao;
\n
import\u00a0com.sf.dao.UsersDao;
\n
\u00a0
\n
public\u00a0class\u00a0Doregservlet\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Constructor of the object.
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0Doregservlet() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Destruction of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setContentType(\"text/html\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0PrintWriter out = response.getWriter();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = request.getParameter(\"uname\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String upwd = request.getParameter(\"upwd\");
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0UsersDao ud =\u00a0new\u00a0UsersDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md =\u00a0new\u00a0MsgDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if\u00a0(ud.register(uname, upwd) >\u00a00) {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.getSession().setAttribute(\"uname\", uname);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.getRequestDispatcher(\"denglu.jsp\").forward(request,
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0else\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0out.print(\"\u6ce8\u518c\u5931\u8d25\uff0c\u8bf7\u91cd\u65b0\u6ce8\u518c.......\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"3;url=reg.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
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\u00a0
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doGet(request,response);
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\u00a0\u00a0\u00a0\u00a0}
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\u00a0
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\u00a0\u00a0\u00a0\u00a0/**
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\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
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\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
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\u00a0\u00a0\u00a0\u00a0\u00a0*/
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\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
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\u00a0\u00a0\u00a0\u00a0}
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\u00a0
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}
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\u00a0
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\u00a0\u00a0
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\u00a0
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\u00a0\u00a0
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\u00a0
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package\u00a0com.servlet;
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\u00a0
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import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
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\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
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import\u00a0javax.servlet.http.HttpServletRequest;
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import\u00a0javax.servlet.http.HttpServletResponse;
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\u00a0
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import\u00a0com.dao.MsgDao;
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\u00a0
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public\u00a0class\u00a0servlet5\u00a0extends\u00a0HttpServlet {
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\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0servlet5() {
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
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\u00a0\u00a0\u00a0\u00a0}
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\u00a0
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\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
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\u00a0\u00a0\u00a0\u00a0}
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\u00a0
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\u00a0\u00a0\u00a0\u00a0\u00a0
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\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
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\u00a0
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doPost(request,\u00a0 response);
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\u00a0\u00a0\u00a0\u00a0}
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\u00a0
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\u00a0\u00a0\u00a0\u00a0\u00a0
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\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
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\u00a0
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0int\u00a0id=Integer.parseInt(request.getParameter(\"id\"));
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md=new\u00a0MsgDao();
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0md.delMail(id);\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.getWriter().print(\"\u522a\u9664\u6210\u529f.....\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=main.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.sendRedirect(\"main2.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0}
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\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
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\u00a0\u00a0\u00a0\u00a0\u00a0
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\u00a0\u00a0\u00a0\u00a0}
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\u00a0
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}
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\u00a0
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package\u00a0com.sf.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
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import\u00a0com.sf.dao.MsgDao;
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import\u00a0com.sf.entity.Msg;
\n
\u00a0
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public\u00a0class\u00a0Dowriteservlet\u00a0extends\u00a0HttpServlet {
\n
\u00a0
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\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Constructor of the object.
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0Dowriteservlet() {
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Destruction of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setContentType(\"text/html\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0PrintWriter out = response.getWriter();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = (String) request.getSession().getAttribute(\"uname\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String sendto = request.getParameter(\"receiver\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String title = request.getParameter(\"title\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String content = request.getParameter(\"content\");
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Msg m =\u00a0new\u00a0Msg();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setMsgcontent(content);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setUsername(uname);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setSendto(sendto);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setTitle(title);
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md =\u00a0new\u00a0MsgDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0md.addMsg(m);
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0out.print(\"\u53d1\u9001\u6210\u529f.....\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"3;url=main.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doGet(request,response);\u00a0\u00a0\u00a0\u00a0 }
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n\n
\n\n
posted @\n2022-05-29 20:20\u5f90\u6d9b%\u00a0\n\u9605\u8bfb(70)\u00a0\n\u8bc4\u8bba(0)\u00a0\n\u7f16\u8f91\u6536\u85cf\u4e3e\u62a5
", "statics": {"code": 1, "table": 1, "paragraph": 1, "paragraph.text": 1}, "url": "https://www.cnblogs.com/xu-tao/p/16324793.html", "content": "```\n<%@ page language=\"java\" import=\"java.util.*\" pageEncoding=\"utf-8\"%>\n<%\nString path = request.getContextPath();\nString basePath = request.getScheme()+\"://\"+request.getServerName()+\":\"+request.getServerPort()+path+\"/\";\n%>\n\n\n\n \n My JSP 'register.jsp' starting page\n \n\n \n \n\n
\n\n \u7528\u6237\u540d:
\n \u5bc6 \u7801:
\n \n \u767b\u5f55\n
\n\n \n\n\n\n\npackage com.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.dao.UsersDao;\n\npublic class servlet3 extends HttpServlet {\n\n public servlet3() {\n super();\n }\n\n\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n doPost (request, response);\n\n }\n\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n String uname = request.getParameter(\"uname\");\n String upwd = request.getParameter(\"upwd\");\n UsersDao usersDao = new UsersDao();\n int i=usersDao.reg(uname, upwd);\n if(i>0){\n\n response.setHeader(\"refresh\", \"2;url=login.jsp\");\n }else{\n\n response.setHeader(\"refresh\", \"2;url=reg.jsp\");\n }\n }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n // Put your code here\n }\n\n}\n\n\n\n\n\npackage com.sf.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.sf.dao.MsgDao;\nimport com.sf.dao.UsersDao;\n\npublic class Doregservlet extends HttpServlet {\n\n /**\n * Constructor of the object.\n */\n public Doregservlet() {\n super();\n }\n\n /**\n * Destruction of the servlet.
\n */\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n response.setContentType(\"text/html\");\n PrintWriter out = response.getWriter();\n request.setCharacterEncoding(\"utf-8\");\n String uname = request.getParameter(\"uname\");\n String upwd = request.getParameter(\"upwd\");\n\n UsersDao ud = new UsersDao();\n MsgDao md = new MsgDao();\n if (ud.register(uname, upwd) > 0) {\n request.getSession().setAttribute(\"uname\", uname);\n request.getRequestDispatcher(\"denglu.jsp\").forward(request,\n response);\n } else {\n out.print(\"\u6ce8\u518c\u5931\u8d25\uff0c\u8bf7\u91cd\u65b0\u6ce8\u518c.......\");\n response.setHeader(\"refresh\", \"3;url=reg.jsp\");\n }\n }\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doGet(request,response);\n }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n // Put your code here\n }\n\n}\n\n\n\n\n\npackage com.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.dao.MsgDao;\n\npublic class servlet5 extends HttpServlet {\n\n public servlet5() {\n super();\n }\n\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doPost(request, response);\n }\n\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n request.setCharacterEncoding(\"utf-8\");\n\n int id=Integer.parseInt(request.getParameter(\"id\"));\n MsgDao md=new MsgDao();\n md.delMail(id);\n response.getWriter().print(\"\u522a\u9664\u6210\u529f.....\");\n response.setHeader(\"refresh\", \"2;url=main.jsp\");\n response.sendRedirect(\"main2.jsp\");\n }\n\n\n public void init() throws ServletException {\n\n }\n\n}\n\n\n\n\n\n\n\npackage com.sf.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.sf.dao.MsgDao;\nimport com.sf.entity.Msg;\n\npublic class Dowriteservlet extends HttpServlet {\n\n /**\n * Constructor of the object.\n */\n public Dowriteservlet() {\n super();\n }\n\n /**\n * Destruction of the servlet.
\n */\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n response.setContentType(\"text/html\");\n PrintWriter out = response.getWriter();\n request.setCharacterEncoding(\"utf-8\");\n String uname = (String) request.getSession().getAttribute(\"uname\");\n String sendto = request.getParameter(\"receiver\");\n String title = request.getParameter(\"title\");\n String content = request.getParameter(\"content\");\n\n Msg m = new Msg();\n m.setMsgcontent(content);\n m.setUsername(uname);\n m.setSendto(sendto);\n m.setTitle(title);\n\n MsgDao md = new MsgDao();\n md.addMsg(m);\n\n out.print(\"\u53d1\u9001\u6210\u529f.....\");\n response.setHeader(\"refresh\", \"3;url=main.jsp\");\n }\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doGet(request,response); }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n }\n\n}\n```\n\nposted @ 2022-05-29 20:20 \u5f90\u6d9b% \u9605\u8bfb( 70)\u00a0\n\u8bc4\u8bba( 0) \u7f16\u8f91 \u6536\u85cf \u4e3e\u62a5\n", "html": "\n\n\n\n \n \n \n\n \n \n \n \n \n \u7b2c\u5341\u4e09\u5468\u4f5c\u4e1a - \u5f90\u6d9b% - \u535a\u5ba2\u56ed\n \n \n\n \n\n\n \n \n\n\n \n\n \n \n \n\n \n\n \n \n \n \n\n\n\n \n
\n \n \"\"\n \n \n
\n
\n \n
\n\n\n\n \n
\n
\n\t
\n \"\u8fd4\u56de\u4e3b\u9875\"\n\n\n

\u5f90\u6d9b%\n

\n

\n\n\n\n\n\n\t
\n\t
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\n\t\t\t
\n\t\t
\n\t
\n
\n
\n\t
\n\t
\n\t\t
\n \n
\n
\n

\n \n \u7b2c\u5341\u4e09\u5468\u4f5c\u4e1a\n\n\n\n\n

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\n\n		\n
\n
<%@ page language=\"java\" import=\"java.util.*\" pageEncoding=\"utf-8\"%>
\n
<%
\n
String path = request.getContextPath();
\n
String basePath = request.getScheme()+\"://\"+request.getServerName()+\":\"+request.getServerPort()+path+\"/\";
\n
%>
\n
 
\n
<!DOCTYPE HTML PUBLIC \"-//W3C//DTD HTML 4.01 Transitional//EN\">
\n
<html>
\n
  <head>
\n
    <title>My JSP 'register.jsp' starting page</title>   
\n
  </head>
\n
  
\n
  <body>
\n
  <script type=\"text/javascript\">
\n
        function validate(){
\n
            if(registerForm.uname.value==\"\"){
\n
                alert(\"\u8d26\u53f7\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
                return;
\n
            }
\n
            if(registerForm.upwd.value==\"\"){
\n
                alert(\"\u5bc6\u7801\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
                return;
\n
            }
\n
            registerForm.submit();
\n
        }
\n
    </script>
\n
 
\n
    <form  name=\"registerForm\" action=\"DoregServlet\" method=\"post\">
\n
 
\n
        \u7528\u6237\u540d:<input type=\"text\" name=\"uname\"><br>
\n
        \u5bc6   \u7801: <input type=\"password\" name=\"upwd\"> <br>
\n
        <input type=\"submit\" value=\"\u6ce8\u518c\" >
\n
        <a href=\"denglu.jsp\">\u767b\u5f55</a>
\n
    </form>
\n
     
\n
  </body>
\n
</html>
\n
 
\n
  
\n
 
\n
package com.servlet;
\n
 
\n
import java.io.IOException;
\n
import java.io.PrintWriter;
\n
 
\n
import javax.servlet.ServletException;
\n
import javax.servlet.http.HttpServlet;
\n
import javax.servlet.http.HttpServletRequest;
\n
import javax.servlet.http.HttpServletResponse;
\n
 
\n
import com.dao.UsersDao;
\n
 
\n
public class servlet3 extends HttpServlet {
\n
 
\n
    public servlet3() {
\n
        super();
\n
    }
\n
 
\n
     
\n
    public void destroy() {
\n
        super.destroy(); // Just puts \"destroy\" string in log
\n
        // Put your code here
\n
    }
\n
 
\n
 
\n
    public void doGet(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
        doPost (request, response);
\n
         
\n
    }
\n
 
\n
     
\n
    public void doPost(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        String uname = request.getParameter(\"uname\");
\n
        String upwd = request.getParameter(\"upwd\");
\n
        UsersDao usersDao = new UsersDao();
\n
        int i=usersDao.reg(uname, upwd);
\n
        if(i>0){
\n
         
\n
            response.setHeader(\"refresh\", \"2;url=login.jsp\");
\n
        }else{
\n
     
\n
            response.setHeader(\"refresh\", \"2;url=reg.jsp\");
\n
        }
\n
    }
\n
 
\n
    /**
\n
     * Initialization of the servlet. <br>
\n
     *
\n
     * @throws ServletException if an error occurs
\n
     */
\n
    public void init() throws ServletException {
\n
        // Put your code here
\n
    }
\n
 
\n
}
\n
 
\n
  
\n
 
\n
  
\n
 
\n
package com.sf.servlet;
\n
 
\n
import java.io.IOException;
\n
import java.io.PrintWriter;
\n
 
\n
import javax.servlet.ServletException;
\n
import javax.servlet.http.HttpServlet;
\n
import javax.servlet.http.HttpServletRequest;
\n
import javax.servlet.http.HttpServletResponse;
\n
 
\n
import com.sf.dao.MsgDao;
\n
import com.sf.dao.UsersDao;
\n
 
\n
public class Doregservlet extends HttpServlet {
\n
 
\n
    /**
\n
     * Constructor of the object.
\n
     */
\n
    public Doregservlet() {
\n
        super();
\n
    }
\n
 
\n
    /**
\n
     * Destruction of the servlet. <br>
\n
     */
\n
    public void destroy() {
\n
        super.destroy(); // Just puts \"destroy\" string in log
\n
        // Put your code here
\n
    }
\n
 
\n
    public void doGet(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        response.setContentType(\"text/html\");
\n
        PrintWriter out = response.getWriter();
\n
        request.setCharacterEncoding(\"utf-8\");
\n
        String uname = request.getParameter(\"uname\");
\n
        String upwd = request.getParameter(\"upwd\");
\n
 
\n
        UsersDao ud = new UsersDao();
\n
        MsgDao md = new MsgDao();
\n
        if (ud.register(uname, upwd) > 0) {
\n
            request.getSession().setAttribute(\"uname\", uname);
\n
            request.getRequestDispatcher(\"denglu.jsp\").forward(request,
\n
                    response);
\n
        } else {
\n
            out.print(\"\u6ce8\u518c\u5931\u8d25\uff0c\u8bf7\u91cd\u65b0\u6ce8\u518c.......\");
\n
            response.setHeader(\"refresh\", \"3;url=reg.jsp\");
\n
        }
\n
    }
\n
    public void doPost(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        doGet(request,response);
\n
    }
\n
 
\n
    /**
\n
     * Initialization of the servlet. <br>
\n
     *
\n
     * @throws ServletException if an error occurs
\n
     */
\n
    public void init() throws ServletException {
\n
        // Put your code here
\n
    }
\n
 
\n
}
\n
 
\n
  
\n
 
\n
  
\n
 
\n
package com.servlet;
\n
 
\n
import java.io.IOException;
\n
import java.io.PrintWriter;
\n
 
\n
import javax.servlet.ServletException;
\n
import javax.servlet.http.HttpServlet;
\n
import javax.servlet.http.HttpServletRequest;
\n
import javax.servlet.http.HttpServletResponse;
\n
 
\n
import com.dao.MsgDao;
\n
 
\n
public class servlet5 extends HttpServlet {
\n
 
\n
    public servlet5() {
\n
        super();
\n
    }
\n
 
\n
    public void destroy() {
\n
        super.destroy(); // Just puts \"destroy\" string in log
\n
        // Put your code here
\n
    }
\n
 
\n
     
\n
    public void doGet(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        doPost(request,  response);
\n
    }
\n
 
\n
     
\n
    public void doPost(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        request.setCharacterEncoding(\"utf-8\");
\n
          
\n
        int id=Integer.parseInt(request.getParameter(\"id\"));
\n
        MsgDao md=new MsgDao();
\n
        md.delMail(id);   
\n
        response.getWriter().print(\"\u522a\u9664\u6210\u529f.....\");
\n
        response.setHeader(\"refresh\", \"2;url=main.jsp\");
\n
        response.sendRedirect(\"main2.jsp\");
\n
    }
\n
 
\n
     
\n
    public void init() throws ServletException {
\n
     
\n
    }
\n
 
\n
}
\n
 
\n
  
\n
 
\n
  
\n
 
\n
  
\n
 
\n
package com.sf.servlet;
\n
 
\n
import java.io.IOException;
\n
import java.io.PrintWriter;
\n
 
\n
import javax.servlet.ServletException;
\n
import javax.servlet.http.HttpServlet;
\n
import javax.servlet.http.HttpServletRequest;
\n
import javax.servlet.http.HttpServletResponse;
\n
 
\n
import com.sf.dao.MsgDao;
\n
import com.sf.entity.Msg;
\n
 
\n
public class Dowriteservlet extends HttpServlet {
\n
 
\n
    /**
\n
     * Constructor of the object.
\n
     */
\n
    public Dowriteservlet() {
\n
        super();
\n
    }
\n
 
\n
    /**
\n
     * Destruction of the servlet. <br>
\n
     */
\n
    public void destroy() {
\n
        super.destroy(); // Just puts \"destroy\" string in log
\n
        // Put your code here
\n
    }
\n
 
\n
    public void doGet(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        response.setContentType(\"text/html\");
\n
        PrintWriter out = response.getWriter();
\n
        request.setCharacterEncoding(\"utf-8\");
\n
        String uname = (String) request.getSession().getAttribute(\"uname\");
\n
        String sendto = request.getParameter(\"receiver\");
\n
        String title = request.getParameter(\"title\");
\n
        String content = request.getParameter(\"content\");
\n
 
\n
        Msg m = new Msg();
\n
        m.setMsgcontent(content);
\n
        m.setUsername(uname);
\n
        m.setSendto(sendto);
\n
        m.setTitle(title);
\n
 
\n
        MsgDao md = new MsgDao();
\n
        md.addMsg(m);
\n
 
\n
        out.print(\"\u53d1\u9001\u6210\u529f.....\");
\n
        response.setHeader(\"refresh\", \"3;url=main.jsp\");
\n
    }
\n
 
\n
    public void doPost(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        doGet(request,response);     }
\n
 
\n
    /**
\n
     * Initialization of the servlet. <br>
\n
     *
\n
     * @throws ServletException if an error occurs
\n
     */
\n
    public void init() throws ServletException {
\n
    }
\n
 
\n
}
\n\n
\n\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
posted @\n2022-05-29 20:20 \n\u5f90\u6d9b% \n\u9605\u8bfb(70) \n\u8bc4\u8bba(0) \n\u7f16\u8f91 \n\u6536\u85cf \n\u4e3e\u62a5\n
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\n \n \"\"\n \n \n
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\n \n
\n\n\t
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\n\t\t
\n\t\t\t
\n

\u516c\u544a

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\n \n\t\t\t
\n\t\t
\n\t
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\n\t
\n\t
\n\t
\n\t\t\nCopyright © 2025 \u5f90\u6d9b%\n
Powered by .NET 9.0 on Kubernetes\n\n\n\n\t
\n
\n\n\n\n\n\n \n \n\n\n\n\n"} +{"content_list": [[{"type": "code", "raw_content": "
\n
<%@ page language=\"java\"\u00a0import=\"java.util.*\"\u00a0pageEncoding=\"utf-8\"%>
\n
<%
\n
String path = request.getContextPath();
\n
String basePath = request.getScheme()+\"://\"+request.getServerName()+\":\"+request.getServerPort()+path+\"/\";
\n
%>
\n
\u00a0
\n
<!DOCTYPE HTML PUBLIC\u00a0\"-//W3C//DTD HTML 4.01 Transitional//EN\">
\n
<html>
\n
\u00a0\u00a0<head>
\n
\u00a0\u00a0\u00a0\u00a0<title>My JSP\u00a0'register.jsp'\u00a0starting page</title>\u00a0\u00a0\u00a0
\n
\u00a0\u00a0</head>
\n
\u00a0\u00a0
\n
\u00a0\u00a0<body>
\n
\u00a0\u00a0<script type=\"text/javascript\">
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0function validate(){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(registerForm.uname.value==\"\"){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0alert(\"\u8d26\u53f7\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0return;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(registerForm.upwd.value==\"\"){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0alert(\"\u5bc6\u7801\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0return;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0registerForm.submit();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0</script>
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0<form\u00a0 name=\"registerForm\"\u00a0action=\"DoregServlet\"\u00a0method=\"post\">
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u7528\u6237\u540d:<input type=\"text\"\u00a0name=\"uname\"><br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u5bc6\u00a0\u00a0 \u7801: <input type=\"password\"\u00a0name=\"upwd\"> <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<input type=\"submit\"\u00a0value=\"\u6ce8\u518c\"\u00a0>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<a href=\"denglu.jsp\">\u767b\u5f55</a>
\n
\u00a0\u00a0\u00a0\u00a0</form>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0</body>
\n
</html>
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.dao.UsersDao;
\n
\u00a0
\n
public\u00a0class\u00a0servlet3\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0servlet3() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doPost (request, response);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = request.getParameter(\"uname\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String upwd = request.getParameter(\"upwd\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0UsersDao usersDao =\u00a0new\u00a0UsersDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0int\u00a0i=usersDao.reg(uname, upwd);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(i>0){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=login.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}else{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=reg.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.sf.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.sf.dao.MsgDao;
\n
import\u00a0com.sf.dao.UsersDao;
\n
\u00a0
\n
public\u00a0class\u00a0Doregservlet\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Constructor of the object.
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0Doregservlet() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Destruction of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setContentType(\"text/html\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0PrintWriter out = response.getWriter();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = request.getParameter(\"uname\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String upwd = request.getParameter(\"upwd\");
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0UsersDao ud =\u00a0new\u00a0UsersDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md =\u00a0new\u00a0MsgDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if\u00a0(ud.register(uname, upwd) >\u00a00) {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.getSession().setAttribute(\"uname\", uname);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.getRequestDispatcher(\"denglu.jsp\").forward(request,
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0else\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0out.print(\"\u6ce8\u518c\u5931\u8d25\uff0c\u8bf7\u91cd\u65b0\u6ce8\u518c.......\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"3;url=reg.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doGet(request,response);
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.dao.MsgDao;
\n
\u00a0
\n
public\u00a0class\u00a0servlet5\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0servlet5() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doPost(request,\u00a0 response);
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0int\u00a0id=Integer.parseInt(request.getParameter(\"id\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md=new\u00a0MsgDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0md.delMail(id);\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.getWriter().print(\"\u522a\u9664\u6210\u529f.....\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=main.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.sendRedirect(\"main2.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.sf.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.sf.dao.MsgDao;
\n
import\u00a0com.sf.entity.Msg;
\n
\u00a0
\n
public\u00a0class\u00a0Dowriteservlet\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Constructor of the object.
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0Dowriteservlet() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Destruction of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setContentType(\"text/html\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0PrintWriter out = response.getWriter();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = (String) request.getSession().getAttribute(\"uname\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String sendto = request.getParameter(\"receiver\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String title = request.getParameter(\"title\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String content = request.getParameter(\"content\");
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Msg m =\u00a0new\u00a0Msg();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setMsgcontent(content);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setUsername(uname);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setSendto(sendto);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setTitle(title);
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md =\u00a0new\u00a0MsgDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0md.addMsg(m);
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0out.print(\"\u53d1\u9001\u6210\u529f.....\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"3;url=main.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doGet(request,response);\u00a0\u00a0\u00a0\u00a0 }
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n\n
\n\n", "inline": false, "content": {"code_content": "<%@ page language=\"java\" import=\"java.util.*\" pageEncoding=\"utf-8\"%>\n<%\nString path = request.getContextPath();\nString basePath = request.getScheme()+\"://\"+request.getServerName()+\":\"+request.getServerPort()+path+\"/\";\n%>\n\n\n\n \n My JSP 'register.jsp' starting page\n \n\n \n \n\n
\n\n \u7528\u6237\u540d:
\n \u5bc6 \u7801:
\n \n \u767b\u5f55\n
\n\n \n\n\n\n\npackage com.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.dao.UsersDao;\n\npublic class servlet3 extends HttpServlet {\n\n public servlet3() {\n super();\n }\n\n\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n doPost (request, response);\n\n }\n\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n String uname = request.getParameter(\"uname\");\n String upwd = request.getParameter(\"upwd\");\n UsersDao usersDao = new UsersDao();\n int i=usersDao.reg(uname, upwd);\n if(i>0){\n\n response.setHeader(\"refresh\", \"2;url=login.jsp\");\n }else{\n\n response.setHeader(\"refresh\", \"2;url=reg.jsp\");\n }\n }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n // Put your code here\n }\n\n}\n\n\n\n\n\npackage com.sf.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.sf.dao.MsgDao;\nimport com.sf.dao.UsersDao;\n\npublic class Doregservlet extends HttpServlet {\n\n /**\n * Constructor of the object.\n */\n public Doregservlet() {\n super();\n }\n\n /**\n * Destruction of the servlet.
\n */\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n response.setContentType(\"text/html\");\n PrintWriter out = response.getWriter();\n request.setCharacterEncoding(\"utf-8\");\n String uname = request.getParameter(\"uname\");\n String upwd = request.getParameter(\"upwd\");\n\n UsersDao ud = new UsersDao();\n MsgDao md = new MsgDao();\n if (ud.register(uname, upwd) > 0) {\n request.getSession().setAttribute(\"uname\", uname);\n request.getRequestDispatcher(\"denglu.jsp\").forward(request,\n response);\n } else {\n out.print(\"\u6ce8\u518c\u5931\u8d25\uff0c\u8bf7\u91cd\u65b0\u6ce8\u518c.......\");\n response.setHeader(\"refresh\", \"3;url=reg.jsp\");\n }\n }\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doGet(request,response);\n }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n // Put your code here\n }\n\n}\n\n\n\n\n\npackage com.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.dao.MsgDao;\n\npublic class servlet5 extends HttpServlet {\n\n public servlet5() {\n super();\n }\n\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doPost(request, response);\n }\n\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n request.setCharacterEncoding(\"utf-8\");\n\n int id=Integer.parseInt(request.getParameter(\"id\"));\n MsgDao md=new MsgDao();\n md.delMail(id);\n response.getWriter().print(\"\u522a\u9664\u6210\u529f.....\");\n response.setHeader(\"refresh\", \"2;url=main.jsp\");\n response.sendRedirect(\"main2.jsp\");\n }\n\n\n public void init() throws ServletException {\n\n }\n\n}\n\n\n\n\n\n\n\npackage com.sf.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.sf.dao.MsgDao;\nimport com.sf.entity.Msg;\n\npublic class Dowriteservlet extends HttpServlet {\n\n /**\n * Constructor of the object.\n */\n public Dowriteservlet() {\n super();\n }\n\n /**\n * Destruction of the servlet.
\n */\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n response.setContentType(\"text/html\");\n PrintWriter out = response.getWriter();\n request.setCharacterEncoding(\"utf-8\");\n String uname = (String) request.getSession().getAttribute(\"uname\");\n String sendto = request.getParameter(\"receiver\");\n String title = request.getParameter(\"title\");\n String content = request.getParameter(\"content\");\n\n Msg m = new Msg();\n m.setMsgcontent(content);\n m.setUsername(uname);\n m.setSendto(sendto);\n m.setTitle(title);\n\n MsgDao md = new MsgDao();\n md.addMsg(m);\n\n out.print(\"\u53d1\u9001\u6210\u529f.....\");\n response.setHeader(\"refresh\", \"3;url=main.jsp\");\n }\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doGet(request,response); }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n }\n\n}", "by": "tag_code"}}, {"type": "table", "raw_content": "
", "content": {"html": null, "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "
posted @\n2022-05-29 20:20\u5f90\u6d9b%\u00a0\n\u9605\u8bfb(70)\u00a0\n\u8bc4\u8bba(0)\u00a0\n\u7f16\u8f91\u6536\u85cf\u4e3e\u62a5
", "content": [{"c": "posted @ 2022-05-29 20:20 \u5f90\u6d9b% \u9605\u8bfb( 70)\u00a0\n\u8bc4\u8bba( 0) \u7f16\u8f91 \u6536\u85cf \u4e3e\u62a5", "t": "text"}]}]], "main_html": "
\n
<%@ page language=\"java\"\u00a0import=\"java.util.*\"\u00a0pageEncoding=\"utf-8\"%>
\n
<%
\n
String path = request.getContextPath();
\n
String basePath = request.getScheme()+\"://\"+request.getServerName()+\":\"+request.getServerPort()+path+\"/\";
\n
%>
\n
\u00a0
\n
<!DOCTYPE HTML PUBLIC\u00a0\"-//W3C//DTD HTML 4.01 Transitional//EN\">
\n
<html>
\n
\u00a0\u00a0<head>
\n
\u00a0\u00a0\u00a0\u00a0<title>My JSP\u00a0'register.jsp'\u00a0starting page</title>\u00a0\u00a0\u00a0
\n
\u00a0\u00a0</head>
\n
\u00a0\u00a0
\n
\u00a0\u00a0<body>
\n
\u00a0\u00a0<script type=\"text/javascript\">
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0function validate(){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(registerForm.uname.value==\"\"){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0alert(\"\u8d26\u53f7\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0return;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(registerForm.upwd.value==\"\"){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0alert(\"\u5bc6\u7801\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0return;
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0registerForm.submit();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0</script>
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0<form\u00a0 name=\"registerForm\"\u00a0action=\"DoregServlet\"\u00a0method=\"post\">
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u7528\u6237\u540d:<input type=\"text\"\u00a0name=\"uname\"><br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u5bc6\u00a0\u00a0 \u7801: <input type=\"password\"\u00a0name=\"upwd\"> <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<input type=\"submit\"\u00a0value=\"\u6ce8\u518c\"\u00a0>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<a href=\"denglu.jsp\">\u767b\u5f55</a>
\n
\u00a0\u00a0\u00a0\u00a0</form>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0</body>
\n
</html>
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.dao.UsersDao;
\n
\u00a0
\n
public\u00a0class\u00a0servlet3\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0servlet3() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doPost (request, response);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = request.getParameter(\"uname\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String upwd = request.getParameter(\"upwd\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0UsersDao usersDao =\u00a0new\u00a0UsersDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0int\u00a0i=usersDao.reg(uname, upwd);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if(i>0){
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=login.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}else{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=reg.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.sf.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.sf.dao.MsgDao;
\n
import\u00a0com.sf.dao.UsersDao;
\n
\u00a0
\n
public\u00a0class\u00a0Doregservlet\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Constructor of the object.
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0Doregservlet() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Destruction of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setContentType(\"text/html\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0PrintWriter out = response.getWriter();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = request.getParameter(\"uname\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String upwd = request.getParameter(\"upwd\");
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0UsersDao ud =\u00a0new\u00a0UsersDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md =\u00a0new\u00a0MsgDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if\u00a0(ud.register(uname, upwd) >\u00a00) {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.getSession().setAttribute(\"uname\", uname);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.getRequestDispatcher(\"denglu.jsp\").forward(request,
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0else\u00a0{
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0out.print(\"\u6ce8\u518c\u5931\u8d25\uff0c\u8bf7\u91cd\u65b0\u6ce8\u518c.......\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"3;url=reg.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doGet(request,response);
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.dao.MsgDao;
\n
\u00a0
\n
public\u00a0class\u00a0servlet5\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0servlet5() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doPost(request,\u00a0 response);
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0int\u00a0id=Integer.parseInt(request.getParameter(\"id\"));
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md=new\u00a0MsgDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0md.delMail(id);\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.getWriter().print(\"\u522a\u9664\u6210\u529f.....\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"2;url=main.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.sendRedirect(\"main2.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
\u00a0\u00a0
\n
\u00a0
\n
package\u00a0com.sf.servlet;
\n
\u00a0
\n
import\u00a0java.io.IOException;
\n
import\u00a0java.io.PrintWriter;
\n
\u00a0
\n
import\u00a0javax.servlet.ServletException;
\n
import\u00a0javax.servlet.http.HttpServlet;
\n
import\u00a0javax.servlet.http.HttpServletRequest;
\n
import\u00a0javax.servlet.http.HttpServletResponse;
\n
\u00a0
\n
import\u00a0com.sf.dao.MsgDao;
\n
import\u00a0com.sf.entity.Msg;
\n
\u00a0
\n
public\u00a0class\u00a0Dowriteservlet\u00a0extends\u00a0HttpServlet {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Constructor of the object.
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0Dowriteservlet() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super();
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Destruction of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0destroy() {
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0super.destroy();\u00a0// Just puts \"destroy\" string in log
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0// Put your code here
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doGet(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setContentType(\"text/html\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0PrintWriter out = response.getWriter();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0request.setCharacterEncoding(\"utf-8\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String uname = (String) request.getSession().getAttribute(\"uname\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String sendto = request.getParameter(\"receiver\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String title = request.getParameter(\"title\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0String content = request.getParameter(\"content\");
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Msg m =\u00a0new\u00a0Msg();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setMsgcontent(content);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setUsername(uname);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setSendto(sendto);
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0m.setTitle(title);
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0MsgDao md =\u00a0new\u00a0MsgDao();
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0md.addMsg(m);
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0out.print(\"\u53d1\u9001\u6210\u529f.....\");
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0response.setHeader(\"refresh\",\u00a0\"3;url=main.jsp\");
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0doPost(HttpServletRequest request, HttpServletResponse response)
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0throws\u00a0ServletException, IOException {
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0doGet(request,response);\u00a0\u00a0\u00a0\u00a0 }
\n
\u00a0
\n
\u00a0\u00a0\u00a0\u00a0/**
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* Initialization of the servlet. <br>
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*
\n
\u00a0\u00a0\u00a0\u00a0\u00a0* @throws ServletException if an error occurs
\n
\u00a0\u00a0\u00a0\u00a0\u00a0*/
\n
\u00a0\u00a0\u00a0\u00a0public\u00a0void\u00a0init()\u00a0throws\u00a0ServletException {
\n
\u00a0\u00a0\u00a0\u00a0}
\n
\u00a0
\n
}
\n\n
\n\n
posted @\n2022-05-29 20:20\u5f90\u6d9b%\u00a0\n\u9605\u8bfb(70)\u00a0\n\u8bc4\u8bba(0)\u00a0\n\u7f16\u8f91\u6536\u85cf\u4e3e\u62a5
", "statics": {"code": 1, "table": 1, "paragraph": 1, "paragraph.text": 1}, "url": "https://www.cnblogs.com/xu-tao/p/16324793.html", "content": "```\n<%@ page language=\"java\" import=\"java.util.*\" pageEncoding=\"utf-8\"%>\n<%\nString path = request.getContextPath();\nString basePath = request.getScheme()+\"://\"+request.getServerName()+\":\"+request.getServerPort()+path+\"/\";\n%>\n\n\n\n \n My JSP 'register.jsp' starting page\n \n\n \n \n\n
\n\n \u7528\u6237\u540d:
\n \u5bc6 \u7801:
\n \n \u767b\u5f55\n
\n\n \n\n\n\n\npackage com.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.dao.UsersDao;\n\npublic class servlet3 extends HttpServlet {\n\n public servlet3() {\n super();\n }\n\n\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n doPost (request, response);\n\n }\n\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n String uname = request.getParameter(\"uname\");\n String upwd = request.getParameter(\"upwd\");\n UsersDao usersDao = new UsersDao();\n int i=usersDao.reg(uname, upwd);\n if(i>0){\n\n response.setHeader(\"refresh\", \"2;url=login.jsp\");\n }else{\n\n response.setHeader(\"refresh\", \"2;url=reg.jsp\");\n }\n }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n // Put your code here\n }\n\n}\n\n\n\n\n\npackage com.sf.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.sf.dao.MsgDao;\nimport com.sf.dao.UsersDao;\n\npublic class Doregservlet extends HttpServlet {\n\n /**\n * Constructor of the object.\n */\n public Doregservlet() {\n super();\n }\n\n /**\n * Destruction of the servlet.
\n */\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n response.setContentType(\"text/html\");\n PrintWriter out = response.getWriter();\n request.setCharacterEncoding(\"utf-8\");\n String uname = request.getParameter(\"uname\");\n String upwd = request.getParameter(\"upwd\");\n\n UsersDao ud = new UsersDao();\n MsgDao md = new MsgDao();\n if (ud.register(uname, upwd) > 0) {\n request.getSession().setAttribute(\"uname\", uname);\n request.getRequestDispatcher(\"denglu.jsp\").forward(request,\n response);\n } else {\n out.print(\"\u6ce8\u518c\u5931\u8d25\uff0c\u8bf7\u91cd\u65b0\u6ce8\u518c.......\");\n response.setHeader(\"refresh\", \"3;url=reg.jsp\");\n }\n }\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doGet(request,response);\n }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n // Put your code here\n }\n\n}\n\n\n\n\n\npackage com.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.dao.MsgDao;\n\npublic class servlet5 extends HttpServlet {\n\n public servlet5() {\n super();\n }\n\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doPost(request, response);\n }\n\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n request.setCharacterEncoding(\"utf-8\");\n\n int id=Integer.parseInt(request.getParameter(\"id\"));\n MsgDao md=new MsgDao();\n md.delMail(id);\n response.getWriter().print(\"\u522a\u9664\u6210\u529f.....\");\n response.setHeader(\"refresh\", \"2;url=main.jsp\");\n response.sendRedirect(\"main2.jsp\");\n }\n\n\n public void init() throws ServletException {\n\n }\n\n}\n\n\n\n\n\n\n\npackage com.sf.servlet;\n\nimport java.io.IOException;\nimport java.io.PrintWriter;\n\nimport javax.servlet.ServletException;\nimport javax.servlet.http.HttpServlet;\nimport javax.servlet.http.HttpServletRequest;\nimport javax.servlet.http.HttpServletResponse;\n\nimport com.sf.dao.MsgDao;\nimport com.sf.entity.Msg;\n\npublic class Dowriteservlet extends HttpServlet {\n\n /**\n * Constructor of the object.\n */\n public Dowriteservlet() {\n super();\n }\n\n /**\n * Destruction of the servlet.
\n */\n public void destroy() {\n super.destroy(); // Just puts \"destroy\" string in log\n // Put your code here\n }\n\n public void doGet(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n response.setContentType(\"text/html\");\n PrintWriter out = response.getWriter();\n request.setCharacterEncoding(\"utf-8\");\n String uname = (String) request.getSession().getAttribute(\"uname\");\n String sendto = request.getParameter(\"receiver\");\n String title = request.getParameter(\"title\");\n String content = request.getParameter(\"content\");\n\n Msg m = new Msg();\n m.setMsgcontent(content);\n m.setUsername(uname);\n m.setSendto(sendto);\n m.setTitle(title);\n\n MsgDao md = new MsgDao();\n md.addMsg(m);\n\n out.print(\"\u53d1\u9001\u6210\u529f.....\");\n response.setHeader(\"refresh\", \"3;url=main.jsp\");\n }\n\n public void doPost(HttpServletRequest request, HttpServletResponse response)\n throws ServletException, IOException {\n\n doGet(request,response); }\n\n /**\n * Initialization of the servlet.
\n *\n * @throws ServletException if an error occurs\n */\n public void init() throws ServletException {\n }\n\n}\n```\n\nposted @ 2022-05-29 20:20 \u5f90\u6d9b% \u9605\u8bfb( 70)\u00a0\n\u8bc4\u8bba( 0) \u7f16\u8f91 \u6536\u85cf \u4e3e\u62a5\n", "html": "\n\n\n\n \n \n \n\n \n \n \n \n \n \u7b2c\u5341\u4e09\u5468\u4f5c\u4e1a - \u5f90\u6d9b% - \u535a\u5ba2\u56ed\n \n \n\n \n\n\n \n \n\n\n \n\n \n \n \n\n \n\n \n \n \n \n\n\n\n \n
\n \n \"\"\n \n \n
\n
\n \n
\n\n\n\n \n
\n
\n\t
\n \"\u8fd4\u56de\u4e3b\u9875\"\n\n\n

\u5f90\u6d9b%\n

\n

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\n \n
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\n \n \u7b2c\u5341\u4e09\u5468\u4f5c\u4e1a\n\n\n\n\n

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\n\n		\n
\n
<%@ page language=\"java\" import=\"java.util.*\" pageEncoding=\"utf-8\"%>
\n
<%
\n
String path = request.getContextPath();
\n
String basePath = request.getScheme()+\"://\"+request.getServerName()+\":\"+request.getServerPort()+path+\"/\";
\n
%>
\n
 
\n
<!DOCTYPE HTML PUBLIC \"-//W3C//DTD HTML 4.01 Transitional//EN\">
\n
<html>
\n
  <head>
\n
    <title>My JSP 'register.jsp' starting page</title>   
\n
  </head>
\n
  
\n
  <body>
\n
  <script type=\"text/javascript\">
\n
        function validate(){
\n
            if(registerForm.uname.value==\"\"){
\n
                alert(\"\u8d26\u53f7\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
                return;
\n
            }
\n
            if(registerForm.upwd.value==\"\"){
\n
                alert(\"\u5bc6\u7801\u4e0d\u80fd\u4e3a\u7a7a!\");
\n
                return;
\n
            }
\n
            registerForm.submit();
\n
        }
\n
    </script>
\n
 
\n
    <form  name=\"registerForm\" action=\"DoregServlet\" method=\"post\">
\n
 
\n
        \u7528\u6237\u540d:<input type=\"text\" name=\"uname\"><br>
\n
        \u5bc6   \u7801: <input type=\"password\" name=\"upwd\"> <br>
\n
        <input type=\"submit\" value=\"\u6ce8\u518c\" >
\n
        <a href=\"denglu.jsp\">\u767b\u5f55</a>
\n
    </form>
\n
     
\n
  </body>
\n
</html>
\n
 
\n
  
\n
 
\n
package com.servlet;
\n
 
\n
import java.io.IOException;
\n
import java.io.PrintWriter;
\n
 
\n
import javax.servlet.ServletException;
\n
import javax.servlet.http.HttpServlet;
\n
import javax.servlet.http.HttpServletRequest;
\n
import javax.servlet.http.HttpServletResponse;
\n
 
\n
import com.dao.UsersDao;
\n
 
\n
public class servlet3 extends HttpServlet {
\n
 
\n
    public servlet3() {
\n
        super();
\n
    }
\n
 
\n
     
\n
    public void destroy() {
\n
        super.destroy(); // Just puts \"destroy\" string in log
\n
        // Put your code here
\n
    }
\n
 
\n
 
\n
    public void doGet(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
        doPost (request, response);
\n
         
\n
    }
\n
 
\n
     
\n
    public void doPost(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        String uname = request.getParameter(\"uname\");
\n
        String upwd = request.getParameter(\"upwd\");
\n
        UsersDao usersDao = new UsersDao();
\n
        int i=usersDao.reg(uname, upwd);
\n
        if(i>0){
\n
         
\n
            response.setHeader(\"refresh\", \"2;url=login.jsp\");
\n
        }else{
\n
     
\n
            response.setHeader(\"refresh\", \"2;url=reg.jsp\");
\n
        }
\n
    }
\n
 
\n
    /**
\n
     * Initialization of the servlet. <br>
\n
     *
\n
     * @throws ServletException if an error occurs
\n
     */
\n
    public void init() throws ServletException {
\n
        // Put your code here
\n
    }
\n
 
\n
}
\n
 
\n
  
\n
 
\n
  
\n
 
\n
package com.sf.servlet;
\n
 
\n
import java.io.IOException;
\n
import java.io.PrintWriter;
\n
 
\n
import javax.servlet.ServletException;
\n
import javax.servlet.http.HttpServlet;
\n
import javax.servlet.http.HttpServletRequest;
\n
import javax.servlet.http.HttpServletResponse;
\n
 
\n
import com.sf.dao.MsgDao;
\n
import com.sf.dao.UsersDao;
\n
 
\n
public class Doregservlet extends HttpServlet {
\n
 
\n
    /**
\n
     * Constructor of the object.
\n
     */
\n
    public Doregservlet() {
\n
        super();
\n
    }
\n
 
\n
    /**
\n
     * Destruction of the servlet. <br>
\n
     */
\n
    public void destroy() {
\n
        super.destroy(); // Just puts \"destroy\" string in log
\n
        // Put your code here
\n
    }
\n
 
\n
    public void doGet(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        response.setContentType(\"text/html\");
\n
        PrintWriter out = response.getWriter();
\n
        request.setCharacterEncoding(\"utf-8\");
\n
        String uname = request.getParameter(\"uname\");
\n
        String upwd = request.getParameter(\"upwd\");
\n
 
\n
        UsersDao ud = new UsersDao();
\n
        MsgDao md = new MsgDao();
\n
        if (ud.register(uname, upwd) > 0) {
\n
            request.getSession().setAttribute(\"uname\", uname);
\n
            request.getRequestDispatcher(\"denglu.jsp\").forward(request,
\n
                    response);
\n
        } else {
\n
            out.print(\"\u6ce8\u518c\u5931\u8d25\uff0c\u8bf7\u91cd\u65b0\u6ce8\u518c.......\");
\n
            response.setHeader(\"refresh\", \"3;url=reg.jsp\");
\n
        }
\n
    }
\n
    public void doPost(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        doGet(request,response);
\n
    }
\n
 
\n
    /**
\n
     * Initialization of the servlet. <br>
\n
     *
\n
     * @throws ServletException if an error occurs
\n
     */
\n
    public void init() throws ServletException {
\n
        // Put your code here
\n
    }
\n
 
\n
}
\n
 
\n
  
\n
 
\n
  
\n
 
\n
package com.servlet;
\n
 
\n
import java.io.IOException;
\n
import java.io.PrintWriter;
\n
 
\n
import javax.servlet.ServletException;
\n
import javax.servlet.http.HttpServlet;
\n
import javax.servlet.http.HttpServletRequest;
\n
import javax.servlet.http.HttpServletResponse;
\n
 
\n
import com.dao.MsgDao;
\n
 
\n
public class servlet5 extends HttpServlet {
\n
 
\n
    public servlet5() {
\n
        super();
\n
    }
\n
 
\n
    public void destroy() {
\n
        super.destroy(); // Just puts \"destroy\" string in log
\n
        // Put your code here
\n
    }
\n
 
\n
     
\n
    public void doGet(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        doPost(request,  response);
\n
    }
\n
 
\n
     
\n
    public void doPost(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        request.setCharacterEncoding(\"utf-8\");
\n
          
\n
        int id=Integer.parseInt(request.getParameter(\"id\"));
\n
        MsgDao md=new MsgDao();
\n
        md.delMail(id);   
\n
        response.getWriter().print(\"\u522a\u9664\u6210\u529f.....\");
\n
        response.setHeader(\"refresh\", \"2;url=main.jsp\");
\n
        response.sendRedirect(\"main2.jsp\");
\n
    }
\n
 
\n
     
\n
    public void init() throws ServletException {
\n
     
\n
    }
\n
 
\n
}
\n
 
\n
  
\n
 
\n
  
\n
 
\n
  
\n
 
\n
package com.sf.servlet;
\n
 
\n
import java.io.IOException;
\n
import java.io.PrintWriter;
\n
 
\n
import javax.servlet.ServletException;
\n
import javax.servlet.http.HttpServlet;
\n
import javax.servlet.http.HttpServletRequest;
\n
import javax.servlet.http.HttpServletResponse;
\n
 
\n
import com.sf.dao.MsgDao;
\n
import com.sf.entity.Msg;
\n
 
\n
public class Dowriteservlet extends HttpServlet {
\n
 
\n
    /**
\n
     * Constructor of the object.
\n
     */
\n
    public Dowriteservlet() {
\n
        super();
\n
    }
\n
 
\n
    /**
\n
     * Destruction of the servlet. <br>
\n
     */
\n
    public void destroy() {
\n
        super.destroy(); // Just puts \"destroy\" string in log
\n
        // Put your code here
\n
    }
\n
 
\n
    public void doGet(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        response.setContentType(\"text/html\");
\n
        PrintWriter out = response.getWriter();
\n
        request.setCharacterEncoding(\"utf-8\");
\n
        String uname = (String) request.getSession().getAttribute(\"uname\");
\n
        String sendto = request.getParameter(\"receiver\");
\n
        String title = request.getParameter(\"title\");
\n
        String content = request.getParameter(\"content\");
\n
 
\n
        Msg m = new Msg();
\n
        m.setMsgcontent(content);
\n
        m.setUsername(uname);
\n
        m.setSendto(sendto);
\n
        m.setTitle(title);
\n
 
\n
        MsgDao md = new MsgDao();
\n
        md.addMsg(m);
\n
 
\n
        out.print(\"\u53d1\u9001\u6210\u529f.....\");
\n
        response.setHeader(\"refresh\", \"3;url=main.jsp\");
\n
    }
\n
 
\n
    public void doPost(HttpServletRequest request, HttpServletResponse response)
\n
            throws ServletException, IOException {
\n
 
\n
        doGet(request,response);     }
\n
 
\n
    /**
\n
     * Initialization of the servlet. <br>
\n
     *
\n
     * @throws ServletException if an error occurs
\n
     */
\n
    public void init() throws ServletException {
\n
    }
\n
 
\n
}
\n\n
\n\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
posted @\n2022-05-29 20:20 \n\u5f90\u6d9b% \n\u9605\u8bfb(70) \n\u8bc4\u8bba(0) \n\u7f16\u8f91 \n\u6536\u85cf \n\u4e3e\u62a5\n
\n
\n\n\n
\n
\n\n\n
\n
\n \n
\n \n
\n
\n
\n
\n
\n \n \"\"\n \n \n
\n
\n
\n
\n \n
\n\n\t
\n\t
\n\t
\n\t\t
\n\t\t\t
\n

\u516c\u544a

\n
\n
\n
\n
\n\n\n\n
\n
\n\t\t\t
\n\t\t\t
\n\t\t\t\t
\n \n\t\t\t
\n\t\t
\n\t
\n\t
\n\t
\n\t
\n\t
\n\t\t\nCopyright © 2025 \u5f90\u6d9b%\n
Powered by .NET 9.0 on Kubernetes\n\n\n\n\t
\n
\n\n\n\n\n\n \n \n\n\n\n\n"} diff --git a/bench/data/groundtruth/code_17_csdn.jsonl b/bench/data/groundtruth/code_17_csdn.jsonl index ea745c47..0993ab3d 100644 --- a/bench/data/groundtruth/code_17_csdn.jsonl +++ b/bench/data/groundtruth/code_17_csdn.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "title", "raw_content": "

pandas \u5206\u6279\u8bfb\u5165execl

", "content": {"title_content": "pandas \u5206\u6279\u8bfb\u5165execl", "level": "1"}}, {"type": "paragraph", "raw_content": "
\u65f6\u95f4: 2023-06-29 20:05:48
", "content": [{"c": "\u65f6\u95f4: 2023-06-29 20:05:48", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\u6d4f\u89c8: 99
", "content": [{"c": "\u6d4f\u89c8: 99", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\u4f60\u53ef\u4ee5\u4f7f\u7528 [pandas](https://wenku.csdn.net/doc/6412b725be7fbd1778d4940f?spm=1055.2569.3001.10083&kwd=pandas) \u7684 `read_[excel](https://wenku.csdn.net/doc/640n8f90vm?spm=1055.2569.3001.10083&kwd=excel)()` \u65b9\u6cd5\uff0c\u5e76\u7ed3\u5408 `chunksize` \u53c2\u6570\u6765\u5206\u6279\u8bfb\u5165 Excel \u6587\u4ef6\u3002`chunksize` \u53c2\u6570\u6307\u5b9a\u4e86\u6bcf\u6b21\u8bfb\u5165\u7684\u884c\u6570\uff0c\u53ef\u4ee5\u6839\u636e\u4f60\u7684\u9700\u6c42\u8fdb\u884c\u8c03\u6574\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff1a\n\n```[python](https://wenku.csdn.net/doc/6412b46ebe7fbd1778d3f92a?spm=1055.2569.3001.10083&kwd=python)\n[import](https://wenku.csdn.net/doc/5cpc346q6x?spm=1055.2569.3001.10083&kwd=import) pandas as pd\n\n# \u6bcf\u6b21\u8bfb\u5165 1000 \u884c\u6570\u636e\nchunksize = 1000\n# Excel \u6587\u4ef6\u8def\u5f84\nexcel_path = 'example.xlsx'\n\n# \u904d\u5386\u6bcf\u4e2a\u5206\u5757\u6570\u636e\nfor chunk in pd.read_excel(excel_path, chunksize=chunksize):\n# \u5904\u7406\u5f53\u524d\u5206\u5757\u6570\u636e\n[print](https://wenku.csdn.net/doc/6401acb5cce7214c316ecd5e?spm=1055.2569.3001.10083&kwd=print)(chunk.head())\n```\n\n\u4e0a\u8ff0\u4ee3\u7801\u5c06 Excel \u6587\u4ef6\u6309\u7167\u6bcf\u6b21\u8bfb\u53d6 1000 \u884c\u6570\u636e\u8fdb\u884c\u5206\u5757\uff0c\u7136\u540e\u904d\u5386\u6bcf\u4e2a\u5206\u5757\u5e76\u5bf9\u5176\u8fdb\u884c\u5904\u7406\u3002\u4f60\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4fee\u6539 `chunksize` \u53c2\u6570\u7684\u503c\u3002
", "content": [{"c": "\u4f60\u53ef\u4ee5\u4f7f\u7528 [pandas](https://wenku.csdn.net/doc/6412b725be7fbd1778d4940f?spm=1055.2569.3001.10083&kwd=pandas) \u7684 `read_[excel](https://wenku.csdn.net/doc/640n8f90vm?spm=1055.2569.3001.10083&kwd=excel)()` \u65b9\u6cd5\uff0c\u5e76\u7ed3\u5408 `chunksize` \u53c2\u6570\u6765\u5206\u6279\u8bfb\u5165 Excel \u6587\u4ef6\u3002`chunksize` \u53c2\u6570\u6307\u5b9a\u4e86\u6bcf\u6b21\u8bfb\u5165\u7684\u884c\u6570\uff0c\u53ef\u4ee5\u6839\u636e\u4f60\u7684\u9700\u6c42\u8fdb\u884c\u8c03\u6574\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff1a\n\n```[python](https://wenku.csdn.net/doc/6412b46ebe7fbd1778d3f92a?spm=1055.2569.3001.10083&kwd=python)\n[import](https://wenku.csdn.net/doc/5cpc346q6x?spm=1055.2569.3001.10083&kwd=import) pandas as pd\n\n# \u6bcf\u6b21\u8bfb\u5165 1000 \u884c\u6570\u636e\nchunksize = 1000\n# Excel \u6587\u4ef6\u8def\u5f84\nexcel_path = 'example.xlsx'\n\n# \u904d\u5386\u6bcf\u4e2a\u5206\u5757\u6570\u636e\nfor chunk in pd.read_excel(excel_path, chunksize=chunksize):\n# \u5904\u7406\u5f53\u524d\u5206\u5757\u6570\u636e\n[print](https://wenku.csdn.net/doc/6401acb5cce7214c316ecd5e?spm=1055.2569.3001.10083&kwd=print)(chunk.head())\n```\n\n\u4e0a\u8ff0\u4ee3\u7801\u5c06 Excel \u6587\u4ef6\u6309\u7167\u6bcf\u6b21\u8bfb\u53d6 1000 \u884c\u6570\u636e\u8fdb\u884c\u5206\u5757\uff0c\u7136\u540e\u904d\u5386\u6bcf\u4e2a\u5206\u5757\u5e76\u5bf9\u5176\u8fdb\u884c\u5904\u7406\u3002\u4f60\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4fee\u6539 `chunksize` \u53c2\u6570\u7684\u503c\u3002", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\u9605\u8bfb\u5168\u6587
", "content": [{"c": "\u9605\u8bfb\u5168\u6587", "t": "text"}]}, {"type": "title", "raw_content": "

\u76f8\u5173\u63a8\u8350

", "content": {"title_content": "\u76f8\u5173\u63a8\u8350", "level": "2"}}, {"type": "title", "raw_content": "

\u5927\u5bb6\u5728\u770b

", "content": {"title_content": "\u5927\u5bb6\u5728\u770b", "level": "2"}}, {"type": "title", "raw_content": "

\u6700\u65b0\u63a8\u8350

", "content": {"title_content": "\u6700\u65b0\u63a8\u8350", "level": "2"}}]], "main_html": "

pandas \u5206\u6279\u8bfb\u5165execl

\u65f6\u95f4: 2023-06-29 20:05:48
\u6d4f\u89c8: 99
\u4f60\u53ef\u4ee5\u4f7f\u7528 [pandas](https://wenku.csdn.net/doc/6412b725be7fbd1778d4940f?spm=1055.2569.3001.10083&kwd=pandas) \u7684 `read_[excel](https://wenku.csdn.net/doc/640n8f90vm?spm=1055.2569.3001.10083&kwd=excel)()` \u65b9\u6cd5\uff0c\u5e76\u7ed3\u5408 `chunksize` \u53c2\u6570\u6765\u5206\u6279\u8bfb\u5165 Excel \u6587\u4ef6\u3002`chunksize` \u53c2\u6570\u6307\u5b9a\u4e86\u6bcf\u6b21\u8bfb\u5165\u7684\u884c\u6570\uff0c\u53ef\u4ee5\u6839\u636e\u4f60\u7684\u9700\u6c42\u8fdb\u884c\u8c03\u6574\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff1a\n\n```[python](https://wenku.csdn.net/doc/6412b46ebe7fbd1778d3f92a?spm=1055.2569.3001.10083&kwd=python)\n[import](https://wenku.csdn.net/doc/5cpc346q6x?spm=1055.2569.3001.10083&kwd=import) pandas as pd\n\n# \u6bcf\u6b21\u8bfb\u5165 1000 \u884c\u6570\u636e\nchunksize = 1000\n# Excel \u6587\u4ef6\u8def\u5f84\nexcel_path = 'example.xlsx'\n\n# \u904d\u5386\u6bcf\u4e2a\u5206\u5757\u6570\u636e\nfor chunk in pd.read_excel(excel_path, chunksize=chunksize):\n# \u5904\u7406\u5f53\u524d\u5206\u5757\u6570\u636e\n[print](https://wenku.csdn.net/doc/6401acb5cce7214c316ecd5e?spm=1055.2569.3001.10083&kwd=print)(chunk.head())\n```\n\n\u4e0a\u8ff0\u4ee3\u7801\u5c06 Excel \u6587\u4ef6\u6309\u7167\u6bcf\u6b21\u8bfb\u53d6 1000 \u884c\u6570\u636e\u8fdb\u884c\u5206\u5757\uff0c\u7136\u540e\u904d\u5386\u6bcf\u4e2a\u5206\u5757\u5e76\u5bf9\u5176\u8fdb\u884c\u5904\u7406\u3002\u4f60\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4fee\u6539 `chunksize` \u53c2\u6570\u7684\u503c\u3002
\u9605\u8bfb\u5168\u6587

\u76f8\u5173\u63a8\u8350

\u5927\u5bb6\u5728\u770b

\u6700\u65b0\u63a8\u8350

", "statics": {"title": 4, "paragraph": 4, "paragraph.text": 4}, "url": "https://blog.csdn.net/moon_sky1999/article/details/52744845", "content": "# pandas \u5206\u6279\u8bfb\u5165execl\n\n\u65f6\u95f4: 2023-06-29 20:05:48\n\n\u6d4f\u89c8: 99\n\n\u4f60\u53ef\u4ee5\u4f7f\u7528 [pandas](https://wenku.csdn.net/doc/6412b725be7fbd1778d4940f?spm=1055.2569.3001.10083&kwd=pandas) \u7684 \\`read_[excel](https://wenku.csdn.net/doc/640n8f90vm?spm=1055.2569.3001.10083&kwd=excel)()\\` \u65b9\u6cd5\uff0c\u5e76\u7ed3\u5408 \\`chunksize\\` \u53c2\u6570\u6765\u5206\u6279\u8bfb\u5165 Excel \u6587\u4ef6\u3002\\`chunksize\\` \u53c2\u6570\u6307\u5b9a\u4e86\u6bcf\u6b21\u8bfb\u5165\u7684\u884c\u6570\uff0c\u53ef\u4ee5\u6839\u636e\u4f60\u7684\u9700\u6c42\u8fdb\u884c\u8c03\u6574\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff1a\n\n\\`\\`\\`[python](https://wenku.csdn.net/doc/6412b46ebe7fbd1778d3f92a?spm=1055.2569.3001.10083&kwd=python)\n[import](https://wenku.csdn.net/doc/5cpc346q6x?spm=1055.2569.3001.10083&kwd=import) pandas as pd\n\n\\# \u6bcf\u6b21\u8bfb\u5165 1000 \u884c\u6570\u636e\nchunksize = 1000\n\\# Excel \u6587\u4ef6\u8def\u5f84\nexcel_path = 'example.xlsx'\n\n\\# \u904d\u5386\u6bcf\u4e2a\u5206\u5757\u6570\u636e\nfor chunk in pd.read_excel(excel_path, chunksize=chunksize):\n\\# \u5904\u7406\u5f53\u524d\u5206\u5757\u6570\u636e\n[print](https://wenku.csdn.net/doc/6401acb5cce7214c316ecd5e?spm=1055.2569.3001.10083&kwd=print)(chunk.head())\n\\`\\`\\`\n\n\u4e0a\u8ff0\u4ee3\u7801\u5c06 Excel \u6587\u4ef6\u6309\u7167\u6bcf\u6b21\u8bfb\u53d6 1000 \u884c\u6570\u636e\u8fdb\u884c\u5206\u5757\uff0c\u7136\u540e\u904d\u5386\u6bcf\u4e2a\u5206\u5757\u5e76\u5bf9\u5176\u8fdb\u884c\u5904\u7406\u3002\u4f60\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4fee\u6539 \\`chunksize\\` \u53c2\u6570\u7684\u503c\u3002\n\n\u9605\u8bfb\u5168\u6587\n\n## \u76f8\u5173\u63a8\u8350\n\n## \u5927\u5bb6\u5728\u770b\n\n## \u6700\u65b0\u63a8\u8350\n", "html": "\npandas \u5206\u6279\u8bfb\u5165execl - CSDN\u6587\u5e93 \n \n \n \n\n \n \n \n \n\n \n\n \n\n \n\n \n
\u9996\u9875pandas \u5206\u6279\u8bfb\u5165execl

pandas \u5206\u6279\u8bfb\u5165execl

\u65f6\u95f4: 2023-06-29 20:05:48 \u6d4f\u89c8: 99
\u4f60\u53ef\u4ee5\u4f7f\u7528 [pandas](https://wenku.csdn.net/doc/6412b725be7fbd1778d4940f?spm=1055.2569.3001.10083&kwd=pandas) \u7684 `read_[excel](https://wenku.csdn.net/doc/640n8f90vm?spm=1055.2569.3001.10083&kwd=excel)()` \u65b9\u6cd5\uff0c\u5e76\u7ed3\u5408 `chunksize` \u53c2\u6570\u6765\u5206\u6279\u8bfb\u5165 Excel \u6587\u4ef6\u3002`chunksize` \u53c2\u6570\u6307\u5b9a\u4e86\u6bcf\u6b21\u8bfb\u5165\u7684\u884c\u6570\uff0c\u53ef\u4ee5\u6839\u636e\u4f60\u7684\u9700\u6c42\u8fdb\u884c\u8c03\u6574\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff1a\n\n```[python](https://wenku.csdn.net/doc/6412b46ebe7fbd1778d3f92a?spm=1055.2569.3001.10083&kwd=python)\n[import](https://wenku.csdn.net/doc/5cpc346q6x?spm=1055.2569.3001.10083&kwd=import) pandas as pd\n\n# \u6bcf\u6b21\u8bfb\u5165 1000 \u884c\u6570\u636e\nchunksize = 1000\n# Excel \u6587\u4ef6\u8def\u5f84\nexcel_path = 'example.xlsx'\n\n# \u904d\u5386\u6bcf\u4e2a\u5206\u5757\u6570\u636e\nfor chunk in pd.read_excel(excel_path, chunksize=chunksize):\n# \u5904\u7406\u5f53\u524d\u5206\u5757\u6570\u636e\n[print](https://wenku.csdn.net/doc/6401acb5cce7214c316ecd5e?spm=1055.2569.3001.10083&kwd=print)(chunk.head())\n```\n\n\u4e0a\u8ff0\u4ee3\u7801\u5c06 Excel \u6587\u4ef6\u6309\u7167\u6bcf\u6b21\u8bfb\u53d6 1000 \u884c\u6570\u636e\u8fdb\u884c\u5206\u5757\uff0c\u7136\u540e\u904d\u5386\u6bcf\u4e2a\u5206\u5757\u5e76\u5bf9\u5176\u8fdb\u884c\u5904\u7406\u3002\u4f60\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4fee\u6539 `chunksize` \u53c2\u6570\u7684\u503c\u3002
\u9605\u8bfb\u5168\u6587

\u76f8\u5173\u63a8\u8350

\"pdf\"

\u4f7f\u7528pandas\u8fdb\u884cexcel\u6587\u4ef6\u7684\u8bfb\u53d6\u5199\u5165

pandas\u7684DateFrame\u7c7b\u4f3c\u4e8eexcel\u7684\u6570\u636e\u50a8\u5b58\u7ed3\u6784\uff0c\u6545\u4e3a\u4e86\u4fbf\u4e8epandas\u8fdb\u884c\u6570\u636e\u5904\u7406\uff0c\u8bbe\u8ba1\u8005\u63d0\u4f9b\u4e86\u7b80\u4fbf\u7684\u65b9\u6cd5\u8fdb\u884c\u6570\u636e\u7684\u8bfb\u53d6\uff0c\u4e0b\u9762\u5199\u4e0a\u4ee3\u7801\nimport pandas as pd\nimport numpy as np\ndef save_excel(arr, exc_filename=None): #\u8bfb\u53d6excel\u7684\u51fd\u6570\nSave the data in excel format\nif exc_filename == None:\n exc_filename = excel.xlsx\narr_df = pd.DataFr

pandas\u8bfb\u53d6excel\u6587\u4ef6

pandas\u8bfb\u53d6excel\u6587\u4ef6\uff0c\u5229\u7528pandas\u8bfb\u53d6Excel\u6570\u636e\uff0c\u7136\u540e\u5229\u7528Pycharts\u8fdb\u884c\u53ef\u89c6\u5316\u5c55\u793a\uff0c\u5206\u6790\u4e0d\u540c\u7ad9\u70b9\u4e0d\u540c\u65f6\u95f4\u7684aqi\u5386\u53f2\u6570\u636e\uff0c\u5e76\u8fdb\u884c\u5dee\u503c\u8fd0\u7b97\uff0c\u5206\u6790\u5dee\u503c\u7684\u8d8b\u52bf
\"md\"

python pandas \u8bfb\u53d6excel\u6570\u636e

python\u8bfb\u53d6excel\u6570\u636e\n\n\u8981\u5728Python\u4e2d\u8bfb\u53d6Excel\u6570\u636e\uff0c\u4f60\u53ef\u4ee5\u4f7f\u7528pandas\u5e93
\"pdf\"

pandas\u5206\u6279\u8bfb\u53d6\u5927\u6570\u636e\u96c6\u6559\u7a0b

\u5728\u5904\u7406\u5927\u6570\u636e\u96c6\u65f6\uff0cPandas \u63d0\u4f9b\u4e86\u4e00\u79cd\u9ad8\u6548\u7684\u65b9\u6cd5\uff0c\u5373\u5206\u6279\u8bfb\u53d6\uff08chunking\uff09\uff0c\u6765\u89e3\u51b3\u5185\u5b58\u9650\u5236\u7684\u95ee\u9898\u3002\u8fd9\u4e2a\u6280\u5de7\u5c24\u5176\u9002\u7528\u4e8e\u90a3\u4e9b\u5185\u5b58\u4e0d\u8db3\u4ee5\u4e00\u6b21\u6027\u52a0\u8f7d\u6574\u4e2a\u6570\u636e\u96c6\u7684\u573a\u666f\uff0c\u5982\u5904\u7406\u5927\u578bCSV\u6587\u4ef6\u3002\u5728\u6807\u9898\u63d0\u5230\u7684\u6559\u7a0b\u4e2d\uff0c\u901a\u8fc7\u8bbe\u7f6e...
\"xlsx\"

python\u6570\u636e\u5206\u6790+pandas+\u6587\u4ef6\u8bfb\u5165\u6216\u5199\u5165

\u8f85\u52a9\u6587\u7ae0\u7684\u4f5c\u7528\uff0c\u53ef\u66f4\u5feb\u7684\u53c2\u8003\u6587\u7ae0
\"zip\"

Pandas

\u73b0\u5728\uff0c\u662f\u65f6\u5019\u4e86\u89e3\u60a8\u5bf9Python Pandas\u7684\u4e86\u89e3\u5e76\u5c06\u5176\u5e94\u7528\u4e8e\u65b0\u60c5\u51b5\u4e86\u3002 \u5bf9\u4e8e\u6b64\u4f5c\u4e1a\uff0c\u60a8\u9700\u8981\u5b8c\u6210\u4e24\u4e2a\uff08\u4e0d\u662f\u4e24\u4e2a\uff09\u6570\u636e\u6311\u6218\u4e4b\u4e00\u3002 \u518d\u4e00\u6b21\uff0c\u60a8\u8981\u9762\u5bf9\u7684\u6311\u6218\u662f\u60a8\u7684\u9009\u62e9\u3002 \u53ea\u9700\u786e\u4fdd\u5168\u529b\u4ee5\u8d74-\u56e0\u4e3a\u60a8\u78e8\u7ec3\u7684\u6280\u80fd\u5c06\u6210\u4e3a\u6570\u636e\u5206\u6790...
\"zip\"

pandas

**Pandas\uff1a\u6570\u636e\u5206\u6790\u7684\u6838\u5fc3\u5de5\u5177** Pandas\u662fPython\u7f16\u7a0b\u8bed\u8a00\u4e2d\u4e00\u4e2a\u5f3a\u5927\u7684\u6570\u636e\u5904\u7406\u5e93\uff0c\u5b83\u4e3a\u6570\u636e\u5206\u6790\u63d0\u4f9b\u4e86\u7b80\u5355\u6613\u7528\u4e14\u9ad8\u6548\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5982DataFrame\u548cSeries\u3002\u8fd9\u4e2a\u5e93\u7684\u8bbe\u8ba1\u76ee\u6807\u662f\u4f7f\u5904\u7406\u7ed3\u6784\u5316\u548c\u534a\u7ed3\u6784\u5316\u6570\u636e\u53d8\u5f97\u65b9\u4fbf\uff0c...
\"zip\"

pandas-pandas

pandas pandas pandas pandas pandas
\"zip\"

pandas.zip_pandas_pandas \u901f\u67e5\u624b\u518c_pandas\u5f00\u53d1\u624b\u518c

**Pandas\uff1aPython\u6570\u636e\u5206\u6790\u5e93\u8be6\u89e3** Pandas\u662fPython\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u4e00\u4e2a\u5f3a\u5927\u6570\u636e\u5206\u6790\u5de5\u5177\uff0c\u5b83\u63d0\u4f9b\u4e86\u9ad8\u6548\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5982DataFrame\u548cSeries\uff0c\u4ee5\u53ca\u4e30\u5bcc\u7684\u6570\u636e\u5206\u6790\u529f\u80fd\u3002\u5728\u672c\u7bc7\u4e2d\uff0c\u6211\u4eec\u5c06\u6df1\u5165\u63a2\u8ba8Pandas\u7684\u6838\u5fc3\u6982\u5ff5\u3001\u5e38\u7528...
\"zip\"

LearnPandas\uff1aLearnPandas

\u300aPandas\u6df1\u5ea6\u5b66\u4e60\u6307\u5357\u300b Pandas\uff0c\u4e00\u4e2a\u5728Python\u4e2d\u5e7f\u6cdb\u4f7f\u7528\u7684\u6570\u636e\u5206\u6790\u5e93\uff0c\u56e0\u5176\u5f3a\u5927\u7684\u6570\u636e\u5904\u7406\u80fd\u529b\u548c\u6613\u7528\u6027\uff0c\u5df2\u7ecf\u6210\u4e3a\u6570\u636e\u79d1\u5b66\u5bb6\u548c\u6570\u636e\u5206\u6790\u5e08\u7684\u5fc5\u5907\u5de5\u5177\u3002\"LearnPandas\"\u662f\u9488\u5bf9Pandas\u7684\u5b66\u4e60\u8d44\u6e90\uff0c\u65e8\u5728\u5e2e\u52a9\u7528\u6237\u638c\u63e1\u8fd9\u4e2a...
\"zip\"

pandas\u63d2\u4ef6\uff08pandas\uff09

\u9002\u7528\u4e8epython2.7
\"pdf\"

Learning-pandas \u5b66\u4e60pandas

\u4ece\u63d0\u4f9b\u7684\u6587\u4ef6\u4fe1\u606f\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u63d0\u53d6\u548c\u603b\u7ed3\u4ee5\u4e0b\u5173\u4e8e\u5b66\u4e60pandas\u7684\u77e5\u8bc6\u70b9\u3002 1. pandas\u6982\u8ff0\uff1a pandas\u662f\u4e00\u4e2a\u5f00\u6e90\u7684\u3001\u9762\u5411Python\u7684\u9ad8\u6027\u80fd\u6570\u636e\u5904\u7406\u548c\u5206\u6790\u5e93\u3002\u5b83\u63d0\u4f9b\u4e86\u6570\u636e\u7ed3\u6784\u548c\u64cd\u4f5c\u6570\u636e\u7684\u5de5\u5177\uff0c\u5c24\u5176\u662f\u9488\u5bf9\u8868\u683c\u548c\u65f6\u95f4\u5e8f\u5217...
\"pdf\"

Pandas - Thinking in Pandas

About pandas How pandas helped build an image of a black hole how pandas helps financial institutions make more informed predictions about the future market how pandas helps improve discoverability of...
\"rar\"

pandas\u5b98\u65b9\u6587\u6863\u4e2d\u6587\u7248_pandas_pandas\u6587\u6863_pandas\u4e2d\u6587_python_

**Pandas\uff1aPython\u6570\u636e\u5206\u6790\u7684\u6838\u5fc3\u5e93** Pandas\u662fPython\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u4e00\u4e2a\u5f3a\u5927\u6570\u636e\u5206\u6790\u5de5\u5177\uff0c\u5b83\u63d0\u4f9b\u4e86\u9ad8\u6548\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5982DataFrame\u548cSeries\uff0c\u4f7f\u5f97\u6570\u636e\u6e05\u6d17\u3001\u8f6c\u6362\u3001\u5206\u6790\u53d8\u5f97\u7b80\u5355\u6613\u884c\u3002Pandas\u7684\u8bbe\u8ba1\u76ee\u6807\u662f\u4f7f\u6570\u636e\u5904\u7406\u5bf9...
\"pdf\"

pandas \u624b\u518c

pandas\u662f\u4e00\u4e2a\u5f00\u6e90\u7684Python\u6570\u636e\u5206\u6790\u5e93\uff0c\u63d0\u4f9b\u4e86\u9ad8\u6027\u80fd\u3001\u6613\u7528\u7684\u6570\u636e\u7ed3\u6784\u548c\u6570\u636e\u5206\u6790\u5de5\u5177\u3002pandas\u4e2d\u7684\u57fa\u7840\u6570\u636e\u7ed3\u6784\u662f\u4e24\u79cd\u4e3b\u8981\u7684\u6570\u636e\u7ed3\u6784\uff1aSeries\u548cDataFrame\u3002Series\u662f\u4e00\u7ef4\u7684\u6807\u7b7e\u6570\u7ec4\uff0c\u53ef\u4ee5\u5305\u542b\u4efb\u4f55\u6570\u636e\u7c7b\u578b\uff08\u6574\u6570\u3001\u5b57\u7b26...
\"pdf\"

python pandas

\u6587\u6863\u8fd8\u63d0\u5230\u4e86\u4e00\u4e2a\u540d\u4e3a\u201c10 Minutes to Pandas\u201d\u7684\u6559\u7a0b\uff0c\u8fd9\u4e2a\u6559\u7a0b\u53ef\u80fd\u662f\u4e3a\u4e86\u8ba9\u521d\u5b66\u8005\u5feb\u901f\u4e86\u89e3pandas\u5e93\u7684\u57fa\u672c\u4f7f\u7528\u65b9\u6cd5\uff0c\u901a\u8fc7\u5341\u5206\u949f\u7684\u5b66\u4e60\u5feb\u901f\u638c\u63e1pandas\u5e93\u7684\u7cbe\u9ad3\u3002\u8fd9\u901a\u5e38\u4f1a\u6db5\u76d6\u6570\u636e\u521b\u5efa\u3001\u67e5\u770b\u6570\u636e\u3001\u5904\u7406\u7f3a\u5931\u6570\u636e\u3001\u6267\u884c...
\"pdf\"

pandas\u6587\u6863

Pandas\u662f\u4e00\u4e2a\u529f\u80fd\u5f3a\u5927\u7684Python\u6570\u636e\u5206\u6790\u5de5\u5177\u5305\u3002\u5b83\u4e3a\u6570\u636e\u5206\u6790\u63d0\u4f9b\u4e86\u5feb\u901f\u3001\u7075\u6d3b\u548c\u8868\u8fbe\u80fd\u529b\u5f3a\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7279\u522b\u8bbe\u8ba1\u7528\u6765\u5904\u7406\u7ed3\u6784\u5316\uff08\u8868\u683c\u3001\u591a\u7ef4\u3001\u5f02\u8d28\uff09\u548c\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u3002Pandas \u7684\u76ee\u6807\u662f\u6210\u4e3a Python \u6570\u636e\u5206\u6790\u7684\u6807\u51c6\u5e93\uff0c...
\"pdf\"

pandas\u6559\u7a0b

pandas\u662f\u4e00\u4e2a\u5f00\u6e90\u7684Python\u6570\u636e\u5206\u6790\u5e93\uff0c\u63d0\u4f9b\u4e86\u9ad8\u6027\u80fd\u3001\u6613\u4e8e\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u548c\u6570\u636e\u5206\u6790\u5de5\u5177\u3002\u672c\u6559\u7a0b\u5c06\u4ecb\u7ecd\u5982\u4f55\u4f7f\u7528pandas\u5e93\u7684\u57fa\u672c\u64cd\u4f5c\uff0c\u5305\u62ec\u521b\u5efa\u548c\u5bfc\u5165pandas\u3001\u521b\u5efaSeries\u548cDataFrame\u5bf9\u8c61\u3001\u4f7f\u7528API\u51fd\u6570\u4ee5\u53ca\u67e5\u770b\u6570\u636e\u7684...
\"xls\"

pandas\u8bfb\u53d6\u548c\u4fdd\u5b58excel\u6587\u4ef6

\u6700\u65b0\u56fd\u5bb61\u7ea72\u7ea7\u9e1f\u7c7b\u4fdd\u62a4\u540d\u5f55
\"docx\"

C\u8bed\u8a00\u57fa\u7840\u8bed\u6cd5\u4e0e\u7a0b\u5e8f\u8bbe\u8ba1\u5165\u95e8\u8bb2\u89e3

c\u8bed\u8a00\u5165\u95e8\nC\u8bed\u8a00\u4e00\u7ecf\u51fa\u73b0\u5c31\u4ee5\u5176\u529f\u80fd\u4e30\u5bcc\u3001\u8868\u8fbe\u80fd\u529b\u5f3a\u3001\u7075\u6d3b\u65b9\u4fbf\u3001\u5e94\u7528\u9762\u5e7f\u7b49\u7279\u70b9\u8fc5\u901f\u5728\u5168\u4e16\u754c\u666e\u53ca\u548c\u63a8\u5e7f\u3002C\u8bed\u8a00\u4e0d\u4f46\u6267\u884c\u6548\u7387\u9ad8\u800c\u4e14\u53ef\u79fb\u690d\u6027\u597d\uff0c\u53ef\u4ee5\u7528\u6765\u5f00\u53d1\u5e94\u7528\u8f6f\u4ef6\u3001\u9a71\u52a8\u3001\u64cd\u4f5c\u7cfb\u7edf\u7b49\u3002C\u8bed\u8a00\u4e5f\u662f\u5176\u5b83\u4f17\u591a\u9ad8\u7ea7\u8bed\u8a00\u7684\u9f3b\u7956\u8bed\u8a00\uff0c\u6240\u4ee5\u8bf4\u5b66\u4e60C\u8bed\u8a00\u662f\u8fdb\u5165\u7f16\u7a0b\u4e16\u754c\u7684\u5fc5\u4fee\u8bfe\u3002hello,world\n#include<stdio.h>\nint main()\n{\n/*\u5728\u53cc\u5f15\u53f7\u4e2d\u95f4\u8f93\u5165Hello World*/\nprintf(\"Hello World\");\nreturn 0;\n}\n\n\u6ce8\uff1a\u5728\u6700\u65b0\u7684C\u6807\u51c6\u4e2d\uff0cmain\u51fd\u6570\u524d\u7684\u7c7b\u578b\u4e3aint\u800c\u4e0d\u662fvoid\nc\u8bed\u8a00\u7684\u5177\u4f53\u7ed3\u6784\u7b80\u5355\u6765\u8bf4\uff0c\u4e00\u4e2aC\u7a0b\u5e8f\u5c31\u662f\u7531\u82e5\u5e72\u5934\u6587\u4ef6\u548c\u51fd\u6570\u7ec4\u6210\u3002#include <stdio.h>\u5c31\u662f\u4e00\u6761\u9884\u5904\u7406\u547d\u4ee4, \u5b83\u7684\u4f5c\u7528\u662f\u901a\u77e5C\u8bed\u8a00\u7f16\u8bd1\u7cfb\u7edf\u5728\u5bf9C\u7a0b\u5e8f\u8fdb\u884c\u6b63\u5f0f\u7f16\u8bd1\u4e4b\u524d\u9700\u505a\u4e00\u4e9b\u9884\u5904\u7406\u5de5\u4f5c\u3002\u51fd\u6570\u5c31\u662f\u5b9e\u73b0\u4ee3\u7801\u903b\u8f91\u7684\u4e00\u4e2a\u5c0f\u7684\u5355\u5143\u3002\u5fc5\u4e0d\u53ef\u5c11\u4e4b\u4e3b\u51fd\u6570\u4e00\u4e2aC\u7a0b\u5e8f\u6709\u4e14\u53ea\u6709\u4e00\u4e2a\u4e3b\u51fd\u6570\uff0c\u5373main\u51fd\u6570\u3002C\u7a0b\u5e8f\u5c31\u662f\u6267\u884c\u4e3b\u51fd\u6570\u91cc\u7684\u4ee3\u7801\uff0c\u4e5f\u53ef\u4ee5\u8bf4\u8fd9\u4e2a\u4e3b\u51fd\u6570\u5c31\u662fC\u8bed\u8a00\u4e2d\u7684\u552f\u4e00\u5165\u53e3\u3002\u800cmain\u524d\u9762\u7684int\u5c31\u662f\u4e3b\u51fd\u6570\u7684\u7c7b\u578b.printf()\u662f\u683c\u5f0f\u8f93\u51fa\u51fd\u6570\uff0c\u8fd9\u91cc

\u5927\u5bb6\u5728\u770b

\"recommend-type\"

SHIMAX_MAC3&MAC50\u901a\u8baf\u624b\u518c

\u65e5\u672cSHIMAX_MAC3&MAC50\u901a\u8baf\u624b\u518c
\"recommend-type\"

\u57fa\u4e8e\u7efc\u5408\u8bc4\u4ef7\u8bed\u4e49\u63cf\u8ff0\u7684\u9886\u57df\u672c\u4f53\u6784\u5efa (2013\u5e74)

\u57fa\u4e8e\u9886\u57df\u7efc\u5408\u8bc4\u4ef7\u7684\u6307\u6807\u4f53\u7cfb\u5206\u6790\u5176\u6240\u63cf\u8ff0\u7684\u8bed\u4e49\u6982\u5ff5,\u5e76\u5bf9\u6307\u6807\u8fdb\u884c\u9884\u5904\u7406,\u63d0\u53d6\u8bc4\u4ef7\u6307\u6807\u6240\u8868\u8ff0\u7684\u6982\u5ff5\u4ee5\u53ca\u6307\u6807\u95f4\u7684\u5173\u7cfb.\u6839\u636e\u8fd9\u4e9b\u6982\u5ff5\u548c\u5173\u7cfb\u6784\u5efa\u57fa\u7840\u672c\u4f53,\u901a\u8fc7\u751f\u6210\u6982\u5ff5\u683c\u5bf9\u8be5\u672c\u4f53\u7684\u6982\u5ff5\u63cf\u8ff0\u8fdb\u4e00\u6b65\u4e30\u5bcc,\u5e76\u751f\u6210\u66f4\u5b8c\u5584\u7684\u6982\u5ff5\u683c.\u6700\u540e\u5229\u7528\u5f62\u5f0f\u6982\u5ff5\u5206\u6790\u7684\u65b9\u6cd5\u548c\u5de5\u5177\u6784\u5efa\u9886\u57df\u672c\u4f53.
\"recommend-type\"

ansys workbench \u975e\u7ebf\u6027\u5206\u6790

ansys workbench \u975e\u7ebf\u6027\u6559\u7a0b
\"recommend-type\"

hw1.rar_C++\u56fe\u50cf\u63d2\u503c_\u4e8c\u7ef4\u63d2\u503c_\u4e8c\u7ef4\u63d2\u503c C++_\u56fe\u50cf_\u6700\u8fd1\u90bb\u63d2\u503c

\u56fe\u50cf\u5904\u7406\uff0c\u5bf9\u6e90\u56fe\u50cf\u8fdb\u884c\u626d\u66f2\uff0c\u91c7\u7528\u6700\u8fd1\u90bb\u3001\u4e8c\u7ef4\u63d2\u503c\u548c\u4e09\u6b21\u6837\u6761\u63d2\u503c\u3002
\"recommend-type\"

Chamber and Station test.pptx

Chamber and Station test.pptx

\u6700\u65b0\u63a8\u8350

\"recommend-type\"

pandas\u5206\u6279\u8bfb\u53d6\u5927\u6570\u636e\u96c6\u6559\u7a0b

`pandas`\u5e93\u63d0\u4f9b\u4e86\u4e00\u79cd\u7075\u6d3b\u7684\u65b9\u6cd5\uff0c\u5373\u5206\u6279\u8bfb\u53d6\u6570\u636e\u96c6\uff0c\u4ee5\u89e3\u51b3\u5185\u5b58\u9650\u5236\u7684\u95ee\u9898\u3002\u672c\u6559\u7a0b\u5c06\u8be6\u7ec6\u4ecb\u7ecd\u5982\u4f55\u5229\u7528`pandas`\u7684\u8fd9\u4e00\u529f\u80fd\u6765\u9ad8\u6548\u5730\u5904\u7406\u5927\u6570\u636e\u3002 \u5728\u5904\u7406\u5927\u578b\u6570\u636e\u96c6\u65f6\uff0c\u76f4\u63a5\u4e00\u6b21\u6027\u52a0\u8f7d\u6240\u6709\u6570\u636e\u53ef\u80fd\u5bfc\u81f4\u5185\u5b58\u6ea2\u51fa\uff0c\u7279\u522b...
\"recommend-type\"

Pandas\u8bfb\u53d6csv\u65f6\u5982\u4f55\u8bbe\u7f6e\u5217\u540d

\u5728\u6570\u636e\u5206\u6790\u9886\u57df\uff0cPandas\u5e93\u662fPython\u4e2d\u4e0d\u53ef\u6216\u7f3a\u7684\u4e00\u90e8\u5206\uff0c\u5b83\u63d0\u4f9b\u4e86\u9ad8\u6548\u7684\u6570\u636e\u5904\u7406\u80fd\u529b\u3002Pandas\u80fd\u591f\u65b9\u4fbf\u5730\u8bfb\u53d6\u548c\u5199\u5165\u591a\u79cd\u683c\u5f0f\u7684\u6570\u636e\u6587\u4ef6\uff0c\u5176\u4e2dCSV\uff08Comma Separated Values\uff09\u662f\u6700\u5e38\u89c1\u7684\u6570\u636e\u5b58\u50a8\u683c\u5f0f\u4e4b\u4e00\u3002\u672c\u7bc7\u5c06\u8be6\u7ec6...
\"recommend-type\"

pandas\u4e2dTimestamp\u7c7b\u7528\u6cd5\u8be6\u89e3

\u5728Python\u7684pandas\u5e93\u4e2d\uff0cTimestamp\u662f\u4e00\u4e2a\u975e\u5e38\u91cd\u8981\u7684\u7c7b\uff0c\u5b83\u7528\u4e8e\u8868\u793a\u7279\u5b9a\u7684\u65e5\u671f\u548c\u65f6\u95f4\u70b9\u3002Timestamp\u7c7b\u63d0\u4f9b\u4e86\u4e30\u5bcc\u7684\u529f\u80fd\uff0c\u53ef\u4ee5\u65b9\u4fbf\u5730\u5904\u7406\u65e5\u671f\u548c\u65f6\u95f4\u6570\u636e\uff0c\u662f\u6570\u636e\u5206\u6790\u4e2d\u4e0d\u53ef\u6216\u7f3a\u7684\u4e00\u90e8\u5206\u3002\u4ee5\u4e0b\u662f\u5bf9pandas\u4e2dTimestamp\u7c7b\u7684...
\"recommend-type\"

python pandas\u751f\u6210\u65f6\u95f4\u5217\u8868

\u5728Python\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0cPandas\u5e93\u662f\u4e00\u4e2a\u5f3a\u5927\u7684\u6570\u636e\u5206\u6790\u5de5\u5177\uff0c\u5b83\u63d0\u4f9b\u4e86\u4e30\u5bcc\u7684\u6570\u636e\u5904\u7406\u529f\u80fd\uff0c\u5305\u62ec\u751f\u6210\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u3002\u5728\u5904\u7406\u6d89\u53ca\u65f6\u95f4\u7684\u6570\u636e\u65f6\uff0cPandas\u7684\u65f6\u95f4\u5217\u8868\uff08\u65f6\u95f4\u5e8f\u5217\uff09\u626e\u6f14\u7740\u81f3\u5173\u91cd\u8981\u7684\u89d2\u8272\u3002\u672c\u7bc7\u5c06\u6df1\u5165\u63a2\u8ba8\u5982\u4f55\u4f7f\u7528...
\"recommend-type\"

pandas\u548cspark dataframe\u4e92\u76f8\u8f6c\u6362\u5b9e\u4f8b\u8be6\u89e3

\u5728\u5927\u6570\u636e\u5904\u7406\u9886\u57df\uff0c`pandas` \u548c `Spark DataFrame` \u662f\u4e24\u4e2a\u91cd\u8981\u7684\u5de5\u5177\u3002`pandas` \u662f Python \u4e2d\u7528\u4e8e\u6570\u636e\u5904\u7406\u548c\u5206\u6790\u7684\u5e93\uff0c\u800c `Spark DataFrame` \u662f Apache Spark \u7684\u6838\u5fc3\u7ec4\u4ef6\uff0c\u63d0\u4f9b\u4e86\u4e00\u79cd\u5206\u5e03\u5f0f\u6570\u636e\u5904\u7406\u80fd\u529b\u3002\u672c\u6587\u5c06\u8be6\u7ec6...
\"recommend-type\"

\u6df1\u5165\u63a2\u7d22CSS\u62c9\u7279\u6d4b\u8bd5\u65b9\u6cd5

\u6839\u636e\u63d0\u4f9b\u7684\u6587\u4ef6\u4fe1\u606f\uff0c\u6211\u4eec\u65e0\u6cd5\u83b7\u53d6\u5177\u4f53\u7684\u6587\u4ef6\u5185\u5bb9\uff0c\u56e0\u6b64\uff0c\u9700\u8981\u4ece\u6587\u4ef6\u7684\u6807\u9898\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\uff0c\u63cf\u8ff0\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\uff0c\u6807\u7b7e\u201cCSS\u201d\uff0c\u4ee5\u53ca\u538b\u7f29\u5305\u5b50\u6587\u4ef6\u7684\u6587\u4ef6\u540d\u79f0\u5217\u8868\u201clat-test-main\u201d\u6765\u63a8\u65ad\u76f8\u5173\u7684\u77e5\u8bc6\u70b9\u3002\u9274\u4e8e\u8fd9\u4e9b\u4fe1\u606f\u91cf\u6709\u9650\uff0c\u6211\u4eec\u5c06\u4e3b\u8981\u56f4\u7ed5\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u8fd9\u4e00\u4e3b\u9898\u8fdb\u884c\u63a2\u8ba8\uff0c\u540c\u65f6\u4e5f\u4f1a\u6d89\u53caCSS\u76f8\u5173\u5185\u5bb9\u3002\n\n\u9996\u5148\uff0c\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u53ef\u80fd\u6307\u7684\u662f\u67d0\u79cd\u7279\u5b9a\u7684\u8f6f\u4ef6\u6d4b\u8bd5\u65b9\u6cd5\u6216\u8005\u6280\u672f\u8bc4\u4f30\u6d41\u7a0b\u3002\u8003\u8651\u5230\u6587\u4ef6\u540d\u201clat-test-main\u201d\u6697\u793a\u5b83\u53ef\u80fd\u662f\u67d0\u4e2a\u9879\u76ee\u7684\u4e3b\u8981\u6d4b\u8bd5\u6587\u4ef6\uff0c\u6211\u4eec\u53ef\u4ee5\u5408\u7406\u63a8\u6d4b\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u53ef\u80fd\u662f\u6d4b\u8bd5\u7684\u4ee3\u7801\u811a\u672c\u3001\u6d4b\u8bd5\u7528\u4f8b\u96c6\u5408\u3001\u6216\u8005\u662f\u4e0e\u6d4b\u8bd5\u76f8\u5173\u7684\u914d\u7f6e\u6587\u4ef6\u3002\u4f46\u5728\u6ca1\u6709\u66f4\u591a\u4e0a\u4e0b\u6587\u7684\u60c5\u51b5\u4e0b\uff0c\u5f88\u96be\u786e\u5b9a\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u5177\u4f53\u6307\u4ee3\u7684\u662f\u4ec0\u4e48\u3002\n\n\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u8ba8\u8bba\u201cCSS\u201d\u3002CSS\u662f\u201c\u5c42\u53e0\u6837\u5f0f\u8868\uff08Cascading Style Sheets\uff09\u201d\u7684\u7f29\u5199\uff0c\u662f\u4e00\u79cd\u7528\u4e8e\u63a7\u5236\u7f51\u9875\u5916\u89c2\u548c\u5e03\u5c40\u7684\u6280\u672f\u6807\u51c6\u3002CSS\u63cf\u8ff0\u4e86\u5982\u4f55\u5728\u5c4f\u5e55\u4e0a\uff0c\u7eb8\u5f20\u4e0a\uff0c\u6216\u5728\u5176\u4ed6\u5a92\u4f53\u4e0a\u5c55\u73b0HTML\u6216XML\uff08\u5305\u62ec\u5404\u79cdXML\u65b9\u8a00\uff0c\u6bd4\u5982SVG\u6216XHTML\uff09\u6587\u6863\u3002\u5b83\u4f7f\u5f00\u53d1\u8005\u80fd\u591f\u5c06\u5185\u5bb9\u4e0e\u8868\u73b0\u5206\u79bb\uff0c\u8fd9\u6709\u52a9\u4e8e\u5bf9\u7f51\u7ad9\u8fdb\u884c\u4fee\u6539\uff0c\u800c\u65e0\u9700\u89e6\u53ca\u5185\u5bb9\u672c\u8eab\u3002CSS\u7684\u89c4\u5219\u7531\u9009\u62e9\u5668\u548c\u58f0\u660e\u5757\u7ec4\u6210\u3002\u9009\u62e9\u5668\u6307\u660e\u4e86\u6837\u5f0f\u89c4\u5219\u5e94\u8be5\u5e94\u7528\u4e8e\u54ea\u4e9bHTML\u5143\u7d20\uff0c\u800c\u58f0\u660e\u5757\u5219\u5305\u542b\u4e86\u4e00\u4e2a\u6216\u591a\u4e2a\u7528\u5206\u53f7\u9694\u5f00\u7684\u5c5e\u6027\u503c\u5bf9\u3002\n\n\u7136\u800c\uff0c\u7531\u4e8e\u6807\u9898\u3001\u63cf\u8ff0\u548c\u6807\u7b7e\u5e76\u6ca1\u6709\u76f4\u63a5\u63d0\u4f9b\u5173\u4e8eCSS\u7684\u5177\u4f53\u4fe1\u606f\uff0c\u6211\u4eec\u4e5f\u65e0\u6cd5\u786e\u5b9aCSS\u5728\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u4e2d\u626e\u6f14\u7684\u5177\u4f53\u89d2\u8272\u3002\u4e0d\u8fc7\uff0c\u5047\u8bbeCSS\u6807\u7b7e\u610f\u5473\u7740\u6d4b\u8bd5\u53ef\u80fd\u4e0e\u7f51\u9875\u7684\u6837\u5f0f\u8868\u6216\u8005\u524d\u7aef\u8bbe\u8ba1\u6709\u5173\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u60f3\u8c61\uff0c\u6d4b\u8bd5\u53ef\u80fd\u6db5\u76d6\u4e86\u5bf9\u7f51\u9875\u6837\u5f0f\u7684\u9a8c\u8bc1\u3001\u5bf9\u5e03\u5c40\u7684\u6d4b\u8bd5\u3001\u5bf9\u4ea4\u4e92\u6548\u679c\u7684\u68c0\u67e5\u7b49\u3002\n\n\u5728\u5f00\u53d1\u548c\u6d4b\u8bd5\u8fc7\u7a0b\u4e2d\uff0cCSS\u7684\u6b63\u786e\u6027\u81f3\u5173\u91cd\u8981\u3002\u4ee5\u4e0b\u662f\u4e00\u4e9b\u4e0eCSS\u76f8\u5173\u7684\u6d4b\u8bd5\u65b9\u6cd5\uff1a\n\n1. CSS\u9a8c\u8bc1\u6d4b\u8bd5\uff1a\u786e\u4fddCSS\u4ee3\u7801\u7b26\u5408\u6807\u51c6\uff0c\u5e76\u4e14\u6ca1\u6709\u8bed\u6cd5\u9519\u8bef\u3002\u53ef\u4ee5\u4f7f\u7528\u5728\u7ebf\u5de5\u5177\u5982W3C\u7684CSS\u9a8c\u8bc1\u670d\u52a1\u8fdb\u884c\u3002\n\n2. \u517c\u5bb9\u6027\u6d4b\u8bd5\uff1a\u68c0\u67e5\u7f51\u7ad9\u5728\u4e0d\u540c\u7684\u6d4f\u89c8\u5668\u548c\u8bbe\u5907\u4e0a\u663e\u793a\u7684\u4e00\u81f4\u6027\u3002\u7531\u4e8e\u6d4f\u89c8\u5668\u5bf9CSS\u7684\u652f\u6301\u5b58\u5728\u5dee\u5f02\uff0c\u8fd9\u4e00\u6b65\u9aa4\u5341\u5206\u91cd\u8981\u3002\n\n3. \u6027\u80fd\u6d4b\u8bd5\uff1a\u5206\u6790CSS\u6587\u4ef6\u7684\u5927\u5c0f\u3001\u590d\u6742\u5ea6\u4ee5\u53ca\u4e0b\u8f7d\u548c\u6e32\u67d3\u65f6\u95f4\uff0c\u4f18\u5316\u8fd9\u4e9b\u6027\u80fd\u6307\u6807\u4ee5\u63d0\u9ad8\u7f51\u9875\u52a0\u8f7d\u901f\u5ea6\u3002\n\n4. \u53ef\u8bbf\u95ee\u6027\u6d4b\u8bd5\uff1a\u786e\u4fdd\u7f51\u7ad9\u5bf9\u4e0d\u540c\u9700\u6c42\u7684\u7528\u6237\uff0c\u5305\u62ec\u6709\u89c6\u89c9\u969c\u788d\u7684\u7528\u6237\uff0c\u662f\u6613\u4e8e\u5bfc\u822a\u548c\u4f7f\u7528\u7684\u3002\n\n5. \u5355\u5143\u6d4b\u8bd5\uff1a\u5bf9\u4e8e\u4f7f\u7528CSS\u9884\u5904\u7406\u5668\u6216\u7f16\u8bd1\u5de5\u5177\u751f\u6210\u6700\u7ec8\u6837\u5f0f\u8868\u7684\u60c5\u51b5\uff0c\u5355\u5143\u6d4b\u8bd5\u53ef\u4ee5\u786e\u4fdd\u8fd9\u4e9b\u5de5\u5177\u7684\u6b63\u786e\u6027\u3002\n\n6. \u529f\u80fd\u6d4b\u8bd5\uff1a\u68c0\u67e5\u7f51\u9875\u4e0a\u7684\u6837\u5f0f\u5143\u7d20\u662f\u5426\u6309\u7167\u8bbe\u8ba1\u5b9e\u73b0\uff0c\u6bd4\u5982\u5b57\u4f53\u3001\u989c\u8272\u3001\u5e03\u5c40\u548c\u5176\u4ed6\u89c6\u89c9\u6548\u679c\u3002\n\n\u7531\u4e8e\u201clat-test-main\u201d\u6697\u793a\u8fd9\u662f\u4e00\u4e2a\u4e3b\u8981\u7684\u6d4b\u8bd5\u6587\u4ef6\uff0c\u5b83\u53ef\u80fd\u5305\u542b\u4e86\u4e0a\u8ff0\u6d4b\u8bd5\u65b9\u6cd5\u4e2d\u7684\u4e00\u79cd\u6216\u591a\u79cd\u7684\u5b9e\u73b0\u3002\u5728\u5b9e\u9645\u5f00\u53d1\u8fc7\u7a0b\u4e2d\uff0c\u6d4b\u8bd5\u901a\u5e38\u662f\u5728\u7248\u672c\u63a7\u5236\u7cfb\u7edf\u7684\u652f\u6301\u4e0b\u8fdb\u884c\u7684\uff0c\u6bd4\u5982Git\uff0c\u5b83\u53ef\u4ee5\u5e2e\u52a9\u56e2\u961f\u6210\u5458\u7ba1\u7406\u4e0d\u540c\u7684\u6d4b\u8bd5\u7248\u672c\uff0c\u5e76\u8ddf\u8e2a\u4ee3\u7801\u66f4\u6539\u3002\n\n\u7efc\u4e0a\u6240\u8ff0\uff0c\u5173\u4e8e\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u548c\u201cCSS\u201d\u7684\u77e5\u8bc6\u70b9\u96c6\u4e2d\u5728\u6d4b\u8bd5\u65b9\u6cd5\u548c\u6837\u5f0f\u8868\u7684\u5e94\u7528\u4e0a\u3002\u4e0d\u8fc7\uff0c\u4e3a\u4e86\u66f4\u51c6\u786e\u5730\u63cf\u8ff0\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u7684\u542b\u4e49\uff0c\u6211\u4eec\u9700\u8981\u66f4\u591a\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u6216\u8005\u76f4\u63a5\u8bbf\u95ee\u76f8\u5173\u7684\u6587\u4ef6\u5185\u5bb9\u3002\u5728\u5b9e\u9645\u5de5\u4f5c\u4e2d\uff0c\u4e86\u89e3\u9879\u76ee\u9700\u6c42\u3001\u6d4b\u8bd5\u76ee\u6807\u548c\u73af\u5883\u914d\u7f6e\u5bf9\u4e8e\u6210\u529f\u5730\u5b9e\u65bd\u6d4b\u8bd5\u8ba1\u5212\u81f3\u5173\u91cd\u8981\u3002
\"recommend-type\"

\u65b0\u5510IAP\u6982\u5ff5\u89e3\u6790

\n\n# \u6458\u8981\nIAP\uff08In-Application Programming\uff09\u7f16\u7a0b\u662f\u4e00\u79cd\u5728\u5e94\u7528\u8fd0\u884c\u65f6\u66f4\u65b0\u56fa\u4ef6\u7684\u5148\u8fdb\u65b9\u6cd5\uff0c\u5b83\u63d0\u4f9b\u4e86\u7cfb\u7edf\u66f4\u65b0\u7684\u7075\u6d3b\u6027\u548c\u4fbf\u5229\u6027\u3002\u672c\u6587\u5168\u9762\u4ecb\u7ecd\u4e86IAP\u7f16\u7a0b\u7684\u6982\u5ff5\u3001\u6280\u672f\u57fa\u7840\u548c\u5b9e\u8df5\u5e94\u7528\uff0c\u91cd\u70b9\u5206\u6790\u4e86IAP\u5728\u65b0\u5510\u5fae\u63a7\u5236\u5668\u4e2d\u7684\u5b9e\u73b0\u673a\u5236\uff0c\u5305\u62ec\u5176\u5185\u5b58\u7ed3\u6784\u548c\u5de5\u4f5c\u6d41\u7a0b\uff0c\u5e76\u63a2\u8ba8\u4e86\u8f6f\u4ef6\u5de5\u5177\u548c\u5f00\u53d1\u73af\u5883\u7684\u914d\u7f6e\u3002\u540c\u65f6\uff0c\u672c\u6587\u901a\u8fc7\u5b9e\u9645\u6848\u4f8b\u6df1\u5165\u7814\u7a76\u4e86IAP\u5f00\u53d1\u6d41\u7a0b\u3001\u5b89\u5168\u6027\u548c\u9519\u8bef\u5904\u7406\u7b56\u7565\uff0c\u4ee5\u53ca\u5728\u7269\u8054\u7f51\u8bbe\u5907\u548c\u667a\u80fd\u5bb6\u5c45\u7b49\u9886\u57df\u7684\u9ad8\u7ea7\u5e94\u7528\u3002\u6700\u540e\uff0c\u9488\u5bf9IAP\u9879\u76ee\u7684\u7ba1
\"recommend-type\"

fix_eco_timing \u5199\u51fa\u811a\u672c

`fix_eco_timing`\u8fd9\u4e2a\u540d\u5b57\u770b\u8d77\u6765\u50cf\u662f\u7528\u4e8e\u67d0\u79cd\u7279\u5b9a\u73af\u5883\u4e0b\u7684\u811a\u672c\uff0c\u6bd4\u5982\u53ef\u80fd\u662f\u7528\u4e8e\u8c03\u6574\u7535\u5b50\u7ec4\u4ef6\u6216\u7535\u5b50\u4ea7\u54c1\u7684\u5de5\u4f5c\u5468\u671f\u4f18\u5316\u80fd\u6e90\u6548\u7387\u7684\u4e00\u79cd\u5de5\u5177\u3002\u7136\u800c\uff0c\u6ca1\u6709\u5177\u4f53\u7684\u4e0a\u4e0b\u6587\uff0c\u5f88\u96be\u63d0\u4f9b\u8be6\u7ec6\u7684\u811a\u672c\u5185\u5bb9\u3002\u901a\u5e38\u8fd9\u6837\u7684\u811a\u672c\u53ef\u80fd\u4f1a\u5305\u542b\u4ee5\u4e0b\u51e0\u4e2a\u90e8\u5206\uff1a\n\n```bash\n#!/bin/bash\n# Fix Eco Timing Script\n\n# 1. \u83b7\u53d6\u5f53\u524d\u8bbe\u5907\u72b6\u6001\ndevice_status=$(get_device_status)\n\n# 2. \u68c0\u67e5\u662f\u5426\u8fbe\u5230\u8282\u80fd\u6a21\u5f0f\u6761\u4ef6\nif [ "$device_status" == "idle" ]; then\n# 3. \u8c03\u6574\u5de5\u4f5c\u9891\u7387\u6216\u7535\u6e90\u7ba1\u7406\u8bbe\u7f6e\nad
\"recommend-type\"

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\"recommend-type\"

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\n \n\n \n \n"} +{"content_list": [[{"type": "title", "raw_content": "

pandas \u5206\u6279\u8bfb\u5165execl

", "content": {"title_content": "pandas \u5206\u6279\u8bfb\u5165execl", "level": "1"}}, {"type": "paragraph", "raw_content": "
\u65f6\u95f4: 2023-06-29 20:05:48
", "content": [{"c": "\u65f6\u95f4: 2023-06-29 20:05:48", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\u6d4f\u89c8: 99
", "content": [{"c": "\u6d4f\u89c8: 99", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\u4f60\u53ef\u4ee5\u4f7f\u7528 [pandas](https://wenku.csdn.net/doc/6412b725be7fbd1778d4940f?spm=1055.2569.3001.10083&kwd=pandas) \u7684 `read_[excel](https://wenku.csdn.net/doc/640n8f90vm?spm=1055.2569.3001.10083&kwd=excel)()` \u65b9\u6cd5\uff0c\u5e76\u7ed3\u5408 c h u n k s i z e \u53c2\u6570\u6765\u5206\u6279\u8bfb\u5165 Excel \u6587\u4ef6\u3002c h u n k s i z e \u53c2\u6570\u6307\u5b9a\u4e86\u6bcf\u6b21\u8bfb\u5165\u7684\u884c\u6570\uff0c\u53ef\u4ee5\u6839\u636e\u4f60\u7684\u9700\u6c42\u8fdb\u884c\u8c03\u6574\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff1a\n\n```[python](https://wenku.csdn.net/doc/6412b46ebe7fbd1778d3f92a?spm=1055.2569.3001.10083&kwd=python)\n[import](https://wenku.csdn.net/doc/5cpc346q6x?spm=1055.2569.3001.10083&kwd=import) pandas as pd\n\n# \u6bcf\u6b21\u8bfb\u5165 1000 \u884c\u6570\u636e\nchunksize = 1000\n# Excel \u6587\u4ef6\u8def\u5f84\nexcel_path = 'example.xlsx'\n\n# \u904d\u5386\u6bcf\u4e2a\u5206\u5757\u6570\u636e\nfor chunk in pd.read_excel(excel_path, chunksize=chunksize):\n# \u5904\u7406\u5f53\u524d\u5206\u5757\u6570\u636e\n[print](https://wenku.csdn.net/doc/6401acb5cce7214c316ecd5e?spm=1055.2569.3001.10083&kwd=print)(chunk.head())\n```\n\n\u4e0a\u8ff0\u4ee3\u7801\u5c06 Excel \u6587\u4ef6\u6309\u7167\u6bcf\u6b21\u8bfb\u53d6 1000 \u884c\u6570\u636e\u8fdb\u884c\u5206\u5757\uff0c\u7136\u540e\u904d\u5386\u6bcf\u4e2a\u5206\u5757\u5e76\u5bf9\u5176\u8fdb\u884c\u5904\u7406\u3002\u4f60\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4fee\u6539 c h u n k s i z e \u53c2\u6570\u7684\u503c\u3002
", "content": [{"c": "\u4f60\u53ef\u4ee5\u4f7f\u7528 [pandas](https://wenku.csdn.net/doc/6412b725be7fbd1778d4940f?spm=1055.2569.3001.10083&kwd=pandas) \u7684 `read_[excel](https://wenku.csdn.net/doc/640n8f90vm?spm=1055.2569.3001.10083&kwd=excel)()` \u65b9\u6cd5\uff0c\u5e76\u7ed3\u5408", "t": "text"}, {"c": "c h u n k s i z e", "t": "equation-inline"}, {"c": "\u53c2\u6570\u6765\u5206\u6279\u8bfb\u5165 Excel \u6587\u4ef6\u3002", "t": "text"}, {"c": "c h u n k s i z e", "t": "equation-inline"}, {"c": "\u53c2\u6570\u6307\u5b9a\u4e86\u6bcf\u6b21\u8bfb\u5165\u7684\u884c\u6570\uff0c\u53ef\u4ee5\u6839\u636e\u4f60\u7684\u9700\u6c42\u8fdb\u884c\u8c03\u6574\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff1a\n\n```[python](https://wenku.csdn.net/doc/6412b46ebe7fbd1778d3f92a?spm=1055.2569.3001.10083&kwd=python)\n[import](https://wenku.csdn.net/doc/5cpc346q6x?spm=1055.2569.3001.10083&kwd=import) pandas as pd\n\n# \u6bcf\u6b21\u8bfb\u5165 1000 \u884c\u6570\u636e\nchunksize = 1000\n# Excel \u6587\u4ef6\u8def\u5f84\nexcel_path = 'example.xlsx'\n\n# \u904d\u5386\u6bcf\u4e2a\u5206\u5757\u6570\u636e\nfor chunk in pd.read_excel(excel_path, chunksize=chunksize):\n# \u5904\u7406\u5f53\u524d\u5206\u5757\u6570\u636e\n[print](https://wenku.csdn.net/doc/6401acb5cce7214c316ecd5e?spm=1055.2569.3001.10083&kwd=print)(chunk.head())\n```\n\n\u4e0a\u8ff0\u4ee3\u7801\u5c06 Excel \u6587\u4ef6\u6309\u7167\u6bcf\u6b21\u8bfb\u53d6 1000 \u884c\u6570\u636e\u8fdb\u884c\u5206\u5757\uff0c\u7136\u540e\u904d\u5386\u6bcf\u4e2a\u5206\u5757\u5e76\u5bf9\u5176\u8fdb\u884c\u5904\u7406\u3002\u4f60\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4fee\u6539", "t": "text"}, {"c": "c h u n k s i z e", "t": "equation-inline"}, {"c": "\u53c2\u6570\u7684\u503c\u3002", "t": "text"}]}, {"type": "paragraph", "raw_content": "
\u9605\u8bfb\u5168\u6587
", "content": [{"c": "\u9605\u8bfb\u5168\u6587", "t": "text"}]}, {"type": "title", "raw_content": "

\u76f8\u5173\u63a8\u8350

", "content": {"title_content": "\u76f8\u5173\u63a8\u8350", "level": "2"}}, {"type": "title", "raw_content": "

\u5927\u5bb6\u5728\u770b

", "content": {"title_content": "\u5927\u5bb6\u5728\u770b", "level": "2"}}, {"type": "title", "raw_content": "

\u6700\u65b0\u63a8\u8350

", "content": {"title_content": "\u6700\u65b0\u63a8\u8350", "level": "2"}}]], "main_html": "

pandas \u5206\u6279\u8bfb\u5165execl

\u65f6\u95f4: 2023-06-29 20:05:48
\u6d4f\u89c8: 99
\u4f60\u53ef\u4ee5\u4f7f\u7528 [pandas](https://wenku.csdn.net/doc/6412b725be7fbd1778d4940f?spm=1055.2569.3001.10083&kwd=pandas) \u7684 `read_[excel](https://wenku.csdn.net/doc/640n8f90vm?spm=1055.2569.3001.10083&kwd=excel)()` \u65b9\u6cd5\uff0c\u5e76\u7ed3\u5408 c h u n k s i z e \u53c2\u6570\u6765\u5206\u6279\u8bfb\u5165 Excel \u6587\u4ef6\u3002c h u n k s i z e \u53c2\u6570\u6307\u5b9a\u4e86\u6bcf\u6b21\u8bfb\u5165\u7684\u884c\u6570\uff0c\u53ef\u4ee5\u6839\u636e\u4f60\u7684\u9700\u6c42\u8fdb\u884c\u8c03\u6574\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff1a\n\n```[python](https://wenku.csdn.net/doc/6412b46ebe7fbd1778d3f92a?spm=1055.2569.3001.10083&kwd=python)\n[import](https://wenku.csdn.net/doc/5cpc346q6x?spm=1055.2569.3001.10083&kwd=import) pandas as pd\n\n# \u6bcf\u6b21\u8bfb\u5165 1000 \u884c\u6570\u636e\nchunksize = 1000\n# Excel \u6587\u4ef6\u8def\u5f84\nexcel_path = 'example.xlsx'\n\n# \u904d\u5386\u6bcf\u4e2a\u5206\u5757\u6570\u636e\nfor chunk in pd.read_excel(excel_path, chunksize=chunksize):\n# \u5904\u7406\u5f53\u524d\u5206\u5757\u6570\u636e\n[print](https://wenku.csdn.net/doc/6401acb5cce7214c316ecd5e?spm=1055.2569.3001.10083&kwd=print)(chunk.head())\n```\n\n\u4e0a\u8ff0\u4ee3\u7801\u5c06 Excel \u6587\u4ef6\u6309\u7167\u6bcf\u6b21\u8bfb\u53d6 1000 \u884c\u6570\u636e\u8fdb\u884c\u5206\u5757\uff0c\u7136\u540e\u904d\u5386\u6bcf\u4e2a\u5206\u5757\u5e76\u5bf9\u5176\u8fdb\u884c\u5904\u7406\u3002\u4f60\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4fee\u6539 c h u n k s i z e \u53c2\u6570\u7684\u503c\u3002
\u9605\u8bfb\u5168\u6587

\u76f8\u5173\u63a8\u8350

\u5927\u5bb6\u5728\u770b

\u6700\u65b0\u63a8\u8350

", "statics": {"title": 4, "paragraph": 4, "paragraph.text": 7, "paragraph.equation-inline": 3}, "url": "https://blog.csdn.net/moon_sky1999/article/details/52744845", "content": "# pandas \u5206\u6279\u8bfb\u5165execl\n\n\u65f6\u95f4: 2023-06-29 20:05:48\n\n\u6d4f\u89c8: 99\n\n\u4f60\u53ef\u4ee5\u4f7f\u7528 [pandas](https://wenku.csdn.net/doc/6412b725be7fbd1778d4940f?spm=1055.2569.3001.10083&kwd=pandas) \u7684 \\`read_[excel](https://wenku.csdn.net/doc/640n8f90vm?spm=1055.2569.3001.10083&kwd=excel)()\\` \u65b9\u6cd5\uff0c\u5e76\u7ed3\u5408 $c h u n k s i z e$ \u53c2\u6570\u6765\u5206\u6279\u8bfb\u5165 Excel \u6587\u4ef6\u3002 $c h u n k s i z e$ \u53c2\u6570\u6307\u5b9a\u4e86\u6bcf\u6b21\u8bfb\u5165\u7684\u884c\u6570\uff0c\u53ef\u4ee5\u6839\u636e\u4f60\u7684\u9700\u6c42\u8fdb\u884c\u8c03\u6574\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff1a\n\n\\`\\`\\`[python](https://wenku.csdn.net/doc/6412b46ebe7fbd1778d3f92a?spm=1055.2569.3001.10083&kwd=python)\n[import](https://wenku.csdn.net/doc/5cpc346q6x?spm=1055.2569.3001.10083&kwd=import) pandas as pd\n\n\\# \u6bcf\u6b21\u8bfb\u5165 1000 \u884c\u6570\u636e\nchunksize = 1000\n\\# Excel \u6587\u4ef6\u8def\u5f84\nexcel_path = 'example.xlsx'\n\n\\# \u904d\u5386\u6bcf\u4e2a\u5206\u5757\u6570\u636e\nfor chunk in pd.read_excel(excel_path, chunksize=chunksize):\n\\# \u5904\u7406\u5f53\u524d\u5206\u5757\u6570\u636e\n[print](https://wenku.csdn.net/doc/6401acb5cce7214c316ecd5e?spm=1055.2569.3001.10083&kwd=print)(chunk.head())\n\\`\\`\\`\n\n\u4e0a\u8ff0\u4ee3\u7801\u5c06 Excel \u6587\u4ef6\u6309\u7167\u6bcf\u6b21\u8bfb\u53d6 1000 \u884c\u6570\u636e\u8fdb\u884c\u5206\u5757\uff0c\u7136\u540e\u904d\u5386\u6bcf\u4e2a\u5206\u5757\u5e76\u5bf9\u5176\u8fdb\u884c\u5904\u7406\u3002\u4f60\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4fee\u6539 $c h u n k s i z e$ \u53c2\u6570\u7684\u503c\u3002\n\n\u9605\u8bfb\u5168\u6587\n\n## \u76f8\u5173\u63a8\u8350\n\n## \u5927\u5bb6\u5728\u770b\n\n## \u6700\u65b0\u63a8\u8350\n", "html": "\npandas \u5206\u6279\u8bfb\u5165execl - CSDN\u6587\u5e93 \n \n \n \n\n \n \n \n \n\n \n\n \n\n \n\n \n
\u9996\u9875pandas \u5206\u6279\u8bfb\u5165execl

pandas \u5206\u6279\u8bfb\u5165execl

\u65f6\u95f4: 2023-06-29 20:05:48 \u6d4f\u89c8: 99
\u4f60\u53ef\u4ee5\u4f7f\u7528 [pandas](https://wenku.csdn.net/doc/6412b725be7fbd1778d4940f?spm=1055.2569.3001.10083&kwd=pandas) \u7684 `read_[excel](https://wenku.csdn.net/doc/640n8f90vm?spm=1055.2569.3001.10083&kwd=excel)()` \u65b9\u6cd5\uff0c\u5e76\u7ed3\u5408 `chunksize` \u53c2\u6570\u6765\u5206\u6279\u8bfb\u5165 Excel \u6587\u4ef6\u3002`chunksize` \u53c2\u6570\u6307\u5b9a\u4e86\u6bcf\u6b21\u8bfb\u5165\u7684\u884c\u6570\uff0c\u53ef\u4ee5\u6839\u636e\u4f60\u7684\u9700\u6c42\u8fdb\u884c\u8c03\u6574\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u793a\u4f8b\u4ee3\u7801\uff1a\n\n```[python](https://wenku.csdn.net/doc/6412b46ebe7fbd1778d3f92a?spm=1055.2569.3001.10083&kwd=python)\n[import](https://wenku.csdn.net/doc/5cpc346q6x?spm=1055.2569.3001.10083&kwd=import) pandas as pd\n\n# \u6bcf\u6b21\u8bfb\u5165 1000 \u884c\u6570\u636e\nchunksize = 1000\n# Excel \u6587\u4ef6\u8def\u5f84\nexcel_path = 'example.xlsx'\n\n# \u904d\u5386\u6bcf\u4e2a\u5206\u5757\u6570\u636e\nfor chunk in pd.read_excel(excel_path, chunksize=chunksize):\n# \u5904\u7406\u5f53\u524d\u5206\u5757\u6570\u636e\n[print](https://wenku.csdn.net/doc/6401acb5cce7214c316ecd5e?spm=1055.2569.3001.10083&kwd=print)(chunk.head())\n```\n\n\u4e0a\u8ff0\u4ee3\u7801\u5c06 Excel \u6587\u4ef6\u6309\u7167\u6bcf\u6b21\u8bfb\u53d6 1000 \u884c\u6570\u636e\u8fdb\u884c\u5206\u5757\uff0c\u7136\u540e\u904d\u5386\u6bcf\u4e2a\u5206\u5757\u5e76\u5bf9\u5176\u8fdb\u884c\u5904\u7406\u3002\u4f60\u53ef\u4ee5\u6839\u636e\u9700\u8981\u4fee\u6539 `chunksize` \u53c2\u6570\u7684\u503c\u3002
\u9605\u8bfb\u5168\u6587

\u76f8\u5173\u63a8\u8350

\"pdf\"

\u4f7f\u7528pandas\u8fdb\u884cexcel\u6587\u4ef6\u7684\u8bfb\u53d6\u5199\u5165

pandas\u7684DateFrame\u7c7b\u4f3c\u4e8eexcel\u7684\u6570\u636e\u50a8\u5b58\u7ed3\u6784\uff0c\u6545\u4e3a\u4e86\u4fbf\u4e8epandas\u8fdb\u884c\u6570\u636e\u5904\u7406\uff0c\u8bbe\u8ba1\u8005\u63d0\u4f9b\u4e86\u7b80\u4fbf\u7684\u65b9\u6cd5\u8fdb\u884c\u6570\u636e\u7684\u8bfb\u53d6\uff0c\u4e0b\u9762\u5199\u4e0a\u4ee3\u7801\nimport pandas as pd\nimport numpy as np\ndef save_excel(arr, exc_filename=None): #\u8bfb\u53d6excel\u7684\u51fd\u6570\nSave the data in excel format\nif exc_filename == None:\n exc_filename = excel.xlsx\narr_df = pd.DataFr

pandas\u8bfb\u53d6excel\u6587\u4ef6

pandas\u8bfb\u53d6excel\u6587\u4ef6\uff0c\u5229\u7528pandas\u8bfb\u53d6Excel\u6570\u636e\uff0c\u7136\u540e\u5229\u7528Pycharts\u8fdb\u884c\u53ef\u89c6\u5316\u5c55\u793a\uff0c\u5206\u6790\u4e0d\u540c\u7ad9\u70b9\u4e0d\u540c\u65f6\u95f4\u7684aqi\u5386\u53f2\u6570\u636e\uff0c\u5e76\u8fdb\u884c\u5dee\u503c\u8fd0\u7b97\uff0c\u5206\u6790\u5dee\u503c\u7684\u8d8b\u52bf
\"md\"

python pandas \u8bfb\u53d6excel\u6570\u636e

python\u8bfb\u53d6excel\u6570\u636e\n\n\u8981\u5728Python\u4e2d\u8bfb\u53d6Excel\u6570\u636e\uff0c\u4f60\u53ef\u4ee5\u4f7f\u7528pandas\u5e93
\"pdf\"

pandas\u5206\u6279\u8bfb\u53d6\u5927\u6570\u636e\u96c6\u6559\u7a0b

\u5728\u5904\u7406\u5927\u6570\u636e\u96c6\u65f6\uff0cPandas \u63d0\u4f9b\u4e86\u4e00\u79cd\u9ad8\u6548\u7684\u65b9\u6cd5\uff0c\u5373\u5206\u6279\u8bfb\u53d6\uff08chunking\uff09\uff0c\u6765\u89e3\u51b3\u5185\u5b58\u9650\u5236\u7684\u95ee\u9898\u3002\u8fd9\u4e2a\u6280\u5de7\u5c24\u5176\u9002\u7528\u4e8e\u90a3\u4e9b\u5185\u5b58\u4e0d\u8db3\u4ee5\u4e00\u6b21\u6027\u52a0\u8f7d\u6574\u4e2a\u6570\u636e\u96c6\u7684\u573a\u666f\uff0c\u5982\u5904\u7406\u5927\u578bCSV\u6587\u4ef6\u3002\u5728\u6807\u9898\u63d0\u5230\u7684\u6559\u7a0b\u4e2d\uff0c\u901a\u8fc7\u8bbe\u7f6e...
\"xlsx\"

python\u6570\u636e\u5206\u6790+pandas+\u6587\u4ef6\u8bfb\u5165\u6216\u5199\u5165

\u8f85\u52a9\u6587\u7ae0\u7684\u4f5c\u7528\uff0c\u53ef\u66f4\u5feb\u7684\u53c2\u8003\u6587\u7ae0
\"zip\"

Pandas

\u73b0\u5728\uff0c\u662f\u65f6\u5019\u4e86\u89e3\u60a8\u5bf9Python Pandas\u7684\u4e86\u89e3\u5e76\u5c06\u5176\u5e94\u7528\u4e8e\u65b0\u60c5\u51b5\u4e86\u3002 \u5bf9\u4e8e\u6b64\u4f5c\u4e1a\uff0c\u60a8\u9700\u8981\u5b8c\u6210\u4e24\u4e2a\uff08\u4e0d\u662f\u4e24\u4e2a\uff09\u6570\u636e\u6311\u6218\u4e4b\u4e00\u3002 \u518d\u4e00\u6b21\uff0c\u60a8\u8981\u9762\u5bf9\u7684\u6311\u6218\u662f\u60a8\u7684\u9009\u62e9\u3002 \u53ea\u9700\u786e\u4fdd\u5168\u529b\u4ee5\u8d74-\u56e0\u4e3a\u60a8\u78e8\u7ec3\u7684\u6280\u80fd\u5c06\u6210\u4e3a\u6570\u636e\u5206\u6790...
\"zip\"

pandas

**Pandas\uff1a\u6570\u636e\u5206\u6790\u7684\u6838\u5fc3\u5de5\u5177** Pandas\u662fPython\u7f16\u7a0b\u8bed\u8a00\u4e2d\u4e00\u4e2a\u5f3a\u5927\u7684\u6570\u636e\u5904\u7406\u5e93\uff0c\u5b83\u4e3a\u6570\u636e\u5206\u6790\u63d0\u4f9b\u4e86\u7b80\u5355\u6613\u7528\u4e14\u9ad8\u6548\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5982DataFrame\u548cSeries\u3002\u8fd9\u4e2a\u5e93\u7684\u8bbe\u8ba1\u76ee\u6807\u662f\u4f7f\u5904\u7406\u7ed3\u6784\u5316\u548c\u534a\u7ed3\u6784\u5316\u6570\u636e\u53d8\u5f97\u65b9\u4fbf\uff0c...
\"zip\"

pandas-pandas

pandas pandas pandas pandas pandas
\"zip\"

pandas.zip_pandas_pandas \u901f\u67e5\u624b\u518c_pandas\u5f00\u53d1\u624b\u518c

**Pandas\uff1aPython\u6570\u636e\u5206\u6790\u5e93\u8be6\u89e3** Pandas\u662fPython\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u4e00\u4e2a\u5f3a\u5927\u6570\u636e\u5206\u6790\u5de5\u5177\uff0c\u5b83\u63d0\u4f9b\u4e86\u9ad8\u6548\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5982DataFrame\u548cSeries\uff0c\u4ee5\u53ca\u4e30\u5bcc\u7684\u6570\u636e\u5206\u6790\u529f\u80fd\u3002\u5728\u672c\u7bc7\u4e2d\uff0c\u6211\u4eec\u5c06\u6df1\u5165\u63a2\u8ba8Pandas\u7684\u6838\u5fc3\u6982\u5ff5\u3001\u5e38\u7528...
\"zip\"

LearnPandas\uff1aLearnPandas

\u300aPandas\u6df1\u5ea6\u5b66\u4e60\u6307\u5357\u300b Pandas\uff0c\u4e00\u4e2a\u5728Python\u4e2d\u5e7f\u6cdb\u4f7f\u7528\u7684\u6570\u636e\u5206\u6790\u5e93\uff0c\u56e0\u5176\u5f3a\u5927\u7684\u6570\u636e\u5904\u7406\u80fd\u529b\u548c\u6613\u7528\u6027\uff0c\u5df2\u7ecf\u6210\u4e3a\u6570\u636e\u79d1\u5b66\u5bb6\u548c\u6570\u636e\u5206\u6790\u5e08\u7684\u5fc5\u5907\u5de5\u5177\u3002\"LearnPandas\"\u662f\u9488\u5bf9Pandas\u7684\u5b66\u4e60\u8d44\u6e90\uff0c\u65e8\u5728\u5e2e\u52a9\u7528\u6237\u638c\u63e1\u8fd9\u4e2a...
\"zip\"

pandas\u63d2\u4ef6\uff08pandas\uff09

\u9002\u7528\u4e8epython2.7
\"pdf\"

Learning-pandas \u5b66\u4e60pandas

\u4ece\u63d0\u4f9b\u7684\u6587\u4ef6\u4fe1\u606f\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u63d0\u53d6\u548c\u603b\u7ed3\u4ee5\u4e0b\u5173\u4e8e\u5b66\u4e60pandas\u7684\u77e5\u8bc6\u70b9\u3002 1. pandas\u6982\u8ff0\uff1a pandas\u662f\u4e00\u4e2a\u5f00\u6e90\u7684\u3001\u9762\u5411Python\u7684\u9ad8\u6027\u80fd\u6570\u636e\u5904\u7406\u548c\u5206\u6790\u5e93\u3002\u5b83\u63d0\u4f9b\u4e86\u6570\u636e\u7ed3\u6784\u548c\u64cd\u4f5c\u6570\u636e\u7684\u5de5\u5177\uff0c\u5c24\u5176\u662f\u9488\u5bf9\u8868\u683c\u548c\u65f6\u95f4\u5e8f\u5217...
\"pdf\"

Pandas - Thinking in Pandas

About pandas How pandas helped build an image of a black hole how pandas helps financial institutions make more informed predictions about the future market how pandas helps improve discoverability of...
\"rar\"

pandas\u5b98\u65b9\u6587\u6863\u4e2d\u6587\u7248_pandas_pandas\u6587\u6863_pandas\u4e2d\u6587_python_

**Pandas\uff1aPython\u6570\u636e\u5206\u6790\u7684\u6838\u5fc3\u5e93** Pandas\u662fPython\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u4e00\u4e2a\u5f3a\u5927\u6570\u636e\u5206\u6790\u5de5\u5177\uff0c\u5b83\u63d0\u4f9b\u4e86\u9ad8\u6548\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5982DataFrame\u548cSeries\uff0c\u4f7f\u5f97\u6570\u636e\u6e05\u6d17\u3001\u8f6c\u6362\u3001\u5206\u6790\u53d8\u5f97\u7b80\u5355\u6613\u884c\u3002Pandas\u7684\u8bbe\u8ba1\u76ee\u6807\u662f\u4f7f\u6570\u636e\u5904\u7406\u5bf9...
\"pdf\"

pandas \u624b\u518c

pandas\u662f\u4e00\u4e2a\u5f00\u6e90\u7684Python\u6570\u636e\u5206\u6790\u5e93\uff0c\u63d0\u4f9b\u4e86\u9ad8\u6027\u80fd\u3001\u6613\u7528\u7684\u6570\u636e\u7ed3\u6784\u548c\u6570\u636e\u5206\u6790\u5de5\u5177\u3002pandas\u4e2d\u7684\u57fa\u7840\u6570\u636e\u7ed3\u6784\u662f\u4e24\u79cd\u4e3b\u8981\u7684\u6570\u636e\u7ed3\u6784\uff1aSeries\u548cDataFrame\u3002Series\u662f\u4e00\u7ef4\u7684\u6807\u7b7e\u6570\u7ec4\uff0c\u53ef\u4ee5\u5305\u542b\u4efb\u4f55\u6570\u636e\u7c7b\u578b\uff08\u6574\u6570\u3001\u5b57\u7b26...
\"pdf\"

python pandas

\u6587\u6863\u8fd8\u63d0\u5230\u4e86\u4e00\u4e2a\u540d\u4e3a\u201c10 Minutes to Pandas\u201d\u7684\u6559\u7a0b\uff0c\u8fd9\u4e2a\u6559\u7a0b\u53ef\u80fd\u662f\u4e3a\u4e86\u8ba9\u521d\u5b66\u8005\u5feb\u901f\u4e86\u89e3pandas\u5e93\u7684\u57fa\u672c\u4f7f\u7528\u65b9\u6cd5\uff0c\u901a\u8fc7\u5341\u5206\u949f\u7684\u5b66\u4e60\u5feb\u901f\u638c\u63e1pandas\u5e93\u7684\u7cbe\u9ad3\u3002\u8fd9\u901a\u5e38\u4f1a\u6db5\u76d6\u6570\u636e\u521b\u5efa\u3001\u67e5\u770b\u6570\u636e\u3001\u5904\u7406\u7f3a\u5931\u6570\u636e\u3001\u6267\u884c...
\"pdf\"

pandas\u6587\u6863

Pandas\u662f\u4e00\u4e2a\u529f\u80fd\u5f3a\u5927\u7684Python\u6570\u636e\u5206\u6790\u5de5\u5177\u5305\u3002\u5b83\u4e3a\u6570\u636e\u5206\u6790\u63d0\u4f9b\u4e86\u5feb\u901f\u3001\u7075\u6d3b\u548c\u8868\u8fbe\u80fd\u529b\u5f3a\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7279\u522b\u8bbe\u8ba1\u7528\u6765\u5904\u7406\u7ed3\u6784\u5316\uff08\u8868\u683c\u3001\u591a\u7ef4\u3001\u5f02\u8d28\uff09\u548c\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u3002Pandas \u7684\u76ee\u6807\u662f\u6210\u4e3a Python \u6570\u636e\u5206\u6790\u7684\u6807\u51c6\u5e93\uff0c...
\"pdf\"

pandas\u6559\u7a0b

pandas\u662f\u4e00\u4e2a\u5f00\u6e90\u7684Python\u6570\u636e\u5206\u6790\u5e93\uff0c\u63d0\u4f9b\u4e86\u9ad8\u6027\u80fd\u3001\u6613\u4e8e\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u548c\u6570\u636e\u5206\u6790\u5de5\u5177\u3002\u672c\u6559\u7a0b\u5c06\u4ecb\u7ecd\u5982\u4f55\u4f7f\u7528pandas\u5e93\u7684\u57fa\u672c\u64cd\u4f5c\uff0c\u5305\u62ec\u521b\u5efa\u548c\u5bfc\u5165pandas\u3001\u521b\u5efaSeries\u548cDataFrame\u5bf9\u8c61\u3001\u4f7f\u7528API\u51fd\u6570\u4ee5\u53ca\u67e5\u770b\u6570\u636e\u7684...
\"xls\"

pandas\u8bfb\u53d6\u548c\u4fdd\u5b58excel\u6587\u4ef6

\u6700\u65b0\u56fd\u5bb61\u7ea72\u7ea7\u9e1f\u7c7b\u4fdd\u62a4\u540d\u5f55
\"docx\"

C\u8bed\u8a00\u57fa\u7840\u8bed\u6cd5\u4e0e\u7a0b\u5e8f\u8bbe\u8ba1\u5165\u95e8\u8bb2\u89e3

c\u8bed\u8a00\u5165\u95e8\nC\u8bed\u8a00\u4e00\u7ecf\u51fa\u73b0\u5c31\u4ee5\u5176\u529f\u80fd\u4e30\u5bcc\u3001\u8868\u8fbe\u80fd\u529b\u5f3a\u3001\u7075\u6d3b\u65b9\u4fbf\u3001\u5e94\u7528\u9762\u5e7f\u7b49\u7279\u70b9\u8fc5\u901f\u5728\u5168\u4e16\u754c\u666e\u53ca\u548c\u63a8\u5e7f\u3002C\u8bed\u8a00\u4e0d\u4f46\u6267\u884c\u6548\u7387\u9ad8\u800c\u4e14\u53ef\u79fb\u690d\u6027\u597d\uff0c\u53ef\u4ee5\u7528\u6765\u5f00\u53d1\u5e94\u7528\u8f6f\u4ef6\u3001\u9a71\u52a8\u3001\u64cd\u4f5c\u7cfb\u7edf\u7b49\u3002C\u8bed\u8a00\u4e5f\u662f\u5176\u5b83\u4f17\u591a\u9ad8\u7ea7\u8bed\u8a00\u7684\u9f3b\u7956\u8bed\u8a00\uff0c\u6240\u4ee5\u8bf4\u5b66\u4e60C\u8bed\u8a00\u662f\u8fdb\u5165\u7f16\u7a0b\u4e16\u754c\u7684\u5fc5\u4fee\u8bfe\u3002hello,world\n#include<stdio.h>\nint main()\n{\n/*\u5728\u53cc\u5f15\u53f7\u4e2d\u95f4\u8f93\u5165Hello World*/\nprintf(\"Hello World\");\nreturn 0;\n}\n\n\u6ce8\uff1a\u5728\u6700\u65b0\u7684C\u6807\u51c6\u4e2d\uff0cmain\u51fd\u6570\u524d\u7684\u7c7b\u578b\u4e3aint\u800c\u4e0d\u662fvoid\nc\u8bed\u8a00\u7684\u5177\u4f53\u7ed3\u6784\u7b80\u5355\u6765\u8bf4\uff0c\u4e00\u4e2aC\u7a0b\u5e8f\u5c31\u662f\u7531\u82e5\u5e72\u5934\u6587\u4ef6\u548c\u51fd\u6570\u7ec4\u6210\u3002#include <stdio.h>\u5c31\u662f\u4e00\u6761\u9884\u5904\u7406\u547d\u4ee4, \u5b83\u7684\u4f5c\u7528\u662f\u901a\u77e5C\u8bed\u8a00\u7f16\u8bd1\u7cfb\u7edf\u5728\u5bf9C\u7a0b\u5e8f\u8fdb\u884c\u6b63\u5f0f\u7f16\u8bd1\u4e4b\u524d\u9700\u505a\u4e00\u4e9b\u9884\u5904\u7406\u5de5\u4f5c\u3002\u51fd\u6570\u5c31\u662f\u5b9e\u73b0\u4ee3\u7801\u903b\u8f91\u7684\u4e00\u4e2a\u5c0f\u7684\u5355\u5143\u3002\u5fc5\u4e0d\u53ef\u5c11\u4e4b\u4e3b\u51fd\u6570\u4e00\u4e2aC\u7a0b\u5e8f\u6709\u4e14\u53ea\u6709\u4e00\u4e2a\u4e3b\u51fd\u6570\uff0c\u5373main\u51fd\u6570\u3002C\u7a0b\u5e8f\u5c31\u662f\u6267\u884c\u4e3b\u51fd\u6570\u91cc\u7684\u4ee3\u7801\uff0c\u4e5f\u53ef\u4ee5\u8bf4\u8fd9\u4e2a\u4e3b\u51fd\u6570\u5c31\u662fC\u8bed\u8a00\u4e2d\u7684\u552f\u4e00\u5165\u53e3\u3002\u800cmain\u524d\u9762\u7684int\u5c31\u662f\u4e3b\u51fd\u6570\u7684\u7c7b\u578b.printf()\u662f\u683c\u5f0f\u8f93\u51fa\u51fd\u6570\uff0c\u8fd9\u91cc

\u5927\u5bb6\u5728\u770b

\"recommend-type\"

SHIMAX_MAC3&MAC50\u901a\u8baf\u624b\u518c

\u65e5\u672cSHIMAX_MAC3&MAC50\u901a\u8baf\u624b\u518c
\"recommend-type\"

\u57fa\u4e8e\u7efc\u5408\u8bc4\u4ef7\u8bed\u4e49\u63cf\u8ff0\u7684\u9886\u57df\u672c\u4f53\u6784\u5efa (2013\u5e74)

\u57fa\u4e8e\u9886\u57df\u7efc\u5408\u8bc4\u4ef7\u7684\u6307\u6807\u4f53\u7cfb\u5206\u6790\u5176\u6240\u63cf\u8ff0\u7684\u8bed\u4e49\u6982\u5ff5,\u5e76\u5bf9\u6307\u6807\u8fdb\u884c\u9884\u5904\u7406,\u63d0\u53d6\u8bc4\u4ef7\u6307\u6807\u6240\u8868\u8ff0\u7684\u6982\u5ff5\u4ee5\u53ca\u6307\u6807\u95f4\u7684\u5173\u7cfb.\u6839\u636e\u8fd9\u4e9b\u6982\u5ff5\u548c\u5173\u7cfb\u6784\u5efa\u57fa\u7840\u672c\u4f53,\u901a\u8fc7\u751f\u6210\u6982\u5ff5\u683c\u5bf9\u8be5\u672c\u4f53\u7684\u6982\u5ff5\u63cf\u8ff0\u8fdb\u4e00\u6b65\u4e30\u5bcc,\u5e76\u751f\u6210\u66f4\u5b8c\u5584\u7684\u6982\u5ff5\u683c.\u6700\u540e\u5229\u7528\u5f62\u5f0f\u6982\u5ff5\u5206\u6790\u7684\u65b9\u6cd5\u548c\u5de5\u5177\u6784\u5efa\u9886\u57df\u672c\u4f53.
\"recommend-type\"

ansys workbench \u975e\u7ebf\u6027\u5206\u6790

ansys workbench \u975e\u7ebf\u6027\u6559\u7a0b
\"recommend-type\"

hw1.rar_C++\u56fe\u50cf\u63d2\u503c_\u4e8c\u7ef4\u63d2\u503c_\u4e8c\u7ef4\u63d2\u503c C++_\u56fe\u50cf_\u6700\u8fd1\u90bb\u63d2\u503c

\u56fe\u50cf\u5904\u7406\uff0c\u5bf9\u6e90\u56fe\u50cf\u8fdb\u884c\u626d\u66f2\uff0c\u91c7\u7528\u6700\u8fd1\u90bb\u3001\u4e8c\u7ef4\u63d2\u503c\u548c\u4e09\u6b21\u6837\u6761\u63d2\u503c\u3002
\"recommend-type\"

Chamber and Station test.pptx

Chamber and Station test.pptx

\u6700\u65b0\u63a8\u8350

\"recommend-type\"

pandas\u5206\u6279\u8bfb\u53d6\u5927\u6570\u636e\u96c6\u6559\u7a0b

`pandas`\u5e93\u63d0\u4f9b\u4e86\u4e00\u79cd\u7075\u6d3b\u7684\u65b9\u6cd5\uff0c\u5373\u5206\u6279\u8bfb\u53d6\u6570\u636e\u96c6\uff0c\u4ee5\u89e3\u51b3\u5185\u5b58\u9650\u5236\u7684\u95ee\u9898\u3002\u672c\u6559\u7a0b\u5c06\u8be6\u7ec6\u4ecb\u7ecd\u5982\u4f55\u5229\u7528`pandas`\u7684\u8fd9\u4e00\u529f\u80fd\u6765\u9ad8\u6548\u5730\u5904\u7406\u5927\u6570\u636e\u3002 \u5728\u5904\u7406\u5927\u578b\u6570\u636e\u96c6\u65f6\uff0c\u76f4\u63a5\u4e00\u6b21\u6027\u52a0\u8f7d\u6240\u6709\u6570\u636e\u53ef\u80fd\u5bfc\u81f4\u5185\u5b58\u6ea2\u51fa\uff0c\u7279\u522b...
\"recommend-type\"

Pandas\u8bfb\u53d6csv\u65f6\u5982\u4f55\u8bbe\u7f6e\u5217\u540d

\u5728\u6570\u636e\u5206\u6790\u9886\u57df\uff0cPandas\u5e93\u662fPython\u4e2d\u4e0d\u53ef\u6216\u7f3a\u7684\u4e00\u90e8\u5206\uff0c\u5b83\u63d0\u4f9b\u4e86\u9ad8\u6548\u7684\u6570\u636e\u5904\u7406\u80fd\u529b\u3002Pandas\u80fd\u591f\u65b9\u4fbf\u5730\u8bfb\u53d6\u548c\u5199\u5165\u591a\u79cd\u683c\u5f0f\u7684\u6570\u636e\u6587\u4ef6\uff0c\u5176\u4e2dCSV\uff08Comma Separated Values\uff09\u662f\u6700\u5e38\u89c1\u7684\u6570\u636e\u5b58\u50a8\u683c\u5f0f\u4e4b\u4e00\u3002\u672c\u7bc7\u5c06\u8be6\u7ec6...
\"recommend-type\"

pandas\u4e2dTimestamp\u7c7b\u7528\u6cd5\u8be6\u89e3

\u5728Python\u7684pandas\u5e93\u4e2d\uff0cTimestamp\u662f\u4e00\u4e2a\u975e\u5e38\u91cd\u8981\u7684\u7c7b\uff0c\u5b83\u7528\u4e8e\u8868\u793a\u7279\u5b9a\u7684\u65e5\u671f\u548c\u65f6\u95f4\u70b9\u3002Timestamp\u7c7b\u63d0\u4f9b\u4e86\u4e30\u5bcc\u7684\u529f\u80fd\uff0c\u53ef\u4ee5\u65b9\u4fbf\u5730\u5904\u7406\u65e5\u671f\u548c\u65f6\u95f4\u6570\u636e\uff0c\u662f\u6570\u636e\u5206\u6790\u4e2d\u4e0d\u53ef\u6216\u7f3a\u7684\u4e00\u90e8\u5206\u3002\u4ee5\u4e0b\u662f\u5bf9pandas\u4e2dTimestamp\u7c7b\u7684...
\"recommend-type\"

python pandas\u751f\u6210\u65f6\u95f4\u5217\u8868

\u5728Python\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0cPandas\u5e93\u662f\u4e00\u4e2a\u5f3a\u5927\u7684\u6570\u636e\u5206\u6790\u5de5\u5177\uff0c\u5b83\u63d0\u4f9b\u4e86\u4e30\u5bcc\u7684\u6570\u636e\u5904\u7406\u529f\u80fd\uff0c\u5305\u62ec\u751f\u6210\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u3002\u5728\u5904\u7406\u6d89\u53ca\u65f6\u95f4\u7684\u6570\u636e\u65f6\uff0cPandas\u7684\u65f6\u95f4\u5217\u8868\uff08\u65f6\u95f4\u5e8f\u5217\uff09\u626e\u6f14\u7740\u81f3\u5173\u91cd\u8981\u7684\u89d2\u8272\u3002\u672c\u7bc7\u5c06\u6df1\u5165\u63a2\u8ba8\u5982\u4f55\u4f7f\u7528...
\"recommend-type\"

pandas\u548cspark dataframe\u4e92\u76f8\u8f6c\u6362\u5b9e\u4f8b\u8be6\u89e3

\u5728\u5927\u6570\u636e\u5904\u7406\u9886\u57df\uff0c`pandas` \u548c `Spark DataFrame` \u662f\u4e24\u4e2a\u91cd\u8981\u7684\u5de5\u5177\u3002`pandas` \u662f Python \u4e2d\u7528\u4e8e\u6570\u636e\u5904\u7406\u548c\u5206\u6790\u7684\u5e93\uff0c\u800c `Spark DataFrame` \u662f Apache Spark \u7684\u6838\u5fc3\u7ec4\u4ef6\uff0c\u63d0\u4f9b\u4e86\u4e00\u79cd\u5206\u5e03\u5f0f\u6570\u636e\u5904\u7406\u80fd\u529b\u3002\u672c\u6587\u5c06\u8be6\u7ec6...
\"recommend-type\"

\u6df1\u5165\u63a2\u7d22CSS\u62c9\u7279\u6d4b\u8bd5\u65b9\u6cd5

\u6839\u636e\u63d0\u4f9b\u7684\u6587\u4ef6\u4fe1\u606f\uff0c\u6211\u4eec\u65e0\u6cd5\u83b7\u53d6\u5177\u4f53\u7684\u6587\u4ef6\u5185\u5bb9\uff0c\u56e0\u6b64\uff0c\u9700\u8981\u4ece\u6587\u4ef6\u7684\u6807\u9898\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\uff0c\u63cf\u8ff0\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\uff0c\u6807\u7b7e\u201cCSS\u201d\uff0c\u4ee5\u53ca\u538b\u7f29\u5305\u5b50\u6587\u4ef6\u7684\u6587\u4ef6\u540d\u79f0\u5217\u8868\u201clat-test-main\u201d\u6765\u63a8\u65ad\u76f8\u5173\u7684\u77e5\u8bc6\u70b9\u3002\u9274\u4e8e\u8fd9\u4e9b\u4fe1\u606f\u91cf\u6709\u9650\uff0c\u6211\u4eec\u5c06\u4e3b\u8981\u56f4\u7ed5\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u8fd9\u4e00\u4e3b\u9898\u8fdb\u884c\u63a2\u8ba8\uff0c\u540c\u65f6\u4e5f\u4f1a\u6d89\u53caCSS\u76f8\u5173\u5185\u5bb9\u3002\n\n\u9996\u5148\uff0c\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u53ef\u80fd\u6307\u7684\u662f\u67d0\u79cd\u7279\u5b9a\u7684\u8f6f\u4ef6\u6d4b\u8bd5\u65b9\u6cd5\u6216\u8005\u6280\u672f\u8bc4\u4f30\u6d41\u7a0b\u3002\u8003\u8651\u5230\u6587\u4ef6\u540d\u201clat-test-main\u201d\u6697\u793a\u5b83\u53ef\u80fd\u662f\u67d0\u4e2a\u9879\u76ee\u7684\u4e3b\u8981\u6d4b\u8bd5\u6587\u4ef6\uff0c\u6211\u4eec\u53ef\u4ee5\u5408\u7406\u63a8\u6d4b\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u53ef\u80fd\u662f\u6d4b\u8bd5\u7684\u4ee3\u7801\u811a\u672c\u3001\u6d4b\u8bd5\u7528\u4f8b\u96c6\u5408\u3001\u6216\u8005\u662f\u4e0e\u6d4b\u8bd5\u76f8\u5173\u7684\u914d\u7f6e\u6587\u4ef6\u3002\u4f46\u5728\u6ca1\u6709\u66f4\u591a\u4e0a\u4e0b\u6587\u7684\u60c5\u51b5\u4e0b\uff0c\u5f88\u96be\u786e\u5b9a\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u5177\u4f53\u6307\u4ee3\u7684\u662f\u4ec0\u4e48\u3002\n\n\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u8ba8\u8bba\u201cCSS\u201d\u3002CSS\u662f\u201c\u5c42\u53e0\u6837\u5f0f\u8868\uff08Cascading Style Sheets\uff09\u201d\u7684\u7f29\u5199\uff0c\u662f\u4e00\u79cd\u7528\u4e8e\u63a7\u5236\u7f51\u9875\u5916\u89c2\u548c\u5e03\u5c40\u7684\u6280\u672f\u6807\u51c6\u3002CSS\u63cf\u8ff0\u4e86\u5982\u4f55\u5728\u5c4f\u5e55\u4e0a\uff0c\u7eb8\u5f20\u4e0a\uff0c\u6216\u5728\u5176\u4ed6\u5a92\u4f53\u4e0a\u5c55\u73b0HTML\u6216XML\uff08\u5305\u62ec\u5404\u79cdXML\u65b9\u8a00\uff0c\u6bd4\u5982SVG\u6216XHTML\uff09\u6587\u6863\u3002\u5b83\u4f7f\u5f00\u53d1\u8005\u80fd\u591f\u5c06\u5185\u5bb9\u4e0e\u8868\u73b0\u5206\u79bb\uff0c\u8fd9\u6709\u52a9\u4e8e\u5bf9\u7f51\u7ad9\u8fdb\u884c\u4fee\u6539\uff0c\u800c\u65e0\u9700\u89e6\u53ca\u5185\u5bb9\u672c\u8eab\u3002CSS\u7684\u89c4\u5219\u7531\u9009\u62e9\u5668\u548c\u58f0\u660e\u5757\u7ec4\u6210\u3002\u9009\u62e9\u5668\u6307\u660e\u4e86\u6837\u5f0f\u89c4\u5219\u5e94\u8be5\u5e94\u7528\u4e8e\u54ea\u4e9bHTML\u5143\u7d20\uff0c\u800c\u58f0\u660e\u5757\u5219\u5305\u542b\u4e86\u4e00\u4e2a\u6216\u591a\u4e2a\u7528\u5206\u53f7\u9694\u5f00\u7684\u5c5e\u6027\u503c\u5bf9\u3002\n\n\u7136\u800c\uff0c\u7531\u4e8e\u6807\u9898\u3001\u63cf\u8ff0\u548c\u6807\u7b7e\u5e76\u6ca1\u6709\u76f4\u63a5\u63d0\u4f9b\u5173\u4e8eCSS\u7684\u5177\u4f53\u4fe1\u606f\uff0c\u6211\u4eec\u4e5f\u65e0\u6cd5\u786e\u5b9aCSS\u5728\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u4e2d\u626e\u6f14\u7684\u5177\u4f53\u89d2\u8272\u3002\u4e0d\u8fc7\uff0c\u5047\u8bbeCSS\u6807\u7b7e\u610f\u5473\u7740\u6d4b\u8bd5\u53ef\u80fd\u4e0e\u7f51\u9875\u7684\u6837\u5f0f\u8868\u6216\u8005\u524d\u7aef\u8bbe\u8ba1\u6709\u5173\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u60f3\u8c61\uff0c\u6d4b\u8bd5\u53ef\u80fd\u6db5\u76d6\u4e86\u5bf9\u7f51\u9875\u6837\u5f0f\u7684\u9a8c\u8bc1\u3001\u5bf9\u5e03\u5c40\u7684\u6d4b\u8bd5\u3001\u5bf9\u4ea4\u4e92\u6548\u679c\u7684\u68c0\u67e5\u7b49\u3002\n\n\u5728\u5f00\u53d1\u548c\u6d4b\u8bd5\u8fc7\u7a0b\u4e2d\uff0cCSS\u7684\u6b63\u786e\u6027\u81f3\u5173\u91cd\u8981\u3002\u4ee5\u4e0b\u662f\u4e00\u4e9b\u4e0eCSS\u76f8\u5173\u7684\u6d4b\u8bd5\u65b9\u6cd5\uff1a\n\n1. CSS\u9a8c\u8bc1\u6d4b\u8bd5\uff1a\u786e\u4fddCSS\u4ee3\u7801\u7b26\u5408\u6807\u51c6\uff0c\u5e76\u4e14\u6ca1\u6709\u8bed\u6cd5\u9519\u8bef\u3002\u53ef\u4ee5\u4f7f\u7528\u5728\u7ebf\u5de5\u5177\u5982W3C\u7684CSS\u9a8c\u8bc1\u670d\u52a1\u8fdb\u884c\u3002\n\n2. \u517c\u5bb9\u6027\u6d4b\u8bd5\uff1a\u68c0\u67e5\u7f51\u7ad9\u5728\u4e0d\u540c\u7684\u6d4f\u89c8\u5668\u548c\u8bbe\u5907\u4e0a\u663e\u793a\u7684\u4e00\u81f4\u6027\u3002\u7531\u4e8e\u6d4f\u89c8\u5668\u5bf9CSS\u7684\u652f\u6301\u5b58\u5728\u5dee\u5f02\uff0c\u8fd9\u4e00\u6b65\u9aa4\u5341\u5206\u91cd\u8981\u3002\n\n3. \u6027\u80fd\u6d4b\u8bd5\uff1a\u5206\u6790CSS\u6587\u4ef6\u7684\u5927\u5c0f\u3001\u590d\u6742\u5ea6\u4ee5\u53ca\u4e0b\u8f7d\u548c\u6e32\u67d3\u65f6\u95f4\uff0c\u4f18\u5316\u8fd9\u4e9b\u6027\u80fd\u6307\u6807\u4ee5\u63d0\u9ad8\u7f51\u9875\u52a0\u8f7d\u901f\u5ea6\u3002\n\n4. \u53ef\u8bbf\u95ee\u6027\u6d4b\u8bd5\uff1a\u786e\u4fdd\u7f51\u7ad9\u5bf9\u4e0d\u540c\u9700\u6c42\u7684\u7528\u6237\uff0c\u5305\u62ec\u6709\u89c6\u89c9\u969c\u788d\u7684\u7528\u6237\uff0c\u662f\u6613\u4e8e\u5bfc\u822a\u548c\u4f7f\u7528\u7684\u3002\n\n5. \u5355\u5143\u6d4b\u8bd5\uff1a\u5bf9\u4e8e\u4f7f\u7528CSS\u9884\u5904\u7406\u5668\u6216\u7f16\u8bd1\u5de5\u5177\u751f\u6210\u6700\u7ec8\u6837\u5f0f\u8868\u7684\u60c5\u51b5\uff0c\u5355\u5143\u6d4b\u8bd5\u53ef\u4ee5\u786e\u4fdd\u8fd9\u4e9b\u5de5\u5177\u7684\u6b63\u786e\u6027\u3002\n\n6. \u529f\u80fd\u6d4b\u8bd5\uff1a\u68c0\u67e5\u7f51\u9875\u4e0a\u7684\u6837\u5f0f\u5143\u7d20\u662f\u5426\u6309\u7167\u8bbe\u8ba1\u5b9e\u73b0\uff0c\u6bd4\u5982\u5b57\u4f53\u3001\u989c\u8272\u3001\u5e03\u5c40\u548c\u5176\u4ed6\u89c6\u89c9\u6548\u679c\u3002\n\n\u7531\u4e8e\u201clat-test-main\u201d\u6697\u793a\u8fd9\u662f\u4e00\u4e2a\u4e3b\u8981\u7684\u6d4b\u8bd5\u6587\u4ef6\uff0c\u5b83\u53ef\u80fd\u5305\u542b\u4e86\u4e0a\u8ff0\u6d4b\u8bd5\u65b9\u6cd5\u4e2d\u7684\u4e00\u79cd\u6216\u591a\u79cd\u7684\u5b9e\u73b0\u3002\u5728\u5b9e\u9645\u5f00\u53d1\u8fc7\u7a0b\u4e2d\uff0c\u6d4b\u8bd5\u901a\u5e38\u662f\u5728\u7248\u672c\u63a7\u5236\u7cfb\u7edf\u7684\u652f\u6301\u4e0b\u8fdb\u884c\u7684\uff0c\u6bd4\u5982Git\uff0c\u5b83\u53ef\u4ee5\u5e2e\u52a9\u56e2\u961f\u6210\u5458\u7ba1\u7406\u4e0d\u540c\u7684\u6d4b\u8bd5\u7248\u672c\uff0c\u5e76\u8ddf\u8e2a\u4ee3\u7801\u66f4\u6539\u3002\n\n\u7efc\u4e0a\u6240\u8ff0\uff0c\u5173\u4e8e\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u548c\u201cCSS\u201d\u7684\u77e5\u8bc6\u70b9\u96c6\u4e2d\u5728\u6d4b\u8bd5\u65b9\u6cd5\u548c\u6837\u5f0f\u8868\u7684\u5e94\u7528\u4e0a\u3002\u4e0d\u8fc7\uff0c\u4e3a\u4e86\u66f4\u51c6\u786e\u5730\u63cf\u8ff0\u201c\u62c9\u7279\u6d4b\u8bd5\u201d\u7684\u542b\u4e49\uff0c\u6211\u4eec\u9700\u8981\u66f4\u591a\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u6216\u8005\u76f4\u63a5\u8bbf\u95ee\u76f8\u5173\u7684\u6587\u4ef6\u5185\u5bb9\u3002\u5728\u5b9e\u9645\u5de5\u4f5c\u4e2d\uff0c\u4e86\u89e3\u9879\u76ee\u9700\u6c42\u3001\u6d4b\u8bd5\u76ee\u6807\u548c\u73af\u5883\u914d\u7f6e\u5bf9\u4e8e\u6210\u529f\u5730\u5b9e\u65bd\u6d4b\u8bd5\u8ba1\u5212\u81f3\u5173\u91cd\u8981\u3002
\"recommend-type\"

\u65b0\u5510IAP\u6982\u5ff5\u89e3\u6790

\n\n# \u6458\u8981\nIAP\uff08In-Application Programming\uff09\u7f16\u7a0b\u662f\u4e00\u79cd\u5728\u5e94\u7528\u8fd0\u884c\u65f6\u66f4\u65b0\u56fa\u4ef6\u7684\u5148\u8fdb\u65b9\u6cd5\uff0c\u5b83\u63d0\u4f9b\u4e86\u7cfb\u7edf\u66f4\u65b0\u7684\u7075\u6d3b\u6027\u548c\u4fbf\u5229\u6027\u3002\u672c\u6587\u5168\u9762\u4ecb\u7ecd\u4e86IAP\u7f16\u7a0b\u7684\u6982\u5ff5\u3001\u6280\u672f\u57fa\u7840\u548c\u5b9e\u8df5\u5e94\u7528\uff0c\u91cd\u70b9\u5206\u6790\u4e86IAP\u5728\u65b0\u5510\u5fae\u63a7\u5236\u5668\u4e2d\u7684\u5b9e\u73b0\u673a\u5236\uff0c\u5305\u62ec\u5176\u5185\u5b58\u7ed3\u6784\u548c\u5de5\u4f5c\u6d41\u7a0b\uff0c\u5e76\u63a2\u8ba8\u4e86\u8f6f\u4ef6\u5de5\u5177\u548c\u5f00\u53d1\u73af\u5883\u7684\u914d\u7f6e\u3002\u540c\u65f6\uff0c\u672c\u6587\u901a\u8fc7\u5b9e\u9645\u6848\u4f8b\u6df1\u5165\u7814\u7a76\u4e86IAP\u5f00\u53d1\u6d41\u7a0b\u3001\u5b89\u5168\u6027\u548c\u9519\u8bef\u5904\u7406\u7b56\u7565\uff0c\u4ee5\u53ca\u5728\u7269\u8054\u7f51\u8bbe\u5907\u548c\u667a\u80fd\u5bb6\u5c45\u7b49\u9886\u57df\u7684\u9ad8\u7ea7\u5e94\u7528\u3002\u6700\u540e\uff0c\u9488\u5bf9IAP\u9879\u76ee\u7684\u7ba1
\"recommend-type\"

fix_eco_timing \u5199\u51fa\u811a\u672c

`fix_eco_timing`\u8fd9\u4e2a\u540d\u5b57\u770b\u8d77\u6765\u50cf\u662f\u7528\u4e8e\u67d0\u79cd\u7279\u5b9a\u73af\u5883\u4e0b\u7684\u811a\u672c\uff0c\u6bd4\u5982\u53ef\u80fd\u662f\u7528\u4e8e\u8c03\u6574\u7535\u5b50\u7ec4\u4ef6\u6216\u7535\u5b50\u4ea7\u54c1\u7684\u5de5\u4f5c\u5468\u671f\u4f18\u5316\u80fd\u6e90\u6548\u7387\u7684\u4e00\u79cd\u5de5\u5177\u3002\u7136\u800c\uff0c\u6ca1\u6709\u5177\u4f53\u7684\u4e0a\u4e0b\u6587\uff0c\u5f88\u96be\u63d0\u4f9b\u8be6\u7ec6\u7684\u811a\u672c\u5185\u5bb9\u3002\u901a\u5e38\u8fd9\u6837\u7684\u811a\u672c\u53ef\u80fd\u4f1a\u5305\u542b\u4ee5\u4e0b\u51e0\u4e2a\u90e8\u5206\uff1a\n\n```bash\n#!/bin/bash\n# Fix Eco Timing Script\n\n# 1. \u83b7\u53d6\u5f53\u524d\u8bbe\u5907\u72b6\u6001\ndevice_status=$(get_device_status)\n\n# 2. \u68c0\u67e5\u662f\u5426\u8fbe\u5230\u8282\u80fd\u6a21\u5f0f\u6761\u4ef6\nif [ "$device_status" == "idle" ]; then\n# 3. \u8c03\u6574\u5de5\u4f5c\u9891\u7387\u6216\u7535\u6e90\u7ba1\u7406\u8bbe\u7f6e\nad
\"recommend-type\"

BTS SIO\u57f9\u8bad\u751fYoucef Tarfa\u7684\u4e2a\u4eba\u6295\u8d44\u7ec4\u5408\u7f51\u7ad9

\u6839\u636e\u63d0\u4f9b\u7684\u6587\u4ef6\u4fe1\u606f\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u65ad\u51fa\u4e00\u4e9b\u5173\u952e\u77e5\u8bc6\u70b9\uff1a\n\n### \u6807\u9898\u77e5\u8bc6\u70b9\uff1a\n\n1. **\u4e2a\u4eba\u6295\u8d44\u7ec4\u5408\u7f51\u7ad9**\uff1a\u6807\u9898\u4e2d\u7684\u201cYouceftarfa.github.io\u201d\u8868\u660e\u8fd9\u662f\u4e00\u4e2a\u5728\u7ebf\u7684\u4e2a\u4eba\u6295\u8d44\u7ec4\u5408\u7f51\u7ad9\uff0c\u8fd9\u901a\u5e38\u7528\u4e8e\u5c55\u793a\u4e2a\u4eba\u7684\u9879\u76ee\u3001\u7ecf\u9a8c\u548c\u6280\u80fd\u3002\u4e2a\u4eba\u6295\u8d44\u7ec4\u5408\u7f51\u7ad9\u662f\u4e13\u4e1aIT\u4eba\u58eb\u7528\u6765\u5411\u6f5c\u5728\u96c7\u4e3b\u3001\u5ba2\u6237\u6216\u5408\u4f5c\u4f19\u4f34\u5c55\u793a\u4ed6\u4eec\u4e13\u4e1a\u80fd\u529b\u7684\u91cd\u8981\u5de5\u5177\u3002\n\n2. **GitHub.io\u57df\u540d**\uff1a\u57df\u540d\u4e2d\u7684\u201c.github.io\u201d\u610f\u5473\u7740\u8fd9\u662f\u4e00\u4e2a\u6258\u7ba1\u5728GitHub\u5e73\u53f0\u4e0a\u7684\u4e2a\u4eba\u7f51\u7ad9\u3002GitHub\u4e0d\u4ec5\u63d0\u4f9b\u6e90\u4ee3\u7801\u6258\u7ba1\u670d\u52a1\uff0c\u4e5f\u652f\u6301\u7528\u6237\u901a\u8fc7GitHub Pages\u529f\u80fd\u6765\u53d1\u5e03\u4e2a\u4eba\u7ad9\u70b9\uff0c\u8fd9\u901a\u5e38\u7528\u4e8e\u5f00\u6e90\u9879\u76ee\u5c55\u793a\u3001\u4e2a\u4eba\u7b80\u5386\u5c55\u793a\u3001\u6280\u672f\u535a\u5ba2\u7b49\u591a\u79cd\u7528\u9014\u3002\n\n3. **BTS SIO\u57f9\u8bad\u751f**\uff1a\u8fd9\u53ef\u80fd\u662fYoucef Tarfa\u53c2\u4e0e\u7684\u4e00\u4e2a\u57f9\u8bad\u8ba1\u5212\u6216\u8bfe\u7a0b\u7684\u540d\u79f0\uff0cBTS SIO\uff08Brevet de Technicien Sup\u00e9rieur \u2013 Syst\u00e8mes Informatiques et 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\u7ed3\u8bed\uff1a\n\n\u7efc\u5408\u4ee5\u4e0a\u4fe1\u606f\uff0c\u53ef\u4ee5\u770b\u51fa\u8fd9\u4e2a\u6587\u4ef6\u6d89\u53ca\u4e86\u4e2a\u4eba\u54c1\u724c\u5efa\u8bbe\u3001\u6280\u672f\u5c55\u793a\u3001\u5f00\u6e90\u6587\u5316\u3001\u804c\u4e1a\u53d1\u5c55\u7b49\u591a\u65b9\u9762\u7684\u77e5\u8bc6\u70b9\u3002\u5bf9\u4e8eIT\u4e13\u4e1a\u4eba\u58eb\u6765\u8bf4\uff0c\u7ef4\u62a4\u4e00\u4e2a\u5185\u5bb9\u4e30\u5bcc\u3001\u7ed3\u6784\u826f\u597d\u7684\u4e2a\u4eba\u6295\u8d44\u7ec4\u5408\u7f51\u7ad9\uff0c\u662f\u63d0\u5347\u4e2a\u4eba\u6280\u80fd\u5c55\u793a\u3001\u6269\u5c55\u804c\u4e1a\u7f51\u7edc\u548c\u4fc3\u8fdb\u4e2a\u4eba\u804c\u4e1a\u6210\u957f\u7684\u91cd\u8981\u9014\u5f84\u3002\u540c\u65f6\uff0c\u901a\u8fc7\u53c2\u4e0eGitHub\u8fd9\u6837\u7684\u5f00\u6e90\u5e73\u53f0\uff0c\u4e0d\u4ec5\u53ef\u4ee5\u63d0\u9ad8\u81ea\u8eab\u7684\u6280\u80fd\uff0c\u8fd8\u80fd\u4e0e\u5168\u7403\u7684\u5f00\u53d1\u8005\u5171\u540c\u8fdb\u6b65\uff0c\u4e3a\u8f6f\u4ef6\u884c\u4e1a\u7684\u53d1\u5c55\u4f5c\u51fa\u8d21\u732e\u3002
\"recommend-type\"

\u3010\u533b\u7597\u8bbe\u5907\u7ef4\u4fee\u901f\u6210\u79d8\u7c4d\u3011\uff1a\u4ece\u65b0\u624b\u5230\u4e13\u5bb6\u76845\u5927\u5fc5\u5b66\u6280\u5de7

\n\n# \u6458\u8981\n\u672c\u6587\u8be6\u7ec6\u4ecb\u7ecd\u4e86\u533b\u7597\u8bbe\u5907\u7ef4\u4fee\u7684\u57fa\u7840\u77e5\u8bc6\u3001\u8bbe\u5907\u5206\u7c7b\u548c\u5de5\u4f5c\u539f\u7406\u3001\u65e5\u5e38\u4fdd\u517b\u4e0e\u6545\u969c\u6392\u67e5\u6280\u5de7\u3001\u7ef4\u4fee\u5b9e\u8df5\u64cd\u4f5c\u4ee5\u53ca\u6cd5\u89c4\u9075\u4ece\u4e0e\u4e13\u4e1a\u53d1\u5c55\u3002\u901a\u8fc7\u5bf9\u533b\u7597\u8bbe\u5907\u5206\u7c7b\u548c\u5de5\u4f5c\u539f\u7406\u7684\u9610\u8ff0\uff0c\u4e3a\u7ef4\u4fee\u4eba\u5458\u63d0\u4f9b\u4e86\u6df1\u5165\u7406\u89e3\u8bbe\u5907\u6027\u80fd\u4e0e\u7ef4\u62a4\u8981\u6c42\u7684\u57fa\u7840\u3002\u540c\u65f6\uff0c\u7ed3\u5408\u65e5\u5e38\u4fdd\u517b\u7684\u91cd\u8981\u6027\u548c\u6545\u969c\u6392\u67e5\u7684\u7406
\n \n\n \n \n"} diff --git a/bench/data/groundtruth/code_5.jsonl b/bench/data/groundtruth/code_5.jsonl index 3b75b31d..cbcec0cf 100644 --- a/bench/data/groundtruth/code_5.jsonl +++ b/bench/data/groundtruth/code_5.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "title", "raw_content": "

XmlSuiteResultFormatter

", "content": {"title_content": "XmlSuiteResultFormatter", "level": "1"}}, {"type": "code", "raw_content": "

\n\n public\n\n\n\n class\n XmlSuiteResultFormatter\n\n
\n\n\n\n\n extends Object\n\n\n\n\n\n\n\n\n implements\n\n IFormatterGenerator\n\n\n\n\n\n

", "inline": false, "content": {"code_content": " public class XmlSuiteResultFormatter\n extends Object implements IFormatterGenerator", "by": "tag_code"}}, {"type": "table", "raw_content": "
java.lang.Object \u4e2d\n
\u21b3com.android.tradefed.result.suite.XmlSuiteResultFormatter\n
", "content": {"html": "
java.lang.Object \u4e2d
\u21b3com.android.tradefed.result.suite.XmlSuiteResultFormatter
", "is_complex": true}}, {"type": "paragraph", "raw_content": "

\u7528\u4e8e\u5c06\u5957\u4ef6\u8fd0\u884c\u4f5c\u4e1a\u4fdd\u5b58\u4e3a XML \u7684\u5b9e\u7528\u7a0b\u5e8f\u7c7b\u3002TODO\uff1a\u79fb\u9664\u6240\u6709\u7279\u6b8a\u7684\u517c\u5bb9\u6027\u6d4b\u8bd5\n \u83b7\u5f97\u76f8\u540c\u7684\u683c\u5f0f\u3002

", "content": [{"c": "\u7528\u4e8e\u5c06\u5957\u4ef6\u8fd0\u884c\u4f5c\u4e1a\u4fdd\u5b58\u4e3a XML \u7684\u5b9e\u7528\u7a0b\u5e8f\u7c7b\u3002TODO\uff1a\u79fb\u9664\u6240\u6709\u7279\u6b8a\u7684\u517c\u5bb9\u6027\u6d4b\u8bd5\n \u83b7\u5f97\u76f8\u540c\u7684\u683c\u5f0f\u3002", "t": "text"}]}, {"type": "title", "raw_content": "

\u6458\u8981

", "content": {"title_content": "\u6458\u8981", "level": "2"}}, {"type": "table", "raw_content": "

\u5d4c\u5957\u7c7b

classXmlSuiteResultFormatter.RunHistory

\u7528\u4e8e JSON \u8f6c\u6362\u7684\u8f85\u52a9\u5bf9\u8c61\u3002\n\n\n

", "content": {"html": "
\u5d4c\u5957\u7c7b
classXmlSuiteResultFormatter.RunHistory<br>\u7528\u4e8e JSON \u8f6c\u6362\u7684\u8f85\u52a9\u5bf9\u8c61\u3002
", "is_complex": false}}, {"type": "table", "raw_content": "

\u5b57\u6bb5

public static final StringNS

public static final StringTEST_RESULT_FILE_NAME

", "content": {"html": "
\u5b57\u6bb5
public static final StringNS
public static final StringTEST_RESULT_FILE_NAME
", "is_complex": false}}, {"type": "table", "raw_content": "

\u516c\u5171\u6784\u9020\u51fd\u6570

XmlSuiteResultFormatter()
", "content": {"html": "
\u516c\u5171\u6784\u9020\u51fd\u6570
XmlSuiteResultFormatter()
", "is_complex": false}}, {"type": "table", "raw_content": "

\u516c\u5171\u65b9\u6cd5

void addBuildInfoAttributes(XmlSerializer serializer, SuiteResultHolder holder)

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002\n\n\n

void addSuiteAttributes(XmlSerializer serializer)

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002\n\n\n

void parseBuildInfoAttributes(XmlPullParser parser, IInvocationContext context)

\u4ece addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n\n

SuiteResultHolder parseResults(File resultDir, boolean shallow)

\u5bf9 writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File) \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa SuiteResultHolder\u3002\n\n\n

void parseSuiteAttributes(XmlPullParser parser, IInvocationContext context)

\u4ece addSuiteAttributes(org.xmlpull.v1.XmlSerializer) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n\n

static String sanitizeXmlContent(String s)

\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002\n\n\n

static String truncateStackTrace(String fullStackTrace, String testCaseName)

\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b ERROR(/STACK_TRACE_MAX_SIZE) \u4e2a\u5b57\u7b26\u3002\n\n\n

File writeResults(SuiteResultHolder holder, File resultDir)

\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002\n\n\n

", "content": {"html": "
\u516c\u5171\u65b9\u6cd5
voidaddBuildInfoAttributes(XmlSerializer serializer, SuiteResultHolder holder)<br>\u5141\u8bb8\u901a\u8fc7<br>serializer.attribute<br>\u5411<br>\u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002
voidaddSuiteAttributes(XmlSerializer serializer)<br>\u5141\u8bb8\u901a\u8fc7<br>serializer.attribute<br>\u5411<br>\u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002
voidparseBuildInfoAttributes(XmlPullParser parser, IInvocationContext context)<br>\u4ece<br>addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder)<br>\u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002
SuiteResultHolderparseResults(File resultDir, boolean shallow)<br>\u5bf9<br>writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File)<br>\u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa<br>SuiteResultHolder<br>\u3002
voidparseSuiteAttributes(XmlPullParser parser, IInvocationContext context)<br>\u4ece<br>addSuiteAttributes(org.xmlpull.v1.XmlSerializer)<br>\u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002
static StringsanitizeXmlContent(String s)<br>\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002
static StringtruncateStackTrace(String fullStackTrace, String testCaseName)<br>\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b<br>ERROR(/STACK_TRACE_MAX_SIZE)<br>\u4e2a\u5b57\u7b26\u3002
FilewriteResults(SuiteResultHolder holder, File resultDir)<br>\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002
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TEST_RESULTS_FILE_NAME

", "content": {"title_content": "TEST_RESULTS_FILE_NAME", "level": "3"}}, {"type": "code", "raw_content": "
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\u516c\u5171\u6784\u9020\u51fd\u6570

", "content": {"title_content": "\u516c\u5171\u6784\u9020\u51fd\u6570", "level": "2"}}, {"type": "title", "raw_content": "

XmlSuiteResultFormatter

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\u516c\u5171\u65b9\u6cd5

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addBuildInfoAttributes

", "content": {"title_content": "addBuildInfoAttributes", "level": "3"}}, {"type": "code", "raw_content": "
\npublic void addBuildInfoAttributes (XmlSerializer serializer,\n                SuiteResultHolder holder)
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serializerXmlSerializer<br>\uff1a\u5bf9 XML \u5957\u4ef6\u7ed3\u679c\u8fdb\u884c\u5e8f\u5217\u5316\u7684\u5bf9\u8c61\u3002
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", "content": {"html": "
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serializerXmlSerializer<br>\uff1a\u5bf9 XML \u5957\u4ef6\u7ed3\u679c\u8fdb\u884c\u5e8f\u5217\u5316\u7684\u5bf9\u8c61\u3002
", "is_complex": false}}, {"type": "title", "raw_content": "

parseBuildInfoAttributes

", "content": {"title_content": "parseBuildInfoAttributes", "level": "3"}}, {"type": "code", "raw_content": "
\npublic void parseBuildInfoAttributes (XmlPullParser parser,\n                IInvocationContext context)
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\u4ece addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002

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parserXmlPullParser<br>\uff1a\u4ece\u4e2d\u8bfb\u53d6\u5c5e\u6027\u7684\u89e3\u6790\u5668\u3002
contextIInvocationContext<br>\uff1a\u7528\u4e8e\u653e\u7f6e\u5c5e\u6027\u7684<br>IInvocationContext<br>\u3002
", "is_complex": false}}, {"type": "table", "raw_content": "
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002
", "content": {"html": "
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002
", "is_complex": false}}, {"type": "title", "raw_content": "

parseResults

", "content": {"title_content": "parseResults", "level": "3"}}, {"type": "code", "raw_content": "
\npublic SuiteResultHolder parseResults (File resultDir,\n                boolean shallow)
", "inline": false, "content": {"code_content": "public SuiteResultHolder parseResults (File resultDir,\n boolean shallow)", "by": "tag_pre"}}, {"type": "paragraph", "raw_content": "

\u5bf9 writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File) \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa SuiteResultHolder\u3002

", "content": [{"c": "\u5bf9", "t": "text"}, {"c": "writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File)", "t": "code-inline"}, {"c": "\u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa", "t": "text"}, {"c": "SuiteResultHolder", "t": "code-inline"}, {"c": "\u3002", "t": "text"}]}, {"type": "table", "raw_content": "
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shallowboolean\uff1a\u4ec5\u52a0\u8f7d SuiteResultHolder \u7684\u9876\u7ea7\u4fe1\u606f\u3002
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\u53c2\u6570
resultDirFile<br>\uff1a\u7528\u4e8e\u67e5\u627e\u7ed3\u679c\u7684\u76ee\u5f55\u3002
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", "is_complex": false}}, {"type": "table", "raw_content": "
\u8fd4\u56de
SuiteResultHolder\u5305\u542b\u7ed3\u679c\u8868\u793a\u6cd5\u7684 SuiteResultHolder\u3002\u6216 null\uff0c\u5982\u679c\n \u51fa\u73b0\u4efb\u4f55\u95ee\u9898\u3002
", "content": {"html": "
\u8fd4\u56de
SuiteResultHolder\u5305\u542b\u7ed3\u679c\u8868\u793a\u6cd5\u7684 <br>SuiteResultHolder<br>\u3002\u6216 null\uff0c\u5982\u679c\n \u51fa\u73b0\u4efb\u4f55\u95ee\u9898\u3002
", "is_complex": false}}, {"type": "title", "raw_content": "

parseSuite \u5c5e\u6027

", "content": {"title_content": "parseSuite \u5c5e\u6027", "level": "3"}}, {"type": "code", "raw_content": "
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\u4ece addSuiteAttributes(org.xmlpull.v1.XmlSerializer) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002

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XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002
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XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002
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sanitizeXmlContent

", "content": {"title_content": "sanitizeXmlContent", "level": "3"}}, {"type": "code", "raw_content": "
\npublic static String sanitizeXmlContent (String s)
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\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002

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sString
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String
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String
", "is_complex": false}}, {"type": "title", "raw_content": "

truncateStackTrace

", "content": {"title_content": "truncateStackTrace", "level": "3"}}, {"type": "code", "raw_content": "
\npublic static String truncateStackTrace (String fullStackTrace,\n                String testCaseName)
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\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b ERROR(/STACK_TRACE_MAX_SIZE) \u4e2a\u5b57\u7b26\u3002

", "content": [{"c": "\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b", "t": "text"}, {"c": "ERROR(/STACK_TRACE_MAX_SIZE)", "t": "code-inline"}, {"c": "\u4e2a\u5b57\u7b26\u3002", "t": "text"}]}, {"type": "table", "raw_content": "
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testCaseNameString
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String
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String
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writeResults

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\npublic File writeResults (SuiteResultHolder holder,\n                File resultDir)
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\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002

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resultDirFile\uff1a\u7528\u4e8e\u5b58\u653e\u7ed3\u679c\u7684\u7ed3\u679c\u76ee\u5f55 ERROR(/File)\u3002
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holderSuiteResultHolder<br>\uff1a\u4e00\u4e2a<br>SuiteResultHolder<br>\uff0c\u5305\u542b xml \u6240\u9700\u7684\u6240\u6709\u4fe1\u606f
resultDirFile<br>\uff1a\u7528\u4e8e\u5b58\u653e\u7ed3\u679c\u7684\u7ed3\u679c\u76ee\u5f55<br>ERROR(/File)<br>\u3002
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File\u6307\u5411 xml \u8f93\u51fa\u6587\u4ef6\u7684 ERROR(/File)\u3002
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File\u6307\u5411 xml \u8f93\u51fa\u6587\u4ef6\u7684 <br>ERROR(/File)<br>\u3002
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XmlSuiteResultFormatter

\n\n public\n\n\n\n class\n XmlSuiteResultFormatter\n\n
\n\n\n\n\n extends Object\n\n\n\n\n\n\n\n\n implements\n\n IFormatterGenerator\n\n\n\n\n\n

java.lang.Object \u4e2d\n
\u21b3com.android.tradefed.result.suite.XmlSuiteResultFormatter\n

\u7528\u4e8e\u5c06\u5957\u4ef6\u8fd0\u884c\u4f5c\u4e1a\u4fdd\u5b58\u4e3a XML \u7684\u5b9e\u7528\u7a0b\u5e8f\u7c7b\u3002TODO\uff1a\u79fb\u9664\u6240\u6709\u7279\u6b8a\u7684\u517c\u5bb9\u6027\u6d4b\u8bd5\n \u83b7\u5f97\u76f8\u540c\u7684\u683c\u5f0f\u3002

\u6458\u8981

\u5d4c\u5957\u7c7b

classXmlSuiteResultFormatter.RunHistory

\u7528\u4e8e JSON \u8f6c\u6362\u7684\u8f85\u52a9\u5bf9\u8c61\u3002\n\n\n

\u5b57\u6bb5

public static final StringNS

public static final StringTEST_RESULT_FILE_NAME

\u516c\u5171\u6784\u9020\u51fd\u6570

XmlSuiteResultFormatter()

\u516c\u5171\u65b9\u6cd5

void addBuildInfoAttributes(XmlSerializer serializer, SuiteResultHolder holder)

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002\n\n\n

void addSuiteAttributes(XmlSerializer serializer)

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002\n\n\n

void parseBuildInfoAttributes(XmlPullParser parser, IInvocationContext context)

\u4ece addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n\n

SuiteResultHolder parseResults(File resultDir, boolean shallow)

\u5bf9 writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File) \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa SuiteResultHolder\u3002\n\n\n

void parseSuiteAttributes(XmlPullParser parser, IInvocationContext context)

\u4ece addSuiteAttributes(org.xmlpull.v1.XmlSerializer) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n\n

static String sanitizeXmlContent(String s)

\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002\n\n\n

static String truncateStackTrace(String fullStackTrace, String testCaseName)

\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b ERROR(/STACK_TRACE_MAX_SIZE) \u4e2a\u5b57\u7b26\u3002\n\n\n

File writeResults(SuiteResultHolder holder, File resultDir)

\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002\n\n\n

\u5b57\u6bb5

\u5317\u7f8e\u6d32

\npublic static final String NS

TEST_RESULTS_FILE_NAME

\npublic static final String TEST_RESULT_FILE_NAME

\u516c\u5171\u6784\u9020\u51fd\u6570

XmlSuiteResultFormatter

\npublic XmlSuiteResultFormatter ()

\u516c\u5171\u65b9\u6cd5

addBuildInfoAttributes

\npublic void addBuildInfoAttributes (XmlSerializer serializer,\n                SuiteResultHolder holder)

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002

\u53c2\u6570
serializerXmlSerializer\uff1a\u5bf9 XML \u5957\u4ef6\u7ed3\u679c\u8fdb\u884c\u5e8f\u5217\u5316\u7684\u5bf9\u8c61\u3002
holderSuiteResultHolder\uff1a\u4e00\u4e2a\u5bf9\u8c61\uff0c\u5176\u4e2d\u5305\u542b\u8981\u5199\u5165\u5957\u4ef6\u7ed3\u679c\u7684\u4fe1\u606f\u3002

\u6dfb\u52a0\u5957\u4ef6\u5c5e\u6027

\npublic void addSuiteAttributes (XmlSerializer serializer)

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002

\u53c2\u6570
serializerXmlSerializer\uff1a\u5bf9 XML \u5957\u4ef6\u7ed3\u679c\u8fdb\u884c\u5e8f\u5217\u5316\u7684\u5bf9\u8c61\u3002

parseBuildInfoAttributes

\npublic void parseBuildInfoAttributes (XmlPullParser parser,\n                IInvocationContext context)

\u4ece addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002

\u53c2\u6570
parserXmlPullParser\uff1a\u4ece\u4e2d\u8bfb\u53d6\u5c5e\u6027\u7684\u89e3\u6790\u5668\u3002
contextIInvocationContext\uff1a\u7528\u4e8e\u653e\u7f6e\u5c5e\u6027\u7684 IInvocationContext\u3002
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002

parseResults

\npublic SuiteResultHolder parseResults (File resultDir,\n                boolean shallow)

\u5bf9 writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File) \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa SuiteResultHolder\u3002

\u53c2\u6570
resultDirFile\uff1a\u7528\u4e8e\u67e5\u627e\u7ed3\u679c\u7684\u76ee\u5f55\u3002
shallowboolean\uff1a\u4ec5\u52a0\u8f7d SuiteResultHolder \u7684\u9876\u7ea7\u4fe1\u606f\u3002
\u8fd4\u56de
SuiteResultHolder\u5305\u542b\u7ed3\u679c\u8868\u793a\u6cd5\u7684 SuiteResultHolder\u3002\u6216 null\uff0c\u5982\u679c\n \u51fa\u73b0\u4efb\u4f55\u95ee\u9898\u3002

parseSuite \u5c5e\u6027

\npublic void parseSuiteAttributes (XmlPullParser parser,\n                IInvocationContext context)

\u4ece addSuiteAttributes(org.xmlpull.v1.XmlSerializer) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002

\u53c2\u6570
parserXmlPullParser\uff1a\u4ece\u4e2d\u8bfb\u53d6\u5c5e\u6027\u7684\u89e3\u6790\u5668\u3002
contextIInvocationContext\uff1a\u7528\u4e8e\u653e\u7f6e\u5c5e\u6027\u7684 IInvocationContext\u3002
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002

sanitizeXmlContent

\npublic static String sanitizeXmlContent (String s)

\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002

\u53c2\u6570
sString
\u8fd4\u56de
String

truncateStackTrace

\npublic static String truncateStackTrace (String fullStackTrace,\n                String testCaseName)

\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b ERROR(/STACK_TRACE_MAX_SIZE) \u4e2a\u5b57\u7b26\u3002

\u53c2\u6570
fullStackTraceString
testCaseNameString
\u8fd4\u56de
String

writeResults

\npublic File writeResults (SuiteResultHolder holder,\n                File resultDir)

\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002

\u53c2\u6570
holderSuiteResultHolder\uff1a\u4e00\u4e2a SuiteResultHolder\uff0c\u5305\u542b xml \u6240\u9700\u7684\u6240\u6709\u4fe1\u606f
resultDirFile\uff1a\u7528\u4e8e\u5b58\u653e\u7ed3\u679c\u7684\u7ed3\u679c\u76ee\u5f55 ERROR(/File)\u3002
\u8fd4\u56de
File\u6307\u5411 xml \u8f93\u51fa\u6587\u4ef6\u7684 ERROR(/File)\u3002
", "statics": {"title": 16, "code": 12, "table": 19, "paragraph": 9, "paragraph.text": 16, "paragraph.code-inline": 7}, "url": "https://source.android.com/reference/tradefed/com/android/tradefed/result/suite/XmlSuiteResultFormatter?hl=zh-cn", "content": "# XmlSuiteResultFormatter\n\n```\n public class XmlSuiteResultFormatter\n extends Object implements IFormatterGenerator\n```\n\n
java.lang.Object \u4e2d
\u21b3com.android.tradefed.result.suite.XmlSuiteResultFormatter
\n\n\u7528\u4e8e\u5c06\u5957\u4ef6\u8fd0\u884c\u4f5c\u4e1a\u4fdd\u5b58\u4e3a XML \u7684\u5b9e\u7528\u7a0b\u5e8f\u7c7b\u3002TODO\uff1a\u79fb\u9664\u6240\u6709\u7279\u6b8a\u7684\u517c\u5bb9\u6027\u6d4b\u8bd5\n \u83b7\u5f97\u76f8\u540c\u7684\u683c\u5f0f\u3002\n\n## \u6458\u8981\n\n| \u5d4c\u5957\u7c7b | |\n|---|---|\n| class | XmlSuiteResultFormatter.RunHistory
\u7528\u4e8e JSON \u8f6c\u6362\u7684\u8f85\u52a9\u5bf9\u8c61\u3002 |\n\n| \u5b57\u6bb5 | |\n|---|---|\n| public static final String | NS |\n| public static final String | TEST_RESULT_FILE_NAME |\n\n| \u516c\u5171\u6784\u9020\u51fd\u6570 |\n|---|\n| XmlSuiteResultFormatter() |\n\n| \u516c\u5171\u65b9\u6cd5 | |\n|---|---|\n| void | addBuildInfoAttributes(XmlSerializer serializer, SuiteResultHolder holder)
\u5141\u8bb8\u901a\u8fc7
serializer.attribute
\u5411
\u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002 |\n| void | addSuiteAttributes(XmlSerializer serializer)
\u5141\u8bb8\u901a\u8fc7
serializer.attribute
\u5411
\u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002 |\n| void | parseBuildInfoAttributes(XmlPullParser parser, IInvocationContext context)
\u4ece
addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder)
\u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002 |\n| SuiteResultHolder | parseResults(File resultDir, boolean shallow)
\u5bf9
writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File)
\u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa
SuiteResultHolder
\u3002 |\n| void | parseSuiteAttributes(XmlPullParser parser, IInvocationContext context)
\u4ece
addSuiteAttributes(org.xmlpull.v1.XmlSerializer)
\u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002 |\n| static String | sanitizeXmlContent(String s)
\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002 |\n| static String | truncateStackTrace(String fullStackTrace, String testCaseName)
\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b
ERROR(/STACK_TRACE_MAX_SIZE)
\u4e2a\u5b57\u7b26\u3002 |\n| File | writeResults(SuiteResultHolder holder, File resultDir)
\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002 |\n\n## \u5b57\u6bb5\n\n### \u5317\u7f8e\u6d32\n\n```\npublic static final String NS\n```\n\n### TEST_RESULTS_FILE_NAME\n\n```\npublic static final String TEST_RESULT_FILE_NAME\n```\n\n## \u516c\u5171\u6784\u9020\u51fd\u6570\n\n### XmlSuiteResultFormatter\n\n```\npublic XmlSuiteResultFormatter ()\n```\n\n## \u516c\u5171\u65b9\u6cd5\n\n### addBuildInfoAttributes\n\n```\npublic void addBuildInfoAttributes (XmlSerializer serializer,\n SuiteResultHolder holder)\n```\n\n\u5141\u8bb8\u901a\u8fc7 `serializer.attribute` \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002\n\n| \u53c2\u6570 | |\n|---|---|\n| serializer | XmlSerializer
\uff1a\u5bf9 XML \u5957\u4ef6\u7ed3\u679c\u8fdb\u884c\u5e8f\u5217\u5316\u7684\u5bf9\u8c61\u3002 |\n| holder | SuiteResultHolder
\uff1a\u4e00\u4e2a\u5bf9\u8c61\uff0c\u5176\u4e2d\u5305\u542b\u8981\u5199\u5165\u5957\u4ef6\u7ed3\u679c\u7684\u4fe1\u606f\u3002 |\n\n### \u6dfb\u52a0\u5957\u4ef6\u5c5e\u6027\n\n```\npublic void addSuiteAttributes (XmlSerializer serializer)\n```\n\n\u5141\u8bb8\u901a\u8fc7 `serializer.attribute` \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002\n\n| \u53c2\u6570 | |\n|---|---|\n| serializer | XmlSerializer
\uff1a\u5bf9 XML \u5957\u4ef6\u7ed3\u679c\u8fdb\u884c\u5e8f\u5217\u5316\u7684\u5bf9\u8c61\u3002 |\n\n### parseBuildInfoAttributes\n\n```\npublic void parseBuildInfoAttributes (XmlPullParser parser,\n IInvocationContext context)\n```\n\n\u4ece `addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder)` \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n| \u53c2\u6570 | |\n|---|---|\n| parser | XmlPullParser
\uff1a\u4ece\u4e2d\u8bfb\u53d6\u5c5e\u6027\u7684\u89e3\u6790\u5668\u3002 |\n| context | IInvocationContext
\uff1a\u7528\u4e8e\u653e\u7f6e\u5c5e\u6027\u7684
IInvocationContext
\u3002 |\n\n| \u629b\u51fa | |\n|---|---|\n| XmlPullParserException | \u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002 |\n\n### parseResults\n\n```\npublic SuiteResultHolder parseResults (File resultDir,\n boolean shallow)\n```\n\n\u5bf9 `writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File)` \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa `SuiteResultHolder` \u3002\n\n| \u53c2\u6570 | |\n|---|---|\n| resultDir | File
\uff1a\u7528\u4e8e\u67e5\u627e\u7ed3\u679c\u7684\u76ee\u5f55\u3002 |\n| shallow | boolean
\uff1a\u4ec5\u52a0\u8f7d
SuiteResultHolder
\u7684\u9876\u7ea7\u4fe1\u606f\u3002 |\n\n| \u8fd4\u56de | |\n|---|---|\n| SuiteResultHolder | \u5305\u542b\u7ed3\u679c\u8868\u793a\u6cd5\u7684
SuiteResultHolder
\u3002\u6216 null\uff0c\u5982\u679c\n \u51fa\u73b0\u4efb\u4f55\u95ee\u9898\u3002 |\n\n### parseSuite \u5c5e\u6027\n\n```\npublic void parseSuiteAttributes (XmlPullParser parser,\n IInvocationContext context)\n```\n\n\u4ece `addSuiteAttributes(org.xmlpull.v1.XmlSerializer)` \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n| \u53c2\u6570 | |\n|---|---|\n| parser | XmlPullParser
\uff1a\u4ece\u4e2d\u8bfb\u53d6\u5c5e\u6027\u7684\u89e3\u6790\u5668\u3002 |\n| context | IInvocationContext
\uff1a\u7528\u4e8e\u653e\u7f6e\u5c5e\u6027\u7684
IInvocationContext
\u3002 |\n\n| \u629b\u51fa | |\n|---|---|\n| XmlPullParserException | \u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002 |\n\n### sanitizeXmlContent\n\n```\npublic static String sanitizeXmlContent (String s)\n```\n\n\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002\n\n| \u53c2\u6570 | |\n|---|---|\n| s | String |\n\n| \u8fd4\u56de | |\n|---|---|\n| String | |\n\n### truncateStackTrace\n\n```\npublic static String truncateStackTrace (String fullStackTrace,\n String testCaseName)\n```\n\n\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b `ERROR(/STACK_TRACE_MAX_SIZE)` \u4e2a\u5b57\u7b26\u3002\n\n| \u53c2\u6570 | |\n|---|---|\n| fullStackTrace | String |\n| testCaseName | String |\n\n| \u8fd4\u56de | |\n|---|---|\n| String | |\n\n### writeResults\n\n```\npublic File writeResults (SuiteResultHolder holder,\n File resultDir)\n```\n\n\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002\n\n| \u53c2\u6570 | |\n|---|---|\n| holder | SuiteResultHolder
\uff1a\u4e00\u4e2a
SuiteResultHolder
\uff0c\u5305\u542b xml \u6240\u9700\u7684\u6240\u6709\u4fe1\u606f |\n| resultDir | File
\uff1a\u7528\u4e8e\u5b58\u653e\u7ed3\u679c\u7684\u7ed3\u679c\u76ee\u5f55
ERROR(/File)
\u3002 |\n\n| \u8fd4\u56de | |\n|---|---|\n| File | \u6307\u5411 xml \u8f93\u51fa\u6587\u4ef6\u7684
ERROR(/File)
\u3002 |\n", "html": "\n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n XmlSuiteResultFormatter  |  Android Open Source Project\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n \n \n\n
\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
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\n\n\n \n\n\n \n\n \n\n \u4f7f\u7528\u96c6\u5408\u8ba9\u4e00\u5207\u4e95\u4e95\u6709\u6761\n \n \n\n \u6839\u636e\u60a8\u7684\u504f\u597d\u4fdd\u5b58\u5185\u5bb9\u5e76\u5bf9\u5176\u8fdb\u884c\u5206\u7c7b\u3002\n \n \n\n\n \n \n\n\n\n\n\n\n\n\n\n\n\n
\n\n\n\n\n\n
\n\n \n\n\n\n\n\n
\n\n\n\n

XmlSuiteResultFormatter

\n

\n\n public\n\n\n\n class\n XmlSuiteResultFormatter\n\n
\n\n\n\n\n extends Object\n\n\n\n\n\n\n\n\n implements\n\n IFormatterGenerator\n\n\n\n\n\n

\n\n\n \n\n \n \n\n\n \n\n \n\n \n \n\n\n
java.lang.Object \u4e2d\n
\u21b3com.android.tradefed.result.suite.XmlSuiteResultFormatter\n
\n\n\n\n\n
\n\n\n

\u7528\u4e8e\u5c06\u5957\u4ef6\u8fd0\u884c\u4f5c\u4e1a\u4fdd\u5b58\u4e3a XML \u7684\u5b9e\u7528\u7a0b\u5e8f\u7c7b\u3002TODO\uff1a\u79fb\u9664\u6240\u6709\u7279\u6b8a\u7684\u517c\u5bb9\u6027\u6d4b\u8bd5\n \u83b7\u5f97\u76f8\u540c\u7684\u683c\u5f0f\u3002

\n\n\n\n\n\n\n\n\n\n

\u6458\u8981

\n\n\n\n\n\n\n\n\n\n \n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\u5d4c\u5957\u7c7b

\n\n\n\n\n class\n XmlSuiteResultFormatter.RunHistory\n

\u7528\u4e8e JSON \u8f6c\u6362\u7684\u8f85\u52a9\u5bf9\u8c61\u3002\n\n\n

\n
\n\n\n\n\n \n \n \n \n\n\n \n \n \n \n\n\n\n

\u5b57\u6bb5

\n public\n static\n final\n String\n NS\n

\n\n\n

\n
\n public\n static\n final\n String\n TEST_RESULT_FILE_NAME\n

\n\n\n

\n
\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n \n \n\n\n\n

\u516c\u5171\u6784\u9020\u51fd\u6570

\n \n XmlSuiteResultFormatter()\n \n\n
\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n\n

\u516c\u5171\u65b9\u6cd5

\n\n\n\n\n\n void\n \n \n addBuildInfoAttributes(XmlSerializer serializer, SuiteResultHolder holder)\n \n\n

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\n\n
\n\n\n\n\n\n void\n \n \n addSuiteAttributes(XmlSerializer serializer)\n \n\n

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002\n\n\n

\n\n
\n\n\n\n\n\n void\n \n \n parseBuildInfoAttributes(XmlPullParser parser, IInvocationContext context)\n \n\n

\u4ece addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n\n

\n\n
\n\n\n\n\n\n SuiteResultHolder\n \n \n parseResults(File resultDir, boolean shallow)\n \n\n

\u5bf9 writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File) \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa SuiteResultHolder\u3002\n\n\n

\n\n
\n\n\n\n\n\n void\n \n \n parseSuiteAttributes(XmlPullParser parser, IInvocationContext context)\n \n\n

\u4ece addSuiteAttributes(org.xmlpull.v1.XmlSerializer) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n\n

\n\n
\n\n\n static\n\n\n String\n \n \n sanitizeXmlContent(String s)\n \n\n

\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002\n\n\n

\n\n
\n\n\n static\n\n\n String\n \n \n truncateStackTrace(String fullStackTrace, String testCaseName)\n \n\n

\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b ERROR(/STACK_TRACE_MAX_SIZE) \u4e2a\u5b57\u7b26\u3002\n\n\n

\n\n
\n\n\n\n\n\n File\n \n \n writeResults(SuiteResultHolder holder, File resultDir)\n \n\n

\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002\n\n\n

\n\n
\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\u5b57\u6bb5

\n\n\n
\n

\u5317\u7f8e\u6d32

\n
\n\n\n\n\n\n
\n
\npublic static final String NS
\n\n\n\n\n

\n\n\n
\n\n
\n

TEST_RESULTS_FILE_NAME

\n
\n\n\n\n\n\n
\n
\npublic static final String TEST_RESULT_FILE_NAME
\n\n\n\n\n

\n\n\n
\n\n\n\n\n\n\n\n\n

\u516c\u5171\u6784\u9020\u51fd\u6570

\n\n\n
\n

XmlSuiteResultFormatter

\n
\n
\n
\n\n\n\n
\n
\npublic XmlSuiteResultFormatter ()
\n\n\n\n\n

\n\n
\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\u516c\u5171\u65b9\u6cd5

\n\n\n
\n

addBuildInfoAttributes

\n
\n
\n
\n\n\n\n
\n
\npublic void addBuildInfoAttributes (XmlSerializer serializer,\n                SuiteResultHolder holder)
\n\n\n\n\n

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002

\n \n \n \n \n \n \n \n \n \n \n
\u53c2\u6570
serializer\n XmlSerializer\uff1a\u5bf9 XML \u5957\u4ef6\u7ed3\u679c\u8fdb\u884c\u5e8f\u5217\u5316\u7684\u5bf9\u8c61\u3002

holder\n SuiteResultHolder\uff1a\u4e00\u4e2a\u5bf9\u8c61\uff0c\u5176\u4e2d\u5305\u542b\u8981\u5199\u5165\u5957\u4ef6\u7ed3\u679c\u7684\u4fe1\u606f\u3002

\n\n
\n\n
\n

\u6dfb\u52a0\u5957\u4ef6\u5c5e\u6027

\n
\n
\n
\n\n\n\n
\n
\npublic void addSuiteAttributes (XmlSerializer serializer)
\n\n\n\n\n

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002

\n \n \n \n \n \n \n
\u53c2\u6570
serializer\n XmlSerializer\uff1a\u5bf9 XML \u5957\u4ef6\u7ed3\u679c\u8fdb\u884c\u5e8f\u5217\u5316\u7684\u5bf9\u8c61\u3002

\n\n
\n\n
\n

parseBuildInfoAttributes

\n
\n
\n
\n\n\n\n
\n
\npublic void parseBuildInfoAttributes (XmlPullParser parser,\n                IInvocationContext context)
\n\n\n\n\n

\u4ece addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002

\n \n \n \n \n \n \n \n \n \n \n
\u53c2\u6570
parser\n XmlPullParser\uff1a\u4ece\u4e2d\u8bfb\u53d6\u5c5e\u6027\u7684\u89e3\u6790\u5668\u3002

context\n IInvocationContext\uff1a\u7528\u4e8e\u653e\u7f6e\u5c5e\u6027\u7684 IInvocationContext\u3002

\n \n \n \n \n \n \n
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002
\n\n\n
\n\n
\n

parseResults

\n
\n
\n
\n\n\n\n
\n
\npublic SuiteResultHolder parseResults (File resultDir,\n                boolean shallow)
\n\n\n\n\n

\u5bf9 writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File) \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa SuiteResultHolder\u3002

\n \n \n \n \n \n \n \n \n \n \n
\u53c2\u6570
resultDir\n File\uff1a\u7528\u4e8e\u67e5\u627e\u7ed3\u679c\u7684\u76ee\u5f55\u3002

shallow\n boolean\uff1a\u4ec5\u52a0\u8f7d SuiteResultHolder \u7684\u9876\u7ea7\u4fe1\u606f\u3002

\n \n \n \n \n \n \n
\u8fd4\u56de
SuiteResultHolder\u5305\u542b\u7ed3\u679c\u8868\u793a\u6cd5\u7684 SuiteResultHolder\u3002\u6216 null\uff0c\u5982\u679c\n \u51fa\u73b0\u4efb\u4f55\u95ee\u9898\u3002

\n\n
\n\n
\n

parseSuite \u5c5e\u6027

\n
\n
\n
\n\n\n\n
\n
\npublic void parseSuiteAttributes (XmlPullParser parser,\n                IInvocationContext context)
\n\n\n\n\n

\u4ece addSuiteAttributes(org.xmlpull.v1.XmlSerializer) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002

\n \n \n \n \n \n \n \n \n \n \n
\u53c2\u6570
parser\n XmlPullParser\uff1a\u4ece\u4e2d\u8bfb\u53d6\u5c5e\u6027\u7684\u89e3\u6790\u5668\u3002

context\n IInvocationContext\uff1a\u7528\u4e8e\u653e\u7f6e\u5c5e\u6027\u7684 IInvocationContext\u3002

\n \n \n \n \n \n \n
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002
\n\n\n
\n\n
\n

sanitizeXmlContent

\n
\n
\n
\n\n\n\n
\n
\npublic static String sanitizeXmlContent (String s)
\n\n\n\n\n

\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002

\n \n \n \n \n \n \n
\u53c2\u6570
s\n String

\n \n \n \n \n \n \n
\u8fd4\u56de
String

\n\n
\n\n
\n

truncateStackTrace

\n
\n
\n
\n\n\n\n
\n
\npublic static String truncateStackTrace (String fullStackTrace,\n                String testCaseName)
\n\n\n\n\n

\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b ERROR(/STACK_TRACE_MAX_SIZE) \u4e2a\u5b57\u7b26\u3002

\n \n \n \n \n \n \n \n \n \n \n
\u53c2\u6570
fullStackTrace\n String

testCaseName\n String

\n \n \n \n \n \n \n
\u8fd4\u56de
String

\n\n
\n\n
\n

writeResults

\n
\n
\n
\n\n\n\n
\n
\npublic File writeResults (SuiteResultHolder holder,\n                File resultDir)
\n\n\n\n\n

\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002

\n \n \n \n \n \n \n \n \n \n \n
\u53c2\u6570
holder\n SuiteResultHolder\uff1a\u4e00\u4e2a SuiteResultHolder\uff0c\u5305\u542b xml \u6240\u9700\u7684\u6240\u6709\u4fe1\u606f

resultDir\n File\uff1a\u7528\u4e8e\u5b58\u653e\u7ed3\u679c\u7684\u7ed3\u679c\u76ee\u5f55 ERROR(/File)\u3002

\n \n \n \n \n \n \n
\u8fd4\u56de
File\u6307\u5411 xml \u8f93\u51fa\u6587\u4ef6\u7684 ERROR(/File)\u3002

\n\n
\n\n\n\n\n\n\n\n\n\n\n\n
\n\n\n
\n\n\n\n\n\n\n
\n\n\n\n\n\n\n\n \n \n\n\n\n\n\n\n\n
\n\n\n
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\n\n\n\n

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\n

\u6700\u540e\u66f4\u65b0\u65f6\u95f4 (UTC)\uff1a2024-08-22\u3002

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XmlSuiteResultFormatter

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classXmlSuiteResultFormatter.RunHistory

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voidaddBuildInfoAttributes(XmlSerializer serializer, SuiteResultHolder holder)

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voidaddSuiteAttributes(XmlSerializer serializer)

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voidparseBuildInfoAttributes(XmlPullParser parser, IInvocationContext context)

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SuiteResultHolderparseResults(File resultDir, boolean shallow)

\u5bf9 writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File) \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa SuiteResultHolder\u3002\n\n\n

voidparseSuiteAttributes(XmlPullParser parser, IInvocationContext context)

\u4ece addSuiteAttributes(org.xmlpull.v1.XmlSerializer) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n\n

static StringsanitizeXmlContent(String s)

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static StringtruncateStackTrace(String fullStackTrace, String testCaseName)

\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b ERROR(/STACK_TRACE_MAX_SIZE) \u4e2a\u5b57\u7b26\u3002\n\n\n

FilewriteResults(SuiteResultHolder holder, File resultDir)

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voidaddBuildInfoAttributes(XmlSerializer serializer, SuiteResultHolder holder)<br>serializer.attribute
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voidparseBuildInfoAttributes(XmlPullParser parser, IInvocationContext context)<br>addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder)
SuiteResultHolderparseResults(File resultDir, boolean shallow)<br>writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File)<br>SuiteResultHolder
voidparseSuiteAttributes(XmlPullParser parser, IInvocationContext context)<br>addSuiteAttributes(org.xmlpull.v1.XmlSerializer)
static StringsanitizeXmlContent(String s)
static StringtruncateStackTrace(String fullStackTrace, String testCaseName)<br>ERROR(/STACK_TRACE_MAX_SIZE)
FilewriteResults(SuiteResultHolder holder, File resultDir)
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TEST_RESULTS_FILE_NAME

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XmlSuiteResultFormatter

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addBuildInfoAttributes

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parseBuildInfoAttributes

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parseResults

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SuiteResultHolderSuiteResultHolder
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parseSuite \u5c5e\u6027

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sanitizeXmlContent

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truncateStackTrace

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writeResults

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resultDirFile\uff1a\u7528\u4e8e\u5b58\u653e\u7ed3\u679c\u7684\u7ed3\u679c\u76ee\u5f55 ERROR(/File)\u3002
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XmlSuiteResultFormatter

\n\n public\n\n\n\n class\n XmlSuiteResultFormatter\n\n
\n\n\n\n\n extends Object\n\n\n\n\n\n\n\n\n implements\n\n IFormatterGenerator\n\n\n\n\n\n

java.lang.Object \u4e2d\n
\u21b3com.android.tradefed.result.suite.XmlSuiteResultFormatter\n

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classXmlSuiteResultFormatter.RunHistory

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\u5b57\u6bb5

\n \n public\n static\n final\n String\n \n NS\n

\n\n\n

\n \n \n\n\n \n \n public\n static\n final\n String\n \n TEST_RESULT_FILE_NAME\n

\n\n\n

\n \n \n\n\n\n

\u516c\u5171\u6784\u9020\u51fd\u6570

\n XmlSuiteResultFormatter()\n \n\n

\u516c\u5171\u65b9\u6cd5

voidaddBuildInfoAttributes(XmlSerializer serializer, SuiteResultHolder holder)

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voidaddSuiteAttributes(XmlSerializer serializer)

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voidparseBuildInfoAttributes(XmlPullParser parser, IInvocationContext context)

\u4ece addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n\n

SuiteResultHolderparseResults(File resultDir, boolean shallow)

\u5bf9 writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File) \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa SuiteResultHolder\u3002\n\n\n

voidparseSuiteAttributes(XmlPullParser parser, IInvocationContext context)

\u4ece addSuiteAttributes(org.xmlpull.v1.XmlSerializer) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n\n

static StringsanitizeXmlContent(String s)

\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002\n\n\n

static StringtruncateStackTrace(String fullStackTrace, String testCaseName)

\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b ERROR(/STACK_TRACE_MAX_SIZE) \u4e2a\u5b57\u7b26\u3002\n\n\n

FilewriteResults(SuiteResultHolder holder, File resultDir)

\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002\n\n\n

\u5b57\u6bb5

\u5317\u7f8e\u6d32

\npublic static final String NS

TEST_RESULTS_FILE_NAME

\npublic static final String TEST_RESULT_FILE_NAME

\u516c\u5171\u6784\u9020\u51fd\u6570

XmlSuiteResultFormatter

\npublic XmlSuiteResultFormatter ()

\u516c\u5171\u65b9\u6cd5

addBuildInfoAttributes

\npublic void addBuildInfoAttributes (XmlSerializer serializer,\n                SuiteResultHolder holder)

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002

\u53c2\u6570
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holderSuiteResultHolder\uff1a\u4e00\u4e2a\u5bf9\u8c61\uff0c\u5176\u4e2d\u5305\u542b\u8981\u5199\u5165\u5957\u4ef6\u7ed3\u679c\u7684\u4fe1\u606f\u3002

\u6dfb\u52a0\u5957\u4ef6\u5c5e\u6027

\npublic void addSuiteAttributes (XmlSerializer serializer)

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\u53c2\u6570
serializerXmlSerializer\uff1a\u5bf9 XML \u5957\u4ef6\u7ed3\u679c\u8fdb\u884c\u5e8f\u5217\u5316\u7684\u5bf9\u8c61\u3002

parseBuildInfoAttributes

\npublic void parseBuildInfoAttributes (XmlPullParser parser,\n                IInvocationContext context)

\u4ece addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002

\u53c2\u6570
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contextIInvocationContext\uff1a\u7528\u4e8e\u653e\u7f6e\u5c5e\u6027\u7684 IInvocationContext\u3002
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002

parseResults

\npublic SuiteResultHolder parseResults (File resultDir,\n                boolean shallow)

\u5bf9 writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File) \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa SuiteResultHolder\u3002

\u53c2\u6570
resultDirFile\uff1a\u7528\u4e8e\u67e5\u627e\u7ed3\u679c\u7684\u76ee\u5f55\u3002
shallowboolean\uff1a\u4ec5\u52a0\u8f7d SuiteResultHolder \u7684\u9876\u7ea7\u4fe1\u606f\u3002
\u8fd4\u56de
SuiteResultHolder\u5305\u542b\u7ed3\u679c\u8868\u793a\u6cd5\u7684 SuiteResultHolder\u3002\u6216 null\uff0c\u5982\u679c\n \u51fa\u73b0\u4efb\u4f55\u95ee\u9898\u3002

parseSuite \u5c5e\u6027

\npublic void parseSuiteAttributes (XmlPullParser parser,\n                IInvocationContext context)

\u4ece addSuiteAttributes(org.xmlpull.v1.XmlSerializer) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002

\u53c2\u6570
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contextIInvocationContext\uff1a\u7528\u4e8e\u653e\u7f6e\u5c5e\u6027\u7684 IInvocationContext\u3002
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002

sanitizeXmlContent

\npublic static String sanitizeXmlContent (String s)

\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002

\u53c2\u6570
\n s\n \n String \n \n
\u8fd4\u56de
String

truncateStackTrace

\npublic static String truncateStackTrace (String fullStackTrace,\n                String testCaseName)

\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b ERROR(/STACK_TRACE_MAX_SIZE) \u4e2a\u5b57\u7b26\u3002

\u53c2\u6570
\n fullStackTrace\n \n String \n \n \n testCaseName\n \n String \n \n
\u8fd4\u56de
String

writeResults

\npublic File writeResults (SuiteResultHolder holder,\n                File resultDir)

\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002

\u53c2\u6570
holderSuiteResultHolder\uff1a\u4e00\u4e2a SuiteResultHolder\uff0c\u5305\u542b xml \u6240\u9700\u7684\u6240\u6709\u4fe1\u606f
resultDirFile\uff1a\u7528\u4e8e\u5b58\u653e\u7ed3\u679c\u7684\u7ed3\u679c\u76ee\u5f55 ERROR(/File)\u3002
\u8fd4\u56de
File\u6307\u5411 xml \u8f93\u51fa\u6587\u4ef6\u7684 ERROR(/File)\u3002
", "statics": {"title": 16, "code": 20, "table": 25, "table.complex": 19, "paragraph": 9, "paragraph.text": 16, "paragraph.code-inline": 7}, "url": "https://source.android.com/reference/tradefed/com/android/tradefed/result/suite/XmlSuiteResultFormatter?hl=zh-cn", "content": "# XmlSuiteResultFormatter\n\n```\npublic class XmlSuiteResultFormatter\nextends Object implements IFormatterGenerator\n```\n\n
java.lang.Object \u4e2d
\u21b3com.android.tradefed.result.suite.XmlSuiteResultFormatter
\n\n\u7528\u4e8e\u5c06\u5957\u4ef6\u8fd0\u884c\u4f5c\u4e1a\u4fdd\u5b58\u4e3a XML \u7684\u5b9e\u7528\u7a0b\u5e8f\u7c7b\u3002TODO\uff1a\u79fb\u9664\u6240\u6709\u7279\u6b8a\u7684\u517c\u5bb9\u6027\u6d4b\u8bd5\n \u83b7\u5f97\u76f8\u540c\u7684\u683c\u5f0f\u3002\n\n## \u6458\u8981\n\n
\u5d4c\u5957\u7c7b
classXmlSuiteResultFormatter.RunHistory
\n\n\u5b57\u6bb5\n\n```\npublic static final String NS\n```\n\n```\npublic static final String TEST_RESULT_FILE_NAME\n```\n\n
\u516c\u5171\u6784\u9020\u51fd\u6570
\n\n```\nXmlSuiteResultFormatter()\n```\n\n
\u516c\u5171\u65b9\u6cd5
voidaddBuildInfoAttributes(XmlSerializer serializer, SuiteResultHolder holder)<br>serializer.attribute
voidaddSuiteAttributes(XmlSerializer serializer)<br>serializer.attribute
voidparseBuildInfoAttributes(XmlPullParser parser, IInvocationContext context)<br>addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder)
SuiteResultHolderparseResults(File resultDir, boolean shallow)<br>writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File)<br>SuiteResultHolder
voidparseSuiteAttributes(XmlPullParser parser, IInvocationContext context)<br>addSuiteAttributes(org.xmlpull.v1.XmlSerializer)
static StringsanitizeXmlContent(String s)
static StringtruncateStackTrace(String fullStackTrace, String testCaseName)<br>ERROR(/STACK_TRACE_MAX_SIZE)
FilewriteResults(SuiteResultHolder holder, File resultDir)
\n\n## \u5b57\u6bb5\n\n### \u5317\u7f8e\u6d32\n\n```\npublic static final String NS\n```\n\n### TEST_RESULTS_FILE_NAME\n\n```\npublic static final String TEST_RESULT_FILE_NAME\n```\n\n## \u516c\u5171\u6784\u9020\u51fd\u6570\n\n### XmlSuiteResultFormatter\n\n```\npublic XmlSuiteResultFormatter ()\n```\n\n## \u516c\u5171\u65b9\u6cd5\n\n### addBuildInfoAttributes\n\n```\npublic void addBuildInfoAttributes (XmlSerializer serializer,\n SuiteResultHolder holder)\n```\n\n\u5141\u8bb8\u901a\u8fc7 `serializer.attribute` \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002\n\n
\u53c2\u6570
serializerXmlSerializer
holderSuiteResultHolder
\n\n### \u6dfb\u52a0\u5957\u4ef6\u5c5e\u6027\n\n```\npublic void addSuiteAttributes (XmlSerializer serializer)\n```\n\n\u5141\u8bb8\u901a\u8fc7 `serializer.attribute` \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002\n\n
\u53c2\u6570
serializerXmlSerializer
\n\n### parseBuildInfoAttributes\n\n```\npublic void parseBuildInfoAttributes (XmlPullParser parser,\n IInvocationContext context)\n```\n\n\u4ece `addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder)` \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n
\u53c2\u6570
parserXmlPullParser
contextIInvocationContext<br>IInvocationContext
\n\n
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002
\n\n### parseResults\n\n```\npublic SuiteResultHolder parseResults (File resultDir,\n boolean shallow)\n```\n\n\u5bf9 `writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File)` \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa `SuiteResultHolder` \u3002\n\n
\u53c2\u6570
resultDirFile
shallowboolean<br>SuiteResultHolder
\n\n
\u8fd4\u56de
SuiteResultHolderSuiteResultHolder
\n\n### parseSuite \u5c5e\u6027\n\n```\npublic void parseSuiteAttributes (XmlPullParser parser,\n IInvocationContext context)\n```\n\n\u4ece `addSuiteAttributes(org.xmlpull.v1.XmlSerializer)` \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n
\u53c2\u6570
parserXmlPullParser
contextIInvocationContext<br>IInvocationContext
\n\n
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002
\n\n### sanitizeXmlContent\n\n```\npublic static String sanitizeXmlContent (String s)\n```\n\n\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002\n\n\u53c2\u6570\n\n```\ns String\n```\n\n
\u8fd4\u56de
\n\n```\nString\n```\n\n### truncateStackTrace\n\n```\npublic static String truncateStackTrace (String fullStackTrace,\n String testCaseName)\n```\n\n\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b `ERROR(/STACK_TRACE_MAX_SIZE)` \u4e2a\u5b57\u7b26\u3002\n\n\u53c2\u6570\n\n```\nfullStackTrace String\n```\n\n```\ntestCaseName String\n```\n\n
\u8fd4\u56de
\n\n```\nString\n```\n\n### writeResults\n\n```\npublic File writeResults (SuiteResultHolder holder,\n File resultDir)\n```\n\n\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002\n\n
\u53c2\u6570
holderSuiteResultHolder<br>SuiteResultHolder
resultDirFile<br>ERROR(/File)
\n\n
\u8fd4\u56de
FileERROR(/File)
\n", "html": "\n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n XmlSuiteResultFormatter  |  Android Open Source Project\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n \n \n\n
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\n\n\n \n\n\n \n\n \n\n \u4f7f\u7528\u96c6\u5408\u8ba9\u4e00\u5207\u4e95\u4e95\u6709\u6761\n \n \n\n \u6839\u636e\u60a8\u7684\u504f\u597d\u4fdd\u5b58\u5185\u5bb9\u5e76\u5bf9\u5176\u8fdb\u884c\u5206\u7c7b\u3002\n \n \n\n\n \n \n\n\n\n\n\n\n\n\n\n\n\n
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\n\n\n\n

XmlSuiteResultFormatter

\n

\n\n public\n\n\n\n class\n XmlSuiteResultFormatter\n\n
\n\n\n\n\n extends Object\n\n\n\n\n\n\n\n\n implements\n\n IFormatterGenerator\n\n\n\n\n\n

\n\n\n \n\n \n \n\n\n \n\n \n\n \n \n\n\n
java.lang.Object \u4e2d\n
\u21b3com.android.tradefed.result.suite.XmlSuiteResultFormatter\n
\n\n\n\n\n
\n\n\n

\u7528\u4e8e\u5c06\u5957\u4ef6\u8fd0\u884c\u4f5c\u4e1a\u4fdd\u5b58\u4e3a XML \u7684\u5b9e\u7528\u7a0b\u5e8f\u7c7b\u3002TODO\uff1a\u79fb\u9664\u6240\u6709\u7279\u6b8a\u7684\u517c\u5bb9\u6027\u6d4b\u8bd5\n \u83b7\u5f97\u76f8\u540c\u7684\u683c\u5f0f\u3002

\n\n\n\n\n\n\n\n\n\n

\u6458\u8981

\n\n\n\n\n\n\n\n\n\n \n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\u5d4c\u5957\u7c7b

\n\n\n\n\n class\n XmlSuiteResultFormatter.RunHistory\n

\u7528\u4e8e JSON \u8f6c\u6362\u7684\u8f85\u52a9\u5bf9\u8c61\u3002\n\n\n

\n
\n\n\n\n\n \n \n \n \n\n\n \n \n \n \n\n\n\n

\u5b57\u6bb5

\n public\n static\n final\n String\n NS\n

\n\n\n

\n
\n public\n static\n final\n String\n TEST_RESULT_FILE_NAME\n

\n\n\n

\n
\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n \n \n\n\n\n

\u516c\u5171\u6784\u9020\u51fd\u6570

\n \n XmlSuiteResultFormatter()\n \n\n
\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n \n\n\n \n\n \n \n\n\n\n

\u516c\u5171\u65b9\u6cd5

\n\n\n\n\n\n void\n \n \n addBuildInfoAttributes(XmlSerializer serializer, SuiteResultHolder holder)\n \n\n

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002\n\n\n

\n\n
\n\n\n\n\n\n void\n \n \n addSuiteAttributes(XmlSerializer serializer)\n \n\n

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002\n\n\n

\n\n
\n\n\n\n\n\n void\n \n \n parseBuildInfoAttributes(XmlPullParser parser, IInvocationContext context)\n \n\n

\u4ece addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n\n

\n\n
\n\n\n\n\n\n SuiteResultHolder\n \n \n parseResults(File resultDir, boolean shallow)\n \n\n

\u5bf9 writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File) \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa SuiteResultHolder\u3002\n\n\n

\n\n
\n\n\n\n\n\n void\n \n \n parseSuiteAttributes(XmlPullParser parser, IInvocationContext context)\n \n\n

\u4ece addSuiteAttributes(org.xmlpull.v1.XmlSerializer) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002\n\n\n

\n\n
\n\n\n static\n\n\n String\n \n \n sanitizeXmlContent(String s)\n \n\n

\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002\n\n\n

\n\n
\n\n\n static\n\n\n String\n \n \n truncateStackTrace(String fullStackTrace, String testCaseName)\n \n\n

\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b ERROR(/STACK_TRACE_MAX_SIZE) \u4e2a\u5b57\u7b26\u3002\n\n\n

\n\n
\n\n\n\n\n\n File\n \n \n writeResults(SuiteResultHolder holder, File resultDir)\n \n\n

\u4ee5 XML \u683c\u5f0f\u7f16\u5199\u8c03\u7528\u7ed3\u679c\u3002\n\n\n

\n\n
\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\u5b57\u6bb5

\n\n\n
\n

\u5317\u7f8e\u6d32

\n
\n\n\n\n\n\n
\n
\npublic static final String NS
\n\n\n\n\n

\n\n\n
\n\n
\n

TEST_RESULTS_FILE_NAME

\n
\n\n\n\n\n\n
\n
\npublic static final String TEST_RESULT_FILE_NAME
\n\n\n\n\n

\n\n\n
\n\n\n\n\n\n\n\n\n

\u516c\u5171\u6784\u9020\u51fd\u6570

\n\n\n
\n

XmlSuiteResultFormatter

\n
\n
\n
\n\n\n\n
\n
\npublic XmlSuiteResultFormatter ()
\n\n\n\n\n

\n\n
\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\u516c\u5171\u65b9\u6cd5

\n\n\n
\n

addBuildInfoAttributes

\n
\n
\n
\n\n\n\n
\n
\npublic void addBuildInfoAttributes (XmlSerializer serializer,\n                SuiteResultHolder holder)
\n\n\n\n\n

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002

\n \n \n \n \n \n \n \n \n \n \n
\u53c2\u6570
serializer\n XmlSerializer\uff1a\u5bf9 XML \u5957\u4ef6\u7ed3\u679c\u8fdb\u884c\u5e8f\u5217\u5316\u7684\u5bf9\u8c61\u3002

holder\n SuiteResultHolder\uff1a\u4e00\u4e2a\u5bf9\u8c61\uff0c\u5176\u4e2d\u5305\u542b\u8981\u5199\u5165\u5957\u4ef6\u7ed3\u679c\u7684\u4fe1\u606f\u3002

\n\n
\n\n
\n

\u6dfb\u52a0\u5957\u4ef6\u5c5e\u6027

\n
\n
\n
\n\n\n\n
\n
\npublic void addSuiteAttributes (XmlSerializer serializer)
\n\n\n\n\n

\u5141\u8bb8\u901a\u8fc7 serializer.attribute \u5411 \u6807\u8bb0\u6dfb\u52a0\u4e00\u4e9b\u5c5e\u6027\u3002

\n \n \n \n \n \n \n
\u53c2\u6570
serializer\n XmlSerializer\uff1a\u5bf9 XML \u5957\u4ef6\u7ed3\u679c\u8fdb\u884c\u5e8f\u5217\u5316\u7684\u5bf9\u8c61\u3002

\n\n
\n\n
\n

parseBuildInfoAttributes

\n
\n
\n
\n\n\n\n
\n
\npublic void parseBuildInfoAttributes (XmlPullParser parser,\n                IInvocationContext context)
\n\n\n\n\n

\u4ece addBuildInfoAttributes(org.xmlpull.v1.XmlSerializer, com.android.tradefed.result.suite.SuiteResultHolder) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002

\n \n \n \n \n \n \n \n \n \n \n
\u53c2\u6570
parser\n XmlPullParser\uff1a\u4ece\u4e2d\u8bfb\u53d6\u5c5e\u6027\u7684\u89e3\u6790\u5668\u3002

context\n IInvocationContext\uff1a\u7528\u4e8e\u653e\u7f6e\u5c5e\u6027\u7684 IInvocationContext\u3002

\n \n \n \n \n \n \n
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002
\n\n\n
\n\n
\n

parseResults

\n
\n
\n
\n\n\n\n
\n
\npublic SuiteResultHolder parseResults (File resultDir,\n                boolean shallow)
\n\n\n\n\n

\u5bf9 writeResults(com.android.tradefed.result.suite.SuiteResultHolder, File) \u8fdb\u884c\u53cd\u5411\u8fd0\u7b97\uff0c\u4ee5\u8ba1\u7b97\u7ed3\u679c\n \u76ee\u5f55\uff0c\u7136\u540e\u5728\u8be5\u76ee\u5f55\u4e2d\u521b\u5efa SuiteResultHolder\u3002

\n \n \n \n \n \n \n \n \n \n \n
\u53c2\u6570
resultDir\n File\uff1a\u7528\u4e8e\u67e5\u627e\u7ed3\u679c\u7684\u76ee\u5f55\u3002

shallow\n boolean\uff1a\u4ec5\u52a0\u8f7d SuiteResultHolder \u7684\u9876\u7ea7\u4fe1\u606f\u3002

\n \n \n \n \n \n \n
\u8fd4\u56de
SuiteResultHolder\u5305\u542b\u7ed3\u679c\u8868\u793a\u6cd5\u7684 SuiteResultHolder\u3002\u6216 null\uff0c\u5982\u679c\n \u51fa\u73b0\u4efb\u4f55\u95ee\u9898\u3002

\n\n
\n\n
\n

parseSuite \u5c5e\u6027

\n
\n
\n
\n\n\n\n
\n
\npublic void parseSuiteAttributes (XmlPullParser parser,\n                IInvocationContext context)
\n\n\n\n\n

\u4ece addSuiteAttributes(org.xmlpull.v1.XmlSerializer) \u5f00\u59cb\u9006\u5411\u8fd0\u7b97\u3002

\n \n \n \n \n \n \n \n \n \n \n
\u53c2\u6570
parser\n XmlPullParser\uff1a\u4ece\u4e2d\u8bfb\u53d6\u5c5e\u6027\u7684\u89e3\u6790\u5668\u3002

context\n IInvocationContext\uff1a\u7528\u4e8e\u653e\u7f6e\u5c5e\u6027\u7684 IInvocationContext\u3002

\n \n \n \n \n \n \n
\u629b\u51fa
XmlPullParserException\u5f53 XmlPullParser \u5931\u8d25\u65f6\u3002
\n\n\n
\n\n
\n

sanitizeXmlContent

\n
\n
\n
\n\n\n\n
\n
\npublic static String sanitizeXmlContent (String s)
\n\n\n\n\n

\u6e05\u7406\u5b57\u7b26\u4e32\u4ee5\u8f6c\u4e49\u7279\u6b8a\u5b57\u7b26\u3002

\n \n \n \n \n \n \n
\u53c2\u6570
s\n String

\n \n \n \n \n \n \n
\u8fd4\u56de
String

\n\n
\n\n
\n

truncateStackTrace

\n
\n
\n
\n\n\n\n
\n
\npublic static String truncateStackTrace (String fullStackTrace,\n                String testCaseName)
\n\n\n\n\n

\u622a\u65ad\u5b8c\u6574\u7684\u5806\u6808\u8f68\u8ff9\uff0c\u6700\u591a\u5305\u542b ERROR(/STACK_TRACE_MAX_SIZE) \u4e2a\u5b57\u7b26\u3002

\n \n \n \n \n \n \n \n \n \n \n
\u53c2\u6570
fullStackTrace\n String

testCaseName\n String

\n \n \n \n \n \n \n
\u8fd4\u56de
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writeResults

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\u8fd4\u56de
File\u6307\u5411 xml \u8f93\u51fa\u6587\u4ef6\u7684 ERROR(/File)\u3002

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\u672c\u9875\u9762\u4e0a\u7684\u5185\u5bb9\u548c\u4ee3\u7801\u793a\u4f8b\u53d7\u5185\u5bb9\u8bb8\u53ef\u90e8\u5206\u6240\u8ff0\u8bb8\u53ef\u7684\u9650\u5236\u3002Java \u548c OpenJDK \u662f Oracle \u548c/\u6216\u5176\u5173\u8054\u516c\u53f8\u7684\u6ce8\u518c\u5546\u6807\u3002

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\u6700\u540e\u66f4\u65b0\u65f6\u95f4 (UTC)\uff1a2024-08-22\u3002

\n\n\n\n\n\n\n\n\n
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\n \n \n \n \n \n\n \n\n\n \n\n \n\n\n\n \n \n\n"} diff --git a/bench/data/groundtruth/code_6.jsonl b/bench/data/groundtruth/code_6.jsonl index c6115f23..f0ded27e 100644 --- a/bench/data/groundtruth/code_6.jsonl +++ b/bench/data/groundtruth/code_6.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "paragraph", "raw_content": "
\n RSS\n
", "content": [{"c": "RSS", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Installs and Configures Ark Survival Evolved Dedicated Server

", "content": [{"c": "Installs and Configures Ark Survival Evolved Dedicated Server", "t": "text"}]}, {"type": "list", "raw_content": "
", "content": {"items": [], "ordered": true}}, {"type": "code", "raw_content": "
cookbook 'chef-arkserver', '= 0.1.0', :supermarket
\n ", "inline": false, "content": {"code_content": "cookbook 'chef-arkserver', '= 0.1.0', :supermarket", "by": "tag_pre"}}, {"type": "code", "raw_content": "
cookbook 'chef-arkserver', '= 0.1.0'
\n ", "inline": false, "content": {"code_content": "cookbook 'chef-arkserver', '= 0.1.0'", "by": "tag_pre"}}, {"type": "code", "raw_content": "
knife supermarket install chef-arkserver
\n ", "inline": false, "content": {"code_content": "knife supermarket install chef-arkserver", "by": "tag_pre"}}, {"type": "code", "raw_content": "
knife supermarket download chef-arkserver
\n ", "inline": false, "content": {"code_content": "knife supermarket download chef-arkserver", "by": "tag_pre"}}, {"type": "list", "raw_content": "
README
Dependencies
Quality\n \n 43%
", "content": {"items": [[[{"c": "README", "t": "text"}]], [[{"c": "Dependencies", "t": "text"}]], [[{"c": "Quality\n ", "t": "text"}, {"c": "\n 43%", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "

Ark: Survival Evolved Chef Cookbook

", "content": {"title_content": "Ark: Survival Evolved Chef Cookbook", "level": "1"}}, {"type": "image", "raw_content": "\"Build", "content": {"url": "https://travis-ci.org/f0rkz/chef-arkserver.svg", "data": null, "alt": "Build Status", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

This cookbook will install and configure an Ark: Survival Evolved dedicated
\nserver. Utilize the following README for more information on how to utilize this
\ncookbook.

", "content": [{"c": "This cookbook will install and configure an Ark: Survival Evolved dedicated server. Utilize the following README for more information on how to utilize this cookbook.", "t": "text"}]}, {"type": "title", "raw_content": "

Usage

", "content": {"title_content": "Usage", "level": "2"}}, {"type": "paragraph", "raw_content": "

Include chef-arkserver in your node's runlist

", "content": [{"c": "Include", "t": "text"}, {"c": "chef-arkserver", "t": "code-inline"}, {"c": "in your node's runlist", "t": "text"}]}, {"type": "title", "raw_content": "

Configuration

", "content": {"title_content": "Configuration", "level": "2"}}, {"type": "paragraph", "raw_content": "

In order to properly utilize this cookbook, a suite of default configuration
\noptions are included in the cookbook's attributes/default.rb file. You will
\nneed to override the default options in your cookbook's node configuration.

", "content": [{"c": "In order to properly utilize this cookbook, a suite of default configuration options are included in the cookbook's", "t": "text"}, {"c": "attributes/default.rb", "t": "code-inline"}, {"c": "file. You will need to override the default options in your cookbook's node configuration.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

An example of overriding configuration options is structured as follows:

", "content": [{"c": "An example of overriding configuration options is structured as follows:", "t": "text"}]}, {"type": "code", "raw_content": "
{\n  \"name\": \"my.ark.server.net\",\n  \"chef_environment\": \"_default\",\n  \"normal\": {\n    \"ark\": {\n      \"gameserver\": {\n        \"configuration\": {\n          \"map\": \"ScorchedEarth_P\",\n          \"ServerSettings\": {\n            \"XPMultiplier\": \"5.0000\",\n            \"TamingSpeedMultiplier\": \"2.0000\",\n            \"RCONPort\": 27025\n          },\n          \"SessionSettings\": {\n            \"SessionName\": \"My Ark Dedicated Server\"\n          },\n          \"MessageOfTheDay\": {\n            \"Message\": \"Welcome to my ARK Server dot Net\\nPlease enjoy your stay!\"\n          }\n        }\n      }\n    },\n    \"tags\": [\n\n    ]\n  },\n  \"policy_name\": null,\n  \"policy_group\": null,\n  \"run_list\": [\n  \"role[base]\",\n  \"recipe[]\"\n]\n\n}\n
", "inline": false, "content": {"code_content": "{\n \"name\": \"my.ark.server.net\",\n \"chef_environment\": \"_default\",\n \"normal\": {\n \"ark\": {\n \"gameserver\": {\n \"configuration\": {\n \"map\": \"ScorchedEarth_P\",\n \"ServerSettings\": {\n \"XPMultiplier\": \"5.0000\",\n \"TamingSpeedMultiplier\": \"2.0000\",\n \"RCONPort\": 27025\n },\n \"SessionSettings\": {\n \"SessionName\": \"My Ark Dedicated Server\"\n },\n \"MessageOfTheDay\": {\n \"Message\": \"Welcome to my ARK Server dot Net\\nPlease enjoy your stay!\"\n }\n }\n }\n },\n \"tags\": [\n\n ]\n },\n \"policy_name\": null,\n \"policy_group\": null,\n \"run_list\": [\n \"role[base]\",\n \"recipe[]\"\n]\n\n}", "by": "tag_pre"}}, {"type": "paragraph", "raw_content": "

This configuration will build your GameUserSettings.ini file with the requested
\noverrided options defined above.

", "content": [{"c": "This configuration will build your", "t": "text"}, {"c": "GameUserSettings.ini", "t": "code-inline"}, {"c": "file with the requested overrided options defined above.", "t": "text"}]}, {"type": "title", "raw_content": "

Dependent cookbooks

", "content": {"title_content": "Dependent cookbooks", "level": "3"}}, {"type": "table", "raw_content": "
chef-steamcmd >= 0.0.0
", "content": {"html": "
chef-steamcmd >= 0.0.0
", "is_complex": false}}, {"type": "title", "raw_content": "

Contingent cookbooks

", "content": {"title_content": "Contingent cookbooks", "level": "3"}}, {"type": "paragraph", "raw_content": "

There are no cookbooks that are contingent upon this one.

", "content": [{"c": "There are no cookbooks that are contingent upon this one.", "t": "text"}]}, {"type": "title", "raw_content": "

Collaborator Number Metric\n

", "content": {"title_content": "Collaborator Number Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 failed this metric
\n ", "inline": false, "content": {"code_content": " 0.1.0 failed this metric", "by": "tag_pre"}}, {"type": "code", "raw_content": "
Failure: Cookbook has 0 collaborators. A cookbook must have at least 2 collaborators to pass this metric.
\n \n ", "inline": false, "content": {"code_content": "Failure: Cookbook has 0 collaborators. A cookbook must have at least 2 collaborators to pass this metric.", "by": "tag_pre"}}, {"type": "title", "raw_content": "

Contributing File Metric\n

", "content": {"title_content": "Contributing File Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 failed this metric
\n ", "inline": false, "content": {"code_content": " 0.1.0 failed this metric", "by": "tag_pre"}}, {"type": "code", "raw_content": "
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a CONTRIBUTING.md file
\n \n ", "inline": false, "content": {"code_content": "Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a CONTRIBUTING.md file", "by": "tag_pre"}}, {"type": "title", "raw_content": "

Foodcritic Metric\n

", "content": {"title_content": "Foodcritic Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 passed this metric
\n \n ", "inline": false, "content": {"code_content": " 0.1.0 passed this metric", "by": "tag_pre"}}, {"type": "title", "raw_content": "

License Metric\n

", "content": {"title_content": "License Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 passed this metric
\n \n ", "inline": false, "content": {"code_content": " 0.1.0 passed this metric", "by": "tag_pre"}}, {"type": "title", "raw_content": "

No Binaries Metric\n

", "content": {"title_content": "No Binaries Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 passed this metric
\n \n ", "inline": false, "content": {"code_content": " 0.1.0 passed this metric", "by": "tag_pre"}}, {"type": "title", "raw_content": "

Testing File Metric\n

", "content": {"title_content": "Testing File Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 failed this metric
\n ", "inline": false, "content": {"code_content": " 0.1.0 failed this metric", "by": "tag_pre"}}, {"type": "code", "raw_content": "
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a TESTING.md file
\n \n ", "inline": false, "content": {"code_content": "Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a TESTING.md file", "by": "tag_pre"}}, {"type": "title", "raw_content": "

Version Tag Metric\n

", "content": {"title_content": "Version Tag Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 failed this metric
\n ", "inline": false, "content": {"code_content": " 0.1.0 failed this metric", "by": "tag_pre"}}, {"type": "code", "raw_content": "
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must include a tag that matches this cookbook version number
\n \n ", "inline": false, "content": {"code_content": "Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must include a tag that matches this cookbook version number", "by": "tag_pre"}}]], "main_html": "
\n RSS\n

Installs and Configures Ark Survival Evolved Dedicated Server

cookbook 'chef-arkserver', '= 0.1.0', :supermarket
\n 
cookbook 'chef-arkserver', '= 0.1.0'
\n 
knife supermarket install chef-arkserver
\n 
knife supermarket download chef-arkserver
\n
README
Dependencies
Quality\n \n 43%

Ark: Survival Evolved Chef Cookbook

\"Build

This cookbook will install and configure an Ark: Survival Evolved dedicated
\nserver. Utilize the following README for more information on how to utilize this
\ncookbook.

Usage

Include chef-arkserver in your node's runlist

Configuration

In order to properly utilize this cookbook, a suite of default configuration
\noptions are included in the cookbook's attributes/default.rb file. You will
\nneed to override the default options in your cookbook's node configuration.

An example of overriding configuration options is structured as follows:

{\n  \"name\": \"my.ark.server.net\",\n  \"chef_environment\": \"_default\",\n  \"normal\": {\n    \"ark\": {\n      \"gameserver\": {\n        \"configuration\": {\n          \"map\": \"ScorchedEarth_P\",\n          \"ServerSettings\": {\n            \"XPMultiplier\": \"5.0000\",\n            \"TamingSpeedMultiplier\": \"2.0000\",\n            \"RCONPort\": 27025\n          },\n          \"SessionSettings\": {\n            \"SessionName\": \"My Ark Dedicated Server\"\n          },\n          \"MessageOfTheDay\": {\n            \"Message\": \"Welcome to my ARK Server dot Net\\nPlease enjoy your stay!\"\n          }\n        }\n      }\n    },\n    \"tags\": [\n\n    ]\n  },\n  \"policy_name\": null,\n  \"policy_group\": null,\n  \"run_list\": [\n  \"role[base]\",\n  \"recipe[]\"\n]\n\n}\n

This configuration will build your GameUserSettings.ini file with the requested
\noverrided options defined above.

Dependent cookbooks

chef-steamcmd >= 0.0.0

Contingent cookbooks

There are no cookbooks that are contingent upon this one.

Collaborator Number Metric\n 

\n            
0.1.0 failed this metric
\n 
Failure: Cookbook has 0 collaborators. A cookbook must have at least 2 collaborators to pass this metric.
\n \n

Contributing File Metric\n 

\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a CONTRIBUTING.md file
\n \n

Foodcritic Metric\n 

\n            
0.1.0 passed this metric
\n \n

License Metric\n 

\n            
0.1.0 passed this metric
\n \n

No Binaries Metric\n 

\n            
0.1.0 passed this metric
\n \n

Testing File Metric\n 

\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a TESTING.md file
\n \n

Version Tag Metric\n 

\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must include a tag that matches this cookbook version number
\n \n ", "statics": {"paragraph": 8, "paragraph.text": 11, "list": 2, "code": 16, "list.text": 4, "title": 12, "image": 1, "paragraph.code-inline": 3, "table": 1}, "url": "https://supermarket.chef.io/cookbooks/chef-arkserver/versions/0.1.0", "content": "RSS\n\nInstalls and Configures Ark Survival Evolved Dedicated Server\n\n```\ncookbook 'chef-arkserver', '= 0.1.0', :supermarket\n```\n\n```\ncookbook 'chef-arkserver', '= 0.1.0'\n```\n\n```\nknife supermarket install chef-arkserver\n```\n\n```\nknife supermarket download chef-arkserver\n```\n\n1. README\n2. Dependencies\n3. Quality 43%\n\n# Ark: Survival Evolved Chef Cookbook\n\nThis cookbook will install and configure an Ark: Survival Evolved dedicated server. Utilize the following README for more information on how to utilize this cookbook.\n\n## Usage\n\nInclude `chef-arkserver` in your node's runlist\n\n## Configuration\n\nIn order to properly utilize this cookbook, a suite of default configuration options are included in the cookbook's `attributes/default.rb` file. You will need to override the default options in your cookbook's node configuration.\n\nAn example of overriding configuration options is structured as follows:\n\n```\n{\n \"name\": \"my.ark.server.net\",\n \"chef_environment\": \"_default\",\n \"normal\": {\n \"ark\": {\n \"gameserver\": {\n \"configuration\": {\n \"map\": \"ScorchedEarth_P\",\n \"ServerSettings\": {\n \"XPMultiplier\": \"5.0000\",\n \"TamingSpeedMultiplier\": \"2.0000\",\n \"RCONPort\": 27025\n },\n \"SessionSettings\": {\n \"SessionName\": \"My Ark Dedicated Server\"\n },\n \"MessageOfTheDay\": {\n \"Message\": \"Welcome to my ARK Server dot Net\\nPlease enjoy your stay!\"\n }\n }\n }\n },\n \"tags\": [\n\n ]\n },\n \"policy_name\": null,\n \"policy_group\": null,\n \"run_list\": [\n \"role[base]\",\n \"recipe[]\"\n]\n\n}\n```\n\nThis configuration will build your `GameUserSettings.ini` file with the requested overrided options defined above.\n\n### Dependent cookbooks\n\nchef-steamcmd >= 0.0.0\n\n### Contingent cookbooks\n\nThere are no cookbooks that are contingent upon this one.\n\n# Collaborator Number Metric\n\n```\n 0.1.0 failed this metric\n```\n\n```\nFailure: Cookbook has 0 collaborators. A cookbook must have at least 2 collaborators to pass this metric.\n```\n\n# Contributing File Metric\n\n```\n 0.1.0 failed this metric\n```\n\n```\nFailure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a CONTRIBUTING.md file\n```\n\n# Foodcritic Metric\n\n```\n 0.1.0 passed this metric\n```\n\n# License Metric\n\n```\n 0.1.0 passed this metric\n```\n\n# No Binaries Metric\n\n```\n 0.1.0 passed this metric\n```\n\n# Testing File Metric\n\n```\n 0.1.0 failed this metric\n```\n\n```\nFailure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a TESTING.md file\n```\n\n# Version Tag Metric\n\n```\n 0.1.0 failed this metric\n```\n\n```\nFailure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must include a tag that matches this cookbook version number\n```\n", "html": "\n\n\n\n\n \n \n \n chef-arkserver Cookbook - Chef Supermarket\n \n \n \n \n \n \n \n\n \n\n\n\n \n \n \n
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\n Menu\n
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Chef Supermarket

\n
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Adoptable Cookbooks List

\n\n

\n Looking for a cookbook to adopt? You can now see a list of cookbooks available for adoption!\n
\n List of Adoptable Cookbooks\n

\n
\n\n
\n

Supermarket Belongs to the Community

\n\n

Supermarket belongs to the community. While Chef has the responsibility to keep it running and be stewards of its functionality, what it does and how it works is driven by the community. The chef/supermarket repository will continue to be where development of the Supermarket application takes place. Come be part of shaping the direction of Supermarket by opening issues and pull requests or by joining us on the Chef Mailing List.

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View our collection of guides, documentation, and articles

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\n chef-arkserver\n \n \n \n (15) Versions\n \n 0.1.0\n \n \n\n \n Follow0\n\n\n

\n\n

Installs and Configures Ark Survival Evolved Dedicated Server

\n\n
\n
\n
\n Policyfile\n
\n\n
\n Berkshelf\n
\n\n
\n Knife\n
\n
\n\n
\n
\n 
cookbook 'chef-arkserver', '= 0.1.0', :supermarket
\n
\n\n
\n 
cookbook 'chef-arkserver', '= 0.1.0'
\n
\n\n
\n 
knife supermarket install chef-arkserver
\n 
knife supermarket download chef-arkserver
\n
\n
\n
\n\n\n
\n
README
\n
Dependencies
\n
\n Quality\n \n 43%\n \n
\n
\n
\n
\n

Ark: Survival Evolved Chef Cookbook

\n\n

\"Build

\n\n

This cookbook will install and configure an Ark: Survival Evolved dedicated
\nserver. Utilize the following README for more information on how to utilize this
\ncookbook.

\n\n

Usage

\n\n

Include chef-arkserver in your node's runlist

\n\n

Configuration

\n\n

In order to properly utilize this cookbook, a suite of default configuration
\noptions are included in the cookbook's attributes/default.rb file. You will
\nneed to override the default options in your cookbook's node configuration.

\n\n

An example of overriding configuration options is structured as follows:

\n
\n 
{\n  \"name\": \"my.ark.server.net\",\n  \"chef_environment\": \"_default\",\n  \"normal\": {\n    \"ark\": {\n      \"gameserver\": {\n        \"configuration\": {\n          \"map\": \"ScorchedEarth_P\",\n          \"ServerSettings\": {\n            \"XPMultiplier\": \"5.0000\",\n            \"TamingSpeedMultiplier\": \"2.0000\",\n            \"RCONPort\": 27025\n          },\n          \"SessionSettings\": {\n            \"SessionName\": \"My Ark Dedicated Server\"\n          },\n          \"MessageOfTheDay\": {\n            \"Message\": \"Welcome to my ARK Server dot Net\\nPlease enjoy your stay!\"\n          }\n        }\n      }\n    },\n    \"tags\": [\n\n    ]\n  },\n  \"policy_name\": null,\n  \"policy_group\": null,\n  \"run_list\": [\n  \"role[base]\",\n  \"recipe[]\"\n]\n\n}\n
\n
\n\n

This configuration will build your GameUserSettings.ini file with the requested
\noverrided options defined above.

\n\n
\n
\n

Dependent cookbooks

\n \n \n \n \n\n\n \n
chef-steamcmd >= 0.0.0
\n\n

Contingent cookbooks

\n

There are no cookbooks that are contingent upon this one.

\n
\n
\n

Collaborator Number Metric\n 
\n            
0.1.0 failed this metric
\n 
Failure: Cookbook has 0 collaborators. A cookbook must have at least 2 collaborators to pass this metric.
\n \n

Contributing File Metric\n 
\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a CONTRIBUTING.md file
\n \n

Foodcritic Metric\n 
\n            
0.1.0 passed this metric
\n \n

License Metric\n 
\n            
0.1.0 passed this metric
\n \n

No Binaries Metric\n 
\n            
0.1.0 passed this metric
\n \n

Testing File Metric\n 
\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a TESTING.md file
\n \n

Version Tag Metric\n 
\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must include a tag that matches this cookbook version number
\n \n
\n

\n\n
\n
\n\n
\n
\n \n \"Nick\n f0rkz\n Nick Gray\n
\n\n
\n\n
\n\n\n\n\n

\n Details\n

\n\n
\n\n
\n View Source\n\n View Issues\n
\n\n

\n Updated September 8, 2017\n Created on September 8, 2017\n

\n\n

Supported Platforms

\n
    \n
  • \n \"debian\n
    • \n
  • \n \"ubuntu\n
    • \n
\n\n\n\n

License

\n

Apache-2.0\n

\n\n

Required Chef Infra Client

\n

(>= 12.1)

\n\n\n Download Cookbook\n
\n
\n\n
\n\n
\n \n\n \n
\n
\n\n \n \n\n"} +{"content_list": [[{"type": "paragraph", "raw_content": "
\n RSS\n
", "content": [{"c": "RSS", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Installs and Configures Ark Survival Evolved Dedicated Server

", "content": [{"c": "Installs and Configures Ark Survival Evolved Dedicated Server", "t": "text"}]}, {"type": "list", "raw_content": "
", "content": {"items": [], "ordered": true}}, {"type": "code", "raw_content": "
cookbook 'chef-arkserver', '= 0.1.0', :supermarket
\n ", "inline": false, "content": {"code_content": "cookbook 'chef-arkserver', '= 0.1.0', :supermarket", "by": "tag_pre"}}, {"type": "code", "raw_content": "
cookbook 'chef-arkserver', '= 0.1.0'
\n ", "inline": false, "content": {"code_content": "cookbook 'chef-arkserver', '= 0.1.0'", "by": "tag_pre"}}, {"type": "code", "raw_content": "
knife supermarket install chef-arkserver
\n ", "inline": false, "content": {"code_content": "knife supermarket install chef-arkserver", "by": "tag_pre"}}, {"type": "code", "raw_content": "
knife supermarket download chef-arkserver
\n ", "inline": false, "content": {"code_content": "knife supermarket download chef-arkserver", "by": "tag_pre"}}, {"type": "list", "raw_content": "
README
Dependencies
Quality\n \n 43%
", "content": {"items": [[[{"c": "README", "t": "text"}]], [[{"c": "Dependencies", "t": "text"}]], [[{"c": "Quality\n ", "t": "text"}, {"c": "\n 43%", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "

Ark: Survival Evolved Chef Cookbook

", "content": {"title_content": "Ark: Survival Evolved Chef Cookbook", "level": "1"}}, {"type": "image", "raw_content": "\"Build", "content": {"url": "https://travis-ci.org/f0rkz/chef-arkserver.svg", "data": null, "alt": "Build Status", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

This cookbook will install and configure an Ark: Survival Evolved dedicated
\nserver. Utilize the following README for more information on how to utilize this
\ncookbook.

", "content": [{"c": "This cookbook will install and configure an Ark: Survival Evolved dedicated server. Utilize the following README for more information on how to utilize this cookbook.", "t": "text"}]}, {"type": "title", "raw_content": "

Usage

", "content": {"title_content": "Usage", "level": "2"}}, {"type": "paragraph", "raw_content": "

Include chef-arkserver in your node's runlist

", "content": [{"c": "Include", "t": "text"}, {"c": "chef-arkserver", "t": "code-inline"}, {"c": "in your node's runlist", "t": "text"}]}, {"type": "title", "raw_content": "

Configuration

", "content": {"title_content": "Configuration", "level": "2"}}, {"type": "paragraph", "raw_content": "

In order to properly utilize this cookbook, a suite of default configuration
\noptions are included in the cookbook's attributes/default.rb file. You will
\nneed to override the default options in your cookbook's node configuration.

", "content": [{"c": "In order to properly utilize this cookbook, a suite of default configuration options are included in the cookbook's", "t": "text"}, {"c": "attributes/default.rb", "t": "code-inline"}, {"c": "file. You will need to override the default options in your cookbook's node configuration.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

An example of overriding configuration options is structured as follows:

", "content": [{"c": "An example of overriding configuration options is structured as follows:", "t": "text"}]}, {"type": "code", "raw_content": "
{\n  \"name\": \"my.ark.server.net\",\n  \"chef_environment\": \"_default\",\n  \"normal\": {\n    \"ark\": {\n      \"gameserver\": {\n        \"configuration\": {\n          \"map\": \"ScorchedEarth_P\",\n          \"ServerSettings\": {\n            \"XPMultiplier\": \"5.0000\",\n            \"TamingSpeedMultiplier\": \"2.0000\",\n            \"RCONPort\": 27025\n          },\n          \"SessionSettings\": {\n            \"SessionName\": \"My Ark Dedicated Server\"\n          },\n          \"MessageOfTheDay\": {\n            \"Message\": \"Welcome to my ARK Server dot Net\\nPlease enjoy your stay!\"\n          }\n        }\n      }\n    },\n    \"tags\": [\n\n    ]\n  },\n  \"policy_name\": null,\n  \"policy_group\": null,\n  \"run_list\": [\n  \"role[base]\",\n  \"recipe[]\"\n]\n\n}\n
", "inline": false, "content": {"code_content": "{\n \"name\": \"my.ark.server.net\",\n \"chef_environment\": \"_default\",\n \"normal\": {\n \"ark\": {\n \"gameserver\": {\n \"configuration\": {\n \"map\": \"ScorchedEarth_P\",\n \"ServerSettings\": {\n \"XPMultiplier\": \"5.0000\",\n \"TamingSpeedMultiplier\": \"2.0000\",\n \"RCONPort\": 27025\n },\n \"SessionSettings\": {\n \"SessionName\": \"My Ark Dedicated Server\"\n },\n \"MessageOfTheDay\": {\n \"Message\": \"Welcome to my ARK Server dot Net\\nPlease enjoy your stay!\"\n }\n }\n }\n },\n \"tags\": [\n\n ]\n },\n \"policy_name\": null,\n \"policy_group\": null,\n \"run_list\": [\n \"role[base]\",\n \"recipe[]\"\n]\n\n}", "by": "tag_pre"}}, {"type": "paragraph", "raw_content": "

This configuration will build your GameUserSettings.ini file with the requested
\noverrided options defined above.

", "content": [{"c": "This configuration will build your", "t": "text"}, {"c": "GameUserSettings.ini", "t": "code-inline"}, {"c": "file with the requested overrided options defined above.", "t": "text"}]}, {"type": "title", "raw_content": "

Dependent cookbooks

", "content": {"title_content": "Dependent cookbooks", "level": "3"}}, {"type": "table", "raw_content": "
chef-steamcmd >= 0.0.0
", "content": {"html": "
chef-steamcmd >= 0.0.0
", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

Contingent cookbooks

", "content": {"title_content": "Contingent cookbooks", "level": "3"}}, {"type": "paragraph", "raw_content": "

There are no cookbooks that are contingent upon this one.

", "content": [{"c": "There are no cookbooks that are contingent upon this one.", "t": "text"}]}, {"type": "title", "raw_content": "

Collaborator Number Metric\n

", "content": {"title_content": "Collaborator Number Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 failed this metric
\n ", "inline": false, "content": {"code_content": "0.1.0 failed this metric", "by": "tag_pre"}}, {"type": "code", "raw_content": "
Failure: Cookbook has 0 collaborators. A cookbook must have at least 2 collaborators to pass this metric.
\n \n ", "inline": false, "content": {"code_content": "Failure: Cookbook has 0 collaborators. A cookbook must have at least 2 collaborators to pass this metric.", "by": "tag_pre"}}, {"type": "title", "raw_content": "

Contributing File Metric\n

", "content": {"title_content": "Contributing File Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 failed this metric
\n ", "inline": false, "content": {"code_content": "0.1.0 failed this metric", "by": "tag_pre"}}, {"type": "code", "raw_content": "
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a CONTRIBUTING.md file
\n \n ", "inline": false, "content": {"code_content": "Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a CONTRIBUTING.md file", "by": "tag_pre"}}, {"type": "title", "raw_content": "

Foodcritic Metric\n

", "content": {"title_content": "Foodcritic Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 passed this metric
\n \n ", "inline": false, "content": {"code_content": "0.1.0 passed this metric", "by": "tag_pre"}}, {"type": "title", "raw_content": "

License Metric\n

", "content": {"title_content": "License Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 passed this metric
\n \n ", "inline": false, "content": {"code_content": "0.1.0 passed this metric", "by": "tag_pre"}}, {"type": "title", "raw_content": "

No Binaries Metric\n

", "content": {"title_content": "No Binaries Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 passed this metric
\n \n ", "inline": false, "content": {"code_content": "0.1.0 passed this metric", "by": "tag_pre"}}, {"type": "title", "raw_content": "

Testing File Metric\n

", "content": {"title_content": "Testing File Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 failed this metric
\n ", "inline": false, "content": {"code_content": "0.1.0 failed this metric", "by": "tag_pre"}}, {"type": "code", "raw_content": "
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a TESTING.md file
\n \n ", "inline": false, "content": {"code_content": "Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a TESTING.md file", "by": "tag_pre"}}, {"type": "title", "raw_content": "

Version Tag Metric\n

", "content": {"title_content": "Version Tag Metric", "level": "1"}}, {"type": "code", "raw_content": "
\n            
0.1.0 failed this metric
\n ", "inline": false, "content": {"code_content": "0.1.0 failed this metric", "by": "tag_pre"}}, {"type": "code", "raw_content": "
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must include a tag that matches this cookbook version number
\n \n ", "inline": false, "content": {"code_content": "Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must include a tag that matches this cookbook version number", "by": "tag_pre"}}]], "main_html": "
\n RSS\n

Installs and Configures Ark Survival Evolved Dedicated Server

cookbook 'chef-arkserver', '= 0.1.0', :supermarket
\n 
cookbook 'chef-arkserver', '= 0.1.0'
\n 
knife supermarket install chef-arkserver
\n 
knife supermarket download chef-arkserver
\n
README
Dependencies
Quality\n \n 43%

Ark: Survival Evolved Chef Cookbook

\"Build

This cookbook will install and configure an Ark: Survival Evolved dedicated
\nserver. Utilize the following README for more information on how to utilize this
\ncookbook.

Usage

Include chef-arkserver in your node's runlist

Configuration

In order to properly utilize this cookbook, a suite of default configuration
\noptions are included in the cookbook's attributes/default.rb file. You will
\nneed to override the default options in your cookbook's node configuration.

An example of overriding configuration options is structured as follows:

{\n  \"name\": \"my.ark.server.net\",\n  \"chef_environment\": \"_default\",\n  \"normal\": {\n    \"ark\": {\n      \"gameserver\": {\n        \"configuration\": {\n          \"map\": \"ScorchedEarth_P\",\n          \"ServerSettings\": {\n            \"XPMultiplier\": \"5.0000\",\n            \"TamingSpeedMultiplier\": \"2.0000\",\n            \"RCONPort\": 27025\n          },\n          \"SessionSettings\": {\n            \"SessionName\": \"My Ark Dedicated Server\"\n          },\n          \"MessageOfTheDay\": {\n            \"Message\": \"Welcome to my ARK Server dot Net\\nPlease enjoy your stay!\"\n          }\n        }\n      }\n    },\n    \"tags\": [\n\n    ]\n  },\n  \"policy_name\": null,\n  \"policy_group\": null,\n  \"run_list\": [\n  \"role[base]\",\n  \"recipe[]\"\n]\n\n}\n

This configuration will build your GameUserSettings.ini file with the requested
\noverrided options defined above.

Dependent cookbooks

chef-steamcmd >= 0.0.0

Contingent cookbooks

There are no cookbooks that are contingent upon this one.

Collaborator Number Metric\n 

\n            
0.1.0 failed this metric
\n 
Failure: Cookbook has 0 collaborators. A cookbook must have at least 2 collaborators to pass this metric.
\n \n

Contributing File Metric\n 

\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a CONTRIBUTING.md file
\n \n

Foodcritic Metric\n 

\n            
0.1.0 passed this metric
\n \n

License Metric\n 

\n            
0.1.0 passed this metric
\n \n

No Binaries Metric\n 

\n            
0.1.0 passed this metric
\n \n

Testing File Metric\n 

\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a TESTING.md file
\n \n

Version Tag Metric\n 

\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must include a tag that matches this cookbook version number
\n \n ", "statics": {"paragraph": 8, "paragraph.text": 11, "list": 2, "code": 16, "list.text": 4, "title": 12, "image": 1, "paragraph.code-inline": 3, "table": 1}, "url": "https://supermarket.chef.io/cookbooks/chef-arkserver/versions/0.1.0", "content": "RSS\n\nInstalls and Configures Ark Survival Evolved Dedicated Server\n\n```\ncookbook 'chef-arkserver', '= 0.1.0', :supermarket\n```\n\n```\ncookbook 'chef-arkserver', '= 0.1.0'\n```\n\n```\nknife supermarket install chef-arkserver\n```\n\n```\nknife supermarket download chef-arkserver\n```\n\n1. README\n2. Dependencies\n3. Quality 43%\n\n# Ark: Survival Evolved Chef Cookbook\n\nThis cookbook will install and configure an Ark: Survival Evolved dedicated server. Utilize the following README for more information on how to utilize this cookbook.\n\n## Usage\n\nInclude `chef-arkserver` in your node's runlist\n\n## Configuration\n\nIn order to properly utilize this cookbook, a suite of default configuration options are included in the cookbook's `attributes/default.rb` file. You will need to override the default options in your cookbook's node configuration.\n\nAn example of overriding configuration options is structured as follows:\n\n```\n{\n \"name\": \"my.ark.server.net\",\n \"chef_environment\": \"_default\",\n \"normal\": {\n \"ark\": {\n \"gameserver\": {\n \"configuration\": {\n \"map\": \"ScorchedEarth_P\",\n \"ServerSettings\": {\n \"XPMultiplier\": \"5.0000\",\n \"TamingSpeedMultiplier\": \"2.0000\",\n \"RCONPort\": 27025\n },\n \"SessionSettings\": {\n \"SessionName\": \"My Ark Dedicated Server\"\n },\n \"MessageOfTheDay\": {\n \"Message\": \"Welcome to my ARK Server dot Net\\nPlease enjoy your stay!\"\n }\n }\n }\n },\n \"tags\": [\n\n ]\n },\n \"policy_name\": null,\n \"policy_group\": null,\n \"run_list\": [\n \"role[base]\",\n \"recipe[]\"\n]\n\n}\n```\n\nThis configuration will build your `GameUserSettings.ini` file with the requested overrided options defined above.\n\n### Dependent cookbooks\n\nchef-steamcmd >= 0.0.0\n\n### Contingent cookbooks\n\nThere are no cookbooks that are contingent upon this one.\n\n# Collaborator Number Metric\n\n```\n0.1.0 failed this metric\n```\n\n```\nFailure: Cookbook has 0 collaborators. A cookbook must have at least 2 collaborators to pass this metric.\n```\n\n# Contributing File Metric\n\n```\n0.1.0 failed this metric\n```\n\n```\nFailure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a CONTRIBUTING.md file\n```\n\n# Foodcritic Metric\n\n```\n0.1.0 passed this metric\n```\n\n# License Metric\n\n```\n0.1.0 passed this metric\n```\n\n# No Binaries Metric\n\n```\n0.1.0 passed this metric\n```\n\n# Testing File Metric\n\n```\n0.1.0 failed this metric\n```\n\n```\nFailure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a TESTING.md file\n```\n\n# Version Tag Metric\n\n```\n0.1.0 failed this metric\n```\n\n```\nFailure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must include a tag that matches this cookbook version number\n```\n", "html": "\n\n\n\n\n \n \n \n chef-arkserver Cookbook - Chef Supermarket\n \n \n \n \n \n \n \n\n \n\n\n\n \n \n \n
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\n Menu\n
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Chef Supermarket

\n
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Adoptable Cookbooks List

\n\n

\n Looking for a cookbook to adopt? You can now see a list of cookbooks available for adoption!\n
\n List of Adoptable Cookbooks\n

\n
\n\n
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Supermarket Belongs to the Community

\n\n

Supermarket belongs to the community. While Chef has the responsibility to keep it running and be stewards of its functionality, what it does and how it works is driven by the community. The chef/supermarket repository will continue to be where development of the Supermarket application takes place. Come be part of shaping the direction of Supermarket by opening issues and pull requests or by joining us on the Chef Mailing List.

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View our collection of guides, documentation, and articles

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\n chef-arkserver\n \n \n \n (15) Versions\n \n 0.1.0\n \n \n\n \n Follow0\n\n\n

\n\n

Installs and Configures Ark Survival Evolved Dedicated Server

\n\n
\n
\n
\n Policyfile\n
\n\n
\n Berkshelf\n
\n\n
\n Knife\n
\n
\n\n
\n
\n 
cookbook 'chef-arkserver', '= 0.1.0', :supermarket
\n
\n\n
\n 
cookbook 'chef-arkserver', '= 0.1.0'
\n
\n\n
\n 
knife supermarket install chef-arkserver
\n 
knife supermarket download chef-arkserver
\n
\n
\n
\n\n\n
\n
README
\n
Dependencies
\n
\n Quality\n \n 43%\n \n
\n
\n
\n
\n

Ark: Survival Evolved Chef Cookbook

\n\n

\"Build

\n\n

This cookbook will install and configure an Ark: Survival Evolved dedicated
\nserver. Utilize the following README for more information on how to utilize this
\ncookbook.

\n\n

Usage

\n\n

Include chef-arkserver in your node's runlist

\n\n

Configuration

\n\n

In order to properly utilize this cookbook, a suite of default configuration
\noptions are included in the cookbook's attributes/default.rb file. You will
\nneed to override the default options in your cookbook's node configuration.

\n\n

An example of overriding configuration options is structured as follows:

\n
\n 
{\n  \"name\": \"my.ark.server.net\",\n  \"chef_environment\": \"_default\",\n  \"normal\": {\n    \"ark\": {\n      \"gameserver\": {\n        \"configuration\": {\n          \"map\": \"ScorchedEarth_P\",\n          \"ServerSettings\": {\n            \"XPMultiplier\": \"5.0000\",\n            \"TamingSpeedMultiplier\": \"2.0000\",\n            \"RCONPort\": 27025\n          },\n          \"SessionSettings\": {\n            \"SessionName\": \"My Ark Dedicated Server\"\n          },\n          \"MessageOfTheDay\": {\n            \"Message\": \"Welcome to my ARK Server dot Net\\nPlease enjoy your stay!\"\n          }\n        }\n      }\n    },\n    \"tags\": [\n\n    ]\n  },\n  \"policy_name\": null,\n  \"policy_group\": null,\n  \"run_list\": [\n  \"role[base]\",\n  \"recipe[]\"\n]\n\n}\n
\n
\n\n

This configuration will build your GameUserSettings.ini file with the requested
\noverrided options defined above.

\n\n
\n
\n

Dependent cookbooks

\n \n \n \n \n\n\n \n
chef-steamcmd >= 0.0.0
\n\n

Contingent cookbooks

\n

There are no cookbooks that are contingent upon this one.

\n
\n
\n

Collaborator Number Metric\n 
\n            
0.1.0 failed this metric
\n 
Failure: Cookbook has 0 collaborators. A cookbook must have at least 2 collaborators to pass this metric.
\n \n

Contributing File Metric\n 
\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a CONTRIBUTING.md file
\n \n

Foodcritic Metric\n 
\n            
0.1.0 passed this metric
\n \n

License Metric\n 
\n            
0.1.0 passed this metric
\n \n

No Binaries Metric\n 
\n            
0.1.0 passed this metric
\n \n

Testing File Metric\n 
\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must contain a TESTING.md file
\n \n

Version Tag Metric\n 
\n            
0.1.0 failed this metric
\n 
Failure: To pass this metric, your cookbook metadata must include a source url, the source url must be in the form of https://github.com/user/repo, and your repo must include a tag that matches this cookbook version number
\n \n
\n

\n\n
\n
\n\n
\n
\n \n \"Nick\n f0rkz\n Nick Gray\n
\n\n
\n\n
\n\n\n\n\n

\n Details\n

\n\n
\n\n
\n View Source\n\n View Issues\n
\n\n

\n Updated September 8, 2017\n Created on September 8, 2017\n

\n\n

Supported Platforms

\n
    \n
  • \n \"debian\n
    • \n
  • \n \"ubuntu\n
    • \n
\n\n\n\n

License

\n

Apache-2.0\n

\n\n

Required Chef Infra Client

\n

(>= 12.1)

\n\n\n Download Cookbook\n
\n
\n\n
\n\n
\n \n\n \n
\n
\n\n \n \n\n"} diff --git a/bench/data/groundtruth/math_1.jsonl b/bench/data/groundtruth/math_1.jsonl index 1ff3c8b1..82cfd9aa 100644 --- a/bench/data/groundtruth/math_1.jsonl +++ b/bench/data/groundtruth/math_1.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "paragraph", "raw_content": "

Consider the following system of equations:

", "content": [{"c": "Consider the following system of equations:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

where \\mathbf{x},\\mathbf{y},\\mathbf{z}, and\\mathbf{u}are1\\times nvectors andD_{\\mathbf{x}},D_{\\mathbf{y}}, andD_{\\mathbf{z}}are diagonaln\\times nmatrices with\\mathbf{x},\\mathbf{y}, and\\mathbf{z}, respectively, along their diagonals (i.e.,D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})).

", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{z}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "\\mathbf{u}", "t": "equation-inline"}, {"c": "are", "t": "text"}, {"c": "1\\times n", "t": "equation-inline"}, {"c": "vectors and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "D_{\\mathbf{z}}", "t": "equation-inline"}, {"c": "are diagonal", "t": "text"}, {"c": "n\\times n", "t": "equation-inline"}, {"c": "matrices with", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "\\mathbf{z}", "t": "equation-inline"}, {"c": ", respectively, along their diagonals (i.e.,", "t": "text"}, {"c": "D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})", "t": "equation-inline"}, {"c": ").", "t": "text"}]}, {"type": "paragraph", "raw_content": "

My question is whether it is possible to solve for \\mathbf{y}here \u2013 both the\\mathbf{y}as a vector and the\\mathbf{y}along the diagonal ofD_{\\mathbf{y}}. The problem is that I do not know of operations to pull\\mathbf{y}out ofD_{\\mathbf{y}}. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussedhere. But I am not sure how this would work in this case, as it would be necessary to distribute \\mathbf{y}out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

", "content": [{"c": "My question is whether it is possible to solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "here \u2013 both the", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as a vector and the", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "along the diagonal of", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". The problem is that I do not know of operations to pull", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "out of", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$", "content": {"math_content": "(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

where \\odotis the Hadamard product ande^T = (1,1,\\ldots)\\in\\mathbb R^n. So, would it be possible to distribute out\\mathbf{y}and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to\\mathbf{y}, notD_{\\mathbf{y}}:

", "content": [{"c": "where", "t": "text"}, {"c": "\\odot", "t": "equation-inline"}, {"c": "is the Hadamard product and", "t": "text"}, {"c": "e^T = (1,1,\\ldots)\\in\\mathbb R^n", "t": "equation-inline"}, {"c": ". So, would it be possible to distribute out", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ", not", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ":", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$", "content": {"math_content": "\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

Edit: Oh, it seems that extracting \\mathbf{y}in this case would be a simple as rewriting the equation above as

", "content": [{"c": "Edit: Oh, it seems that extracting", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in this case would be a simple as rewriting the equation above as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

because rewriting the equation this way would not change the terms along the diagonal of \\mathbf{z}D_{\\mathbf{y}}. Then we can write

", "content": [{"c": "because rewriting the equation this way would not change the terms along the diagonal of", "t": "text"}, {"c": "\\mathbf{z}D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". Then we can write", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$", "content": {"math_content": "\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

But then how would solve for \\mathbf{y}in the following?

", "content": [{"c": "But then how would solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in the following?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

I think that in this case, it would not be possible to solve for a single vector \\mathbf{y}as in the previous case. Instead, we would have a system of polynomials:

", "content": [{"c": "I think that in this case, it would not be possible to solve for a single vector", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as in the previous case. Instead, we would have a system of polynomials:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$", "content": {"math_content": "\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

where \\mathbf{y^2}is a vector where the elements are the squares of the corresponding elements of\\mathbf{y}\u2013 that is,\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}.

", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{y^2}", "t": "equation-inline"}, {"c": "is a vector where the elements are the squares of the corresponding elements of", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "\u2013 that is,", "t": "text"}, {"c": "\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

Next, what if we had an equation as follows?

", "content": [{"c": "Next, what if we had an equation as follows?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

where \\mathbf{M}is ann \\times nmatrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal ofD_{\\mathbf{yM}}, we have some linear combination.

", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "is an", "t": "text"}, {"c": "n \\times n", "t": "equation-inline"}, {"c": "matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of", "t": "text"}, {"c": "D_{\\mathbf{yM}}", "t": "equation-inline"}, {"c": ", we have some linear combination.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

I think in this case, the polynomial system of equations would be rather complicated. We would have the vector \\mathbf{y^2}again, but this time multiplied by some diagonal matrix based on the elements of\\mathbf{M}andD_{\\mathbf{x}}that is multiplied by a scalar (n, I believe). Then, we would have a set of vectors (n-1, I believe) that each have as elements different products of the elements in\\mathbf{y}(e.g.,y_1 y_3) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in\\mathbf{M}andD_{\\mathbf{x}}.

", "content": [{"c": "I think in this case, the polynomial system of equations would be rather complicated. We would have the vector", "t": "text"}, {"c": "\\mathbf{y^2}", "t": "equation-inline"}, {"c": "again, but this time multiplied by some diagonal matrix based on the elements of", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": "that is multiplied by a scalar (", "t": "text"}, {"c": "n", "t": "equation-inline"}, {"c": ", I believe). Then, we would have a set of vectors (", "t": "text"}, {"c": "n-1", "t": "equation-inline"}, {"c": ", I believe) that each have as elements different products of the elements in", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "(e.g.,", "t": "text"}, {"c": "y_1 y_3", "t": "equation-inline"}, {"c": ") and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

So, I have two questions here:

", "content": [{"c": "So, I have two questions here:", "t": "text"}]}, {"type": "list", "raw_content": "
  1. Is my general intuition about what this equation would look like correct?
  2. Are there techniques to solve for the elements of \\mathbf{y}in this system?
", "content": {"items": [[[{"c": "Is my general intuition about what this equation would look like correct?", "t": "text"}]], [[{"c": "Are there techniques to solve for the elements of ", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in this system?", "t": "text"}]]], "ordered": true}}]], "main_html": "

Consider the following system of equations:

$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

where \\mathbf{x},\\mathbf{y},\\mathbf{z}, and\\mathbf{u}are1\\times nvectors andD_{\\mathbf{x}},D_{\\mathbf{y}}, andD_{\\mathbf{z}}are diagonaln\\times nmatrices with\\mathbf{x},\\mathbf{y}, and\\mathbf{z}, respectively, along their diagonals (i.e.,D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})).

My question is whether it is possible to solve for \\mathbf{y}here \u2013 both the\\mathbf{y}as a vector and the\\mathbf{y}along the diagonal ofD_{\\mathbf{y}}. The problem is that I do not know of operations to pull\\mathbf{y}out ofD_{\\mathbf{y}}. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussedhere. But I am not sure how this would work in this case, as it would be necessary to distribute \\mathbf{y}out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

$$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$

where \\odotis the Hadamard product ande^T = (1,1,\\ldots)\\in\\mathbb R^n. So, would it be possible to distribute out\\mathbf{y}and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to\\mathbf{y}, notD_{\\mathbf{y}}:

$$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$

Edit: Oh, it seems that extracting \\mathbf{y}in this case would be a simple as rewriting the equation above as

$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$

because rewriting the equation this way would not change the terms along the diagonal of \\mathbf{z}D_{\\mathbf{y}}. Then we can write

$$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$

But then how would solve for \\mathbf{y}in the following?

$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

I think that in this case, it would not be possible to solve for a single vector \\mathbf{y}as in the previous case. Instead, we would have a system of polynomials:

$$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$

where \\mathbf{y^2}is a vector where the elements are the squares of the corresponding elements of\\mathbf{y}\u2013 that is,\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}.

Next, what if we had an equation as follows?

$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$

where \\mathbf{M}is ann \\times nmatrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal ofD_{\\mathbf{yM}}, we have some linear combination.

I think in this case, the polynomial system of equations would be rather complicated. We would have the vector \\mathbf{y^2}again, but this time multiplied by some diagonal matrix based on the elements of\\mathbf{M}andD_{\\mathbf{x}}that is multiplied by a scalar (n, I believe). Then, we would have a set of vectors (n-1, I believe) that each have as elements different products of the elements in\\mathbf{y}(e.g.,y_1 y_3) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in\\mathbf{M}andD_{\\mathbf{x}}.

So, I have two questions here:

  1. Is my general intuition about what this equation would look like correct?
  2. Are there techniques to solve for the elements of \\mathbf{y}in this system?
", "statics": {"paragraph": 13, "paragraph.text": 57, "equation-interline": 8, "paragraph.equation-inline": 44, "list": 1, "list.text": 3, "list.equation-inline": 1}, "url": "https://stackoverflow.com/questions/18327314/how-to-allow-http-content-within-an-iframe-on-a-https-site", "content": "Consider the following system of equations:\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{x}$ , $\\mathbf{y}$ , $\\mathbf{z}$ , and $\\mathbf{u}$ are $1\\times n$ vectors and $D_{\\mathbf{x}}$ , $D_{\\mathbf{y}}$ , and $D_{\\mathbf{z}}$ are diagonal $n\\times n$ matrices with $\\mathbf{x}$ , $\\mathbf{y}$ , and $\\mathbf{z}$ , respectively, along their diagonals (i.e., $D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})$ ).\n\nMy question is whether it is possible to solve for $\\mathbf{y}$ here \u2013 both the $\\mathbf{y}$ as a vector and the $\\mathbf{y}$ along the diagonal of $D_{\\mathbf{y}}$ . The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$ . The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write\n\n$$\n(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}\n$$\n\nwhere $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$ . So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$ , not $D_{\\mathbf{y}}$ :\n\n$$\n\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}\n$$\n\nEdit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nbecause rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$ . Then we can write\n\n$$\n\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}\n$$\n\nBut then how would solve for $\\mathbf{y}$ in the following?\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nI think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:\n\n$$\n\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ \u2013 that is, $\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}$ .\n\nNext, what if we had an equation as follows?\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$ , we have some linear combination.\n\nI think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ( $n$ , I believe). Then, we would have a set of vectors ( $n-1$ , I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$ ) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$ .\n\nSo, I have two questions here:\n\n1. Is my general intuition about what this equation would look like correct?\n2. Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?\n", "html": "\r\n\r\n\r\n\r\n\r\n \r\n\r\n \r\n\r\n linear algebra - Solving for vector contained in a diagonal matrix - Mathematics Stack Exchange\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n\r\n \r\n \r\n\r\n\r\n \r\n \r\n \r\n\r\n \r\n \r\n \r\n \r\n\r\n \r\n \r\n\r\n \r\n \r\n \r\n\r\n\r\n \r\n \r\n\r\n\r\n\r\n \r\n \r\n \r\n \r\n\r\n \r\n
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Consider the following system of equations:

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$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

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where $\\mathbf{x}$, $\\mathbf{y}$, $\\mathbf{z}$, and $\\mathbf{u}$ are $1\\times n$ vectors and $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$ are diagonal $n\\times n$ matrices with $\\mathbf{x}$, $\\mathbf{y}$, and $\\mathbf{z}$, respectively, along their diagonals (i.e., $D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})$).

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My question is whether it is possible to solve for $\\mathbf{y}$ here \u2013 both the $\\mathbf{y}$ as a vector and the $\\mathbf{y}$ along the diagonal of $D_{\\mathbf{y}}$. The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

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$$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$

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where $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$. So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$, not $D_{\\mathbf{y}}$:

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$$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$

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Edit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as

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$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$

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because rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$. Then we can write

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$$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$

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But then how would solve for $\\mathbf{y}$ in the following?

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$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

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I think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:

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$$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$

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where $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ \u2013 that is, $\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}$.

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Next, what if we had an equation as follows?

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$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$

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where $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$, we have some linear combination.

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I think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ($n$, I believe). Then, we would have a set of vectors ($n-1$, I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$.

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So, I have two questions here:

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  1. Is my general intuition about what this equation would look like correct?
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  2. Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?
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\r\n asked Mar 29, 2021 at 23:19\r\n
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I am not sure what the notation $D_\\mathbf{x}$, $D_\\mathbf{y}$, and $D_\\mathbf{z}$ represent and if the importance is significant, but assuming $D_{\\mathbf{x}}$ is invertible, you may solve for $\\mathbf{y}$ as follows given the original equation:

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$$ \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}} $$\r\n$$ \\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} $$\r\n$$ \\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1} $$

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Note: I can't verify that your edit is correct because I don't understand what $D_{\\mathbf{x} + \\mathbf{z}}$ represents.

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Edit 1:

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Thank you for the response as I didn't realize the meaning of the matrices $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$. Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.

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Since $D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})$ and similarly for the other matrices, we have

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$$\r\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\r\n$$

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and multiplying through we have

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$$\r\n\\begin{align}\r\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\r\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\r\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\r\n\\end{align}\r\n$$

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Therefore, this can be written in matrix form as

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$$\r\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\r\n$$

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or more concisely as

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$$ \\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}} $$

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which is exactly what is given in your edit:

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$$ \\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}$$

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Edit 2:

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As for the case where you have $\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}$ where $\\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}$, you would not be able to solve for $\\mathbf{y}$ as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):

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$$\r\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\r\n$$

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where we cannot solve for the matrix containing the $y$ variables because we cannot eliminate $\\mathbf{x}$ from the left side of the equation.

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\r\n answered Mar 30, 2021 at 8:19\r\n
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    \r\n $\\begingroup$\r\n My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$. Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$. Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$. My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$.\r\n $\\endgroup$\r\n
    \r\n– Ryan da Silva\r\n
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    \r\n $\\begingroup$\r\n @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck!\r\n $\\endgroup$\r\n
    \r\n– Ralff\r\n
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    \r\n $\\begingroup$\r\n Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on.\r\n $\\endgroup$\r\n
    \r\n– Ryan da Silva\r\n
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    \r\n $\\begingroup$\r\n @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result.\r\n $\\endgroup$\r\n
    \r\n– Ralff\r\n
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In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps to cite. Taking the electromagnetic field \\hat A to be an (operator-valued distributional) 1-form, we can write a smeared operator \\hat A_H in terms of the inner product on the exterior algebra as \\bigl<\\hat A,H\\bigr> (taking the test function H also to be a 1-form, but satisfying Schwartz space-like smoothness conditions in real space and in fourier space instead of being a distribution). \\hat A projects into annihilation and creation parts, \\hat A^+ and \\hat A^- respectively, \\hat A=\\hat A^+ +\\hat A^-, for which the action on the vacuum is defined by \\hat A^+\\left|0\\right>=0, and we have the commutation relations \\Bigl[\\bigl<\\hat A^+,H\\bigr>,\\bigl<\\hat A^-,J\\bigr>\\Bigr]=\\left<H,E(J)\\right>, where \\widetilde{E(J)}(k)=2\\pi\\delta(k^2)\\theta(k_0)\\tilde J(k) projects to the positive frequency forward light-cone.

", "content": [{"c": "In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps to cite. Taking the electromagnetic field", "t": "text"}, {"c": "\\hat A", "t": "equation-inline"}, {"c": "to be an (operator-valued distributional) 1-form, we can write a smeared operator", "t": "text"}, {"c": "\\hat A_H", "t": "equation-inline"}, {"c": "in terms of the inner product on the exterior algebra as", "t": "text"}, {"c": "\\bigl<\\hat A,H\\bigr>", "t": "equation-inline"}, {"c": "(taking the test function", "t": "text"}, {"c": "H", "t": "equation-inline"}, {"c": "also to be a 1-form, but satisfying Schwartz space-like smoothness conditions in real space and in fourier space instead of being a distribution).", "t": "text"}, {"c": "\\hat A", "t": "equation-inline"}, {"c": "projects into annihilation and creation parts,", "t": "text"}, {"c": "\\hat A^+", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "\\hat A^-", "t": "equation-inline"}, {"c": "respectively,", "t": "text"}, {"c": "\\hat A=\\hat A^+ +\\hat A^-", "t": "equation-inline"}, {"c": ", for which the action on the vacuum is defined by", "t": "text"}, {"c": "\\hat A^+\\left|0\\right>=0", "t": "equation-inline"}, {"c": ", and we have the commutation relations", "t": "text"}, {"c": "\\Bigl[\\bigl<\\hat A^+,H\\bigr>,\\bigl<\\hat A^-,J\\bigr>\\Bigr]=\\left", "t": "equation-inline"}, {"c": ", where", "t": "text"}, {"c": "\\widetilde{E(J)}(k)=2\\pi\\delta(k^2)\\theta(k_0)\\tilde J(k)", "t": "equation-inline"}, {"c": "projects to the positive frequency forward light-cone.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

The commutation relations are not positive semi-definite for arbitrary test functions $H$, $\\Bigl[\\bigl<\\hat A^+,H^*\\bigr>,\\bigl<\\hat A^-,H\\bigr>\\Bigr]\\not\\ge 0$, which is fixed by the Gupta-bleuler condition, which can be stated in this formalism as $\\delta\\hat A^+\\left|\\psi\\right>=0$, for all states $\\left|\\psi\\right>$, not just for the vacuum state.

", "content": [{"c": "The commutation relations are not positive semi-definite for arbitrary test functions $H$, $\\Bigl[\\bigl<\\hat A^+,H^*\\bigr>,\\bigl<\\hat A^-,H\\bigr>\\Bigr]\\not\\ge 0$, which is fixed by the Gupta-bleuler condition, which can be stated in this formalism as $\\delta\\hat A^+\\left|\\psi\\right>=0$, for all states $\\left|\\psi\\right>$, not just for the vacuum state.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

By the Hodge decomposition theorem, we can write the test function H uniquely as H=d\\phi+\\delta F+\\omega, where \\phi is a 0-form, F is a 2-form, and \\omega is a harmonic 1-form, so we can write\n$$\\bigl<\\hat A^+\\!,H\\bigr>\\left|\\psi\\right>\\!\n =\\bigl<\\hat A^+\\!,d\\phi+\\delta F+\\omega\\bigr>\\left|\\psi\\right>\\!\n =\\Bigl(\\!\\bigl<\\delta\\hat A^+,\\phi\\bigr>+\\bigl<\\hat A^+\\!,\\delta F+\\omega\\bigr>\\!\\Bigr)\n \\left|\\psi\\right>\\!\n =\\bigl<\\hat A^+\\!,\\delta F+\\omega\\bigr>\\left|\\psi\\right>.$$\nThe harmonic 1-form \\omega has to be zero to satisfy the Schwartz space condition, and by the left action of \\hat A^- on arbitrary states we have the same projection of the arbitrary test function H to \\delta F, so all we are left with is the electromagnetic field observable \\hat\\Phi_F=\\hat A_{\\delta F}=\\bigl<\\hat A,\\delta F\\bigr>. Unsurprisingly, in the free field case, in the absence of charges, the electromagnetic potential observable is exactly equivalent to just the electromagnetic field observable, for which we can verify that \\Bigl[\\bigl<\\hat A^+,\\delta F^*\\bigr>,\\bigl<\\hat A^-,\\delta F\\bigr>\\Bigr]\\ge 0, using which we can use the GNS construction of a free field Hilbert space.

", "content": [{"c": "By the Hodge decomposition theorem, we can write the test function", "t": "text"}, {"c": "H", "t": "equation-inline"}, {"c": "uniquely as", "t": "text"}, {"c": "H=d\\phi+\\delta F+\\omega", "t": "equation-inline"}, {"c": ", where", "t": "text"}, {"c": "\\phi", "t": "equation-inline"}, {"c": "is a 0-form,", "t": "text"}, {"c": "F", "t": "equation-inline"}, {"c": "is a 2-form, and", "t": "text"}, {"c": "\\omega", "t": "equation-inline"}, {"c": "is a harmonic 1-form, so we can write\n$$\\bigl<\\hat A^+\\!,H\\bigr>\\left|\\psi\\right>\\!\n =\\bigl<\\hat A^+\\!,d\\phi+\\delta F+\\omega\\bigr>\\left|\\psi\\right>\\!\n =\\Bigl(\\!\\bigl<\\delta\\hat A^+,\\phi\\bigr>+\\bigl<\\hat A^+\\!,\\delta F+\\omega\\bigr>\\!\\Bigr)\n \\left|\\psi\\right>\\!\n =\\bigl<\\hat A^+\\!,\\delta F+\\omega\\bigr>\\left|\\psi\\right>.$$\nThe harmonic 1-form", "t": "text"}, {"c": "\\omega", "t": "equation-inline"}, {"c": "has to be zero to satisfy the Schwartz space condition, and by the left action of", "t": "text"}, {"c": "\\hat A^-", "t": "equation-inline"}, {"c": "on arbitrary states we have the same projection of the arbitrary test function", "t": "text"}, {"c": "H", "t": "equation-inline"}, {"c": "to", "t": "text"}, {"c": "\\delta F", "t": "equation-inline"}, {"c": ", so all we are left with is the electromagnetic field observable", "t": "text"}, {"c": "\\hat\\Phi_F=\\hat A_{\\delta F}=\\bigl<\\hat A,\\delta F\\bigr>", "t": "equation-inline"}, {"c": ". Unsurprisingly, in the free field case, in the absence of charges, the electromagnetic potential observable is exactly equivalent to just the electromagnetic field observable, for which we can verify that", "t": "text"}, {"c": "\\Bigl[\\bigl<\\hat A^+,\\delta F^*\\bigr>,\\bigl<\\hat A^-,\\delta F\\bigr>\\Bigr]\\ge 0", "t": "equation-inline"}, {"c": ", using which we can use the GNS construction of a free field Hilbert space.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

So this is a reference request. Is there any literature that uses this kind of mathematical formalism for the quantized electromagnetic field? Even vaguely the same! My sense is that AQFT has moved to more abstract methods, while more practical interacting QFT has become historically committed to index notations that are little changed from 50 years ago, even though the methods of interacting quantum fields have changed in many other ways, and that mathematicians who take on the structures of QFT, as Folland does in http://www.amazon.com/Quantum-Theory-Mathematical-Surveys-Monographs/dp/0821847058, make strenuous efforts not to let their notation and methods stray too far from the mainline.

", "content": [{"c": "So this is a reference request. Is there any literature that uses this kind of mathematical formalism for the quantized electromagnetic field? Even vaguely the same! My sense is that AQFT has moved to more abstract methods, while more practical interacting QFT has become historically committed to index notations that are little changed from 50 years ago, even though the methods of interacting quantum fields have changed in many other ways, and that mathematicians who take on the structures of QFT, as Folland does in http://www.amazon.com/Quantum-Theory-Mathematical-Surveys-Monographs/dp/0821847058, make strenuous efforts not to let their notation and methods stray too far from the mainline.", "t": "text"}]}, {"type": "list", "raw_content": "
    ", "content": {"items": [], "ordered": false}}, {"type": "list", "raw_content": "
    • Just stumbled across this again... I'm not familiar with the literature but I'll take a look around and see if I can find anything, since this is a really good question and I think for the good of the site, we need to be able to come up with answers for these sorts of things.
      \n\u2013\u00a0David Z
      CommentedJul 18, 2011 at 3:41
    • 4
      There is no difference between your formalism and any of the standard ones. A is a one form in all of them.CommentedSep 3, 2011 at 6:46
    ", "content": {"items": [[[{"c": "Just stumbled across this again... I'm not familiar with the literature but I'll take a look around and see if I can find anything, since this is a really good question and I think for the good of the site, we need to be able to come up with answers for these sorts of things.", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "David Z", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Jul 18, 2011 at 3:41", "t": "text"}]], [[{"c": "4", "t": "text"}, {"c": "There is no difference between your formalism and any of the standard ones. A is a one form in all of them.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Sep 3, 2011 at 6:46", "t": "text"}]]], "ordered": false}}, {"type": "title", "raw_content": "

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      In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps to cite. Taking the electromagnetic field \\hat A to be an (operator-valued distributional) 1-form, we can write a smeared operator \\hat A_H in terms of the inner product on the exterior algebra as \\bigl<\\hat A,H\\bigr> (taking the test function H also to be a 1-form, but satisfying Schwartz space-like smoothness conditions in real space and in fourier space instead of being a distribution). \\hat A projects into annihilation and creation parts, \\hat A^+ and \\hat A^- respectively, \\hat A=\\hat A^+ +\\hat A^-, for which the action on the vacuum is defined by \\hat A^+\\left|0\\right>=0, and we have the commutation relations \\Bigl[\\bigl<\\hat A^+,H\\bigr>,\\bigl<\\hat A^-,J\\bigr>\\Bigr]=\\left<H,E(J)\\right>, where \\widetilde{E(J)}(k)=2\\pi\\delta(k^2)\\theta(k_0)\\tilde J(k) projects to the positive frequency forward light-cone.

      The commutation relations are not positive semi-definite for arbitrary test functions $H$, $\\Bigl[\\bigl<\\hat A^+,H^*\\bigr>,\\bigl<\\hat A^-,H\\bigr>\\Bigr]\\not\\ge 0$, which is fixed by the Gupta-bleuler condition, which can be stated in this formalism as $\\delta\\hat A^+\\left|\\psi\\right>=0$, for all states $\\left|\\psi\\right>$, not just for the vacuum state.

      By the Hodge decomposition theorem, we can write the test function H uniquely as H=d\\phi+\\delta F+\\omega, where \\phi is a 0-form, F is a 2-form, and \\omega is a harmonic 1-form, so we can write\n$$\\bigl<\\hat A^+\\!,H\\bigr>\\left|\\psi\\right>\\!\n =\\bigl<\\hat A^+\\!,d\\phi+\\delta F+\\omega\\bigr>\\left|\\psi\\right>\\!\n =\\Bigl(\\!\\bigl<\\delta\\hat A^+,\\phi\\bigr>+\\bigl<\\hat A^+\\!,\\delta F+\\omega\\bigr>\\!\\Bigr)\n \\left|\\psi\\right>\\!\n =\\bigl<\\hat A^+\\!,\\delta F+\\omega\\bigr>\\left|\\psi\\right>.$$\nThe harmonic 1-form \\omega has to be zero to satisfy the Schwartz space condition, and by the left action of \\hat A^- on arbitrary states we have the same projection of the arbitrary test function H to \\delta F, so all we are left with is the electromagnetic field observable \\hat\\Phi_F=\\hat A_{\\delta F}=\\bigl<\\hat A,\\delta F\\bigr>. Unsurprisingly, in the free field case, in the absence of charges, the electromagnetic potential observable is exactly equivalent to just the electromagnetic field observable, for which we can verify that \\Bigl[\\bigl<\\hat A^+,\\delta F^*\\bigr>,\\bigl<\\hat A^-,\\delta F\\bigr>\\Bigr]\\ge 0, using which we can use the GNS construction of a free field Hilbert space.

      So this is a reference request. Is there any literature that uses this kind of mathematical formalism for the quantized electromagnetic field? Even vaguely the same! My sense is that AQFT has moved to more abstract methods, while more practical interacting QFT has become historically committed to index notations that are little changed from 50 years ago, even though the methods of interacting quantum fields have changed in many other ways, and that mathematicians who take on the structures of QFT, as Folland does in http://www.amazon.com/Quantum-Theory-Mathematical-Surveys-Monographs/dp/0821847058, make strenuous efforts not to let their notation and methods stray too far from the mainline.

        • Just stumbled across this again... I'm not familiar with the literature but I'll take a look around and see if I can find anything, since this is a really good question and I think for the good of the site, we need to be able to come up with answers for these sorts of things.
          \n\u2013\u00a0David Z
          CommentedJul 18, 2011 at 3:41
        • 4
          There is no difference between your formalism and any of the standard ones. A is a one form in all of them.CommentedSep 3, 2011 at 6:46

        \nKnow someone who can answer? Share a link to this question via email, Twitter, or Facebook.

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          or

          ", "statics": {"paragraph": 4, "paragraph.text": 26, "paragraph.equation-inline": 22, "list": 3, "list.text": 9, "title": 3}, "url": "https://physics.stackexchange.com/questions/10098/1-form-formulation-of-quantized-electromagnetism", "content": "In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps to cite. Taking the electromagnetic field $\\hat A$ to be an (operator-valued distributional) 1-form, we can write a smeared operator $\\hat A_H$ in terms of the inner product on the exterior algebra as $\\bigl<\\hat A,H\\bigr>$ (taking the test function $H$ also to be a 1-form, but satisfying Schwartz space-like smoothness conditions in real space and in fourier space instead of being a distribution). $\\hat A$ projects into annihilation and creation parts, $\\hat A^+$ and $\\hat A^-$ respectively, $\\hat A=\\hat A^+ +\\hat A^-$ , for which the action on the vacuum is defined by $\\hat A^+\\left|0\\right>=0$ , and we have the commutation relations $\\Bigl[\\bigl<\\hat A^+,H\\bigr>,\\bigl<\\hat A^-,J\\bigr>\\Bigr]=\\left$ , where $\\widetilde{E(J)}(k)=2\\pi\\delta(k^2)\\theta(k_0)\\tilde J(k)$ projects to the positive frequency forward light-cone.\n\nThe commutation relations are not positive semi-definite for arbitrary test functions $H$, $\\Bigl[\\bigl<\\hat A^+,H^*\\bigr>,\\bigl<\\hat A^-,H\\bigr>\\Bigr]\\not\\ge 0$, which is fixed by the Gupta-bleuler condition, which can be stated in this formalism as $\\delta\\hat A^+\\left|\\psi\\right>=0$, for all states $\\left|\\psi\\right>$, not just for the vacuum state.\n\nBy the Hodge decomposition theorem, we can write the test function $H$ uniquely as $H=d\\phi+\\delta F+\\omega$ , where $\\phi$ is a 0-form, $F$ is a 2-form, and $\\omega$ is a harmonic 1-form, so we can write\n$$\\bigl<\\hat A^+\\!,H\\bigr>\\left|\\psi\\right>\\!\n =\\bigl<\\hat A^+\\!,d\\phi+\\delta F+\\omega\\bigr>\\left|\\psi\\right>\\!\n =\\Bigl(\\!\\bigl<\\delta\\hat A^+,\\phi\\bigr>+\\bigl<\\hat A^+\\!,\\delta F+\\omega\\bigr>\\!\\Bigr)\n \\left|\\psi\\right>\\!\n =\\bigl<\\hat A^+\\!,\\delta F+\\omega\\bigr>\\left|\\psi\\right>.$$\nThe harmonic 1-form $\\omega$ has to be zero to satisfy the Schwartz space condition, and by the left action of $\\hat A^-$ on arbitrary states we have the same projection of the arbitrary test function $H$ to $\\delta F$ , so all we are left with is the electromagnetic field observable $\\hat\\Phi_F=\\hat A_{\\delta F}=\\bigl<\\hat A,\\delta F\\bigr>$ . Unsurprisingly, in the free field case, in the absence of charges, the electromagnetic potential observable is exactly equivalent to just the electromagnetic field observable, for which we can verify that $\\Bigl[\\bigl<\\hat A^+,\\delta F^*\\bigr>,\\bigl<\\hat A^-,\\delta F\\bigr>\\Bigr]\\ge 0$ , using which we can use the GNS construction of a free field Hilbert space.\n\nSo this is a reference request. Is there any literature that uses this kind of mathematical formalism for the quantized electromagnetic field? Even vaguely the same! My sense is that AQFT has moved to more abstract methods, while more practical interacting QFT has become historically committed to index notations that are little changed from 50 years ago, even though the methods of interacting quantum fields have changed in many other ways, and that mathematicians who take on the structures of QFT, as Folland does in http://www.amazon.com/Quantum-Theory-Mathematical-Surveys-Monographs/dp/0821847058, make strenuous efforts not to let their notation and methods stray too far from the mainline.\n\n- Just stumbled across this again... I'm not familiar with the literature but I'll take a look around and see if I can find anything, since this is a really good question and I think for the good of the site, we need to be able to come up with answers for these sorts of things. \u2013 David Z Commented Jul 18, 2011 at 3:41\n- 4 There is no difference between your formalism and any of the standard ones. A is a one form in all of them. Commented Sep 3, 2011 at 6:46\n\n## Know someone who can answer? Share a link to this question via email , Twitter , or Facebook .\n\n## Browse other questions tagged\n\n## or ask your own question .\n", "html": "\n\n\n\n\n \n\n \n\n quantum field theory - 1-form formulation of quantized electromagnetism - Physics Stack Exchange\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n\n\n\n \n \n\n\n\n\n \n\n\n\n\n\n \n \n\n\n \n \n\n\n\n \n \n \n \n\n\n
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          In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps to cite. Taking the electromagnetic field $\\hat A$ to be an (operator-valued distributional) 1-form, we can write a smeared operator $\\hat A_H$ in terms of the inner product on the exterior algebra as $\\bigl<\\hat A,H\\bigr>$ (taking the test function $H$ also to be a 1-form, but satisfying Schwartz space-like smoothness conditions in real space and in fourier space instead of being a distribution). $\\hat A$ projects into annihilation and creation parts, $\\hat A^+$ and $\\hat A^-$ respectively, $\\hat A=\\hat A^+ +\\hat A^-$, for which the action on the vacuum is defined by $\\hat A^+\\left|0\\right>=0$, and we have the commutation relations $\\Bigl[\\bigl<\\hat A^+,H\\bigr>,\\bigl<\\hat A^-,J\\bigr>\\Bigr]=\\left<H,E(J)\\right>$, where $\\widetilde{E(J)}(k)=2\\pi\\delta(k^2)\\theta(k_0)\\tilde J(k)$ projects to the positive frequency forward light-cone.

          \n\n

          The commutation relations are not positive semi-definite for arbitrary test functions $H$, $\\Bigl[\\bigl<\\hat A^+,H^*\\bigr>,\\bigl<\\hat A^-,H\\bigr>\\Bigr]\\not\\ge 0$, which is fixed by the Gupta-bleuler condition, which can be stated in this formalism as $\\delta\\hat A^+\\left|\\psi\\right>=0$, for all states $\\left|\\psi\\right>$, not just for the vacuum state.

          \n\n

          By the Hodge decomposition theorem, we can write the test function $H$ uniquely as $H=d\\phi+\\delta F+\\omega$, where $\\phi$ is a 0-form, $F$ is a 2-form, and $\\omega$ is a harmonic 1-form, so we can write\n$$\\bigl<\\hat A^+\\!,H\\bigr>\\left|\\psi\\right>\\!\n =\\bigl<\\hat A^+\\!,d\\phi+\\delta F+\\omega\\bigr>\\left|\\psi\\right>\\!\n =\\Bigl(\\!\\bigl<\\delta\\hat A^+,\\phi\\bigr>+\\bigl<\\hat A^+\\!,\\delta F+\\omega\\bigr>\\!\\Bigr)\n \\left|\\psi\\right>\\!\n =\\bigl<\\hat A^+\\!,\\delta F+\\omega\\bigr>\\left|\\psi\\right>.$$\nThe harmonic 1-form $\\omega$ has to be zero to satisfy the Schwartz space condition, and by the left action of $\\hat A^-$ on arbitrary states we have the same projection of the arbitrary test function $H$ to $\\delta F$, so all we are left with is the electromagnetic field observable $\\hat\\Phi_F=\\hat A_{\\delta F}=\\bigl<\\hat A,\\delta F\\bigr>$. Unsurprisingly, in the free field case, in the absence of charges, the electromagnetic potential observable is exactly equivalent to just the electromagnetic field observable, for which we can verify that $\\Bigl[\\bigl<\\hat A^+,\\delta F^*\\bigr>,\\bigl<\\hat A^-,\\delta F\\bigr>\\Bigr]\\ge 0$, using which we can use the GNS construction of a free field Hilbert space.

          \n\n

          So this is a reference request. Is there any literature that uses this kind of mathematical formalism for the quantized electromagnetic field? Even vaguely the same! My sense is that AQFT has moved to more abstract methods, while more practical interacting QFT has become historically committed to index notations that are little changed from 50 years ago, even though the methods of interacting quantum fields have changed in many other ways, and that mathematicians who take on the structures of QFT, as Folland does in http://www.amazon.com/Quantum-Theory-Mathematical-Surveys-Monographs/dp/0821847058, make strenuous efforts not to let their notation and methods stray too far from the mainline.

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          \n asked May 19, 2011 at 17:01\n
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          \"Peter
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          \n Peter MorganPeter Morgan\n
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            \n $\\begingroup$\n Just stumbled across this again... I'm not familiar with the literature but I'll take a look around and see if I can find anything, since this is a really good question and I think for the good of the site, we need to be able to come up with answers for these sorts of things.\n $\\endgroup$\n
            \n– David Z\n
            \n \n Commented\n Jul 18, 2011 at 3:41\n \n
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            \n $\\begingroup$\n There is no difference between your formalism and any of the standard ones. A is a one form in all of them.\n $\\endgroup$\n
            \n– Ron Maimon\n
            \n \n Commented\n Sep 3, 2011 at 6:46\n \n
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          Since the discriminant $$\\Delta >0$$, the cubic equation has one real root and two conjugate complex roots.

          ", "content": [{"c": "Since the discriminant $$\\Delta >0$$, the cubic equation has one real root and two conjugate complex roots.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$ \\Delta=14.472222222222$$

          ", "content": {"math_content": "\\Delta=14.472222222222", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

          $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

          ", "content": {"math_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          In decimals,

          ", "content": [{"c": "In decimals,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

          ", "content": {"math_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Detailed Steps on Solution

          ", "content": [{"c": "Detailed Steps on Solution", "t": "text"}]}, {"type": "title", "raw_content": "

          1. Convert to depressed cubic equation

          ", "content": {"title_content": "1. Convert to depressed cubic equation", "level": "2"}}, {"type": "paragraph", "raw_content": "

          The idea is to convert general form of cubic equation

          ", "content": [{"c": "The idea is to convert general form of cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$ax^3+bx^2+cx+d = 0$$

          ", "content": {"math_content": "ax^3+bx^2+cx+d = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          to the form without quadratic term.

          ", "content": [{"c": "to the form without quadratic term.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$t^3+pt+q = 0$$

          ", "content": {"math_content": "t^3+pt+q = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          By substituting $$x$$ with $$t - \\dfrac{b}{3a}$$, the general cubic equation could be transformed to

          ", "content": [{"c": "By substituting $$x$$ with $$t - \\dfrac{b}{3a}$$, the general cubic equation could be transformed to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0 $$

          ", "content": {"math_content": "t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Compare with the depressed cubic equation. Then,

          ", "content": [{"c": "Compare with the depressed cubic equation. Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$p = \\dfrac{3ac-b^2}{3a^2}$$

          ", "content": {"math_content": "p = \\dfrac{3ac-b^2}{3a^2}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

          $$q = \\dfrac{2b^3-9abc+27a^2d}{27a^3} $$

          ", "content": {"math_content": "q = \\dfrac{2b^3-9abc+27a^2d}{27a^3}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Substitute the values of coefficients, $$p, q$$ is obtained as

          ", "content": [{"c": "Substitute the values of coefficients, $$p, q$$ is obtained as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}$$

          ", "content": {"math_content": "p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

          $$q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}$$

          ", "content": {"math_content": "q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          Use the substitution to transform

          ", "content": {"title_content": "Use the substitution to transform", "level": "3"}}, {"type": "paragraph", "raw_content": "

          Let $$p$$ and $$q$$ being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

          ", "content": [{"c": "Let $$p$$ and $$q$$ being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$t^3 +pt+q=0$$

          ", "content": {"math_content": "t^3 +pt+q=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Let $$x=t-\\dfrac{2}{3}$$

          ", "content": [{"c": "Let $$x=t-\\dfrac{2}{3}$$", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is transformed to

          ", "content": [{"c": "The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is transformed to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

          ", "content": {"math_content": "t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          2. Cardano's solution

          ", "content": {"title_content": "2. Cardano's solution", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Let $$t=u-v$$

          ", "content": [{"c": "Let $$t=u-v$$", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Cube both sides and extract common factor from two middle terms after expanding the bracket.

          ", "content": [{"c": "Cube both sides and extract common factor from two middle terms after expanding the bracket.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}$$

          ", "content": {"math_content": "\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Since $$u-v=t$$, substitution gives a linear term for the equation.\n Rearrange terms.

          ", "content": [{"c": "Since $$u-v=t$$, substitution gives a linear term for the equation.\n Rearrange terms.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$x^3+3uvx-u^3+v^3=0$$

          ", "content": {"math_content": "x^3+3uvx-u^3+v^3=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Compare the cubic equation with the original one (1)

          ", "content": [{"c": "Compare the cubic equation with the original one (1)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}$$

          ", "content": {"math_content": "\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          $$v=\\dfrac{20}{9u}$$ gives relationship between the two variables. Substitute the value of $$v$$ to the second equation

          ", "content": [{"c": "$$v=\\dfrac{20}{9u}$$ gives relationship between the two variables. Substitute the value of $$v$$ to the second equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}$$

          ", "content": {"math_content": "\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Simplifying gives,

          ", "content": [{"c": "Simplifying gives,", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$2

          ", "content": [{"c": "$$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$2", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Let $$m=u^3$$, then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by $$v^3=-\\dfrac{101}{27}+u^3$$.

          ", "content": [{"c": "Let $$m=u^3$$, then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by $$v^3=-\\dfrac{101}{27}+u^3$$.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0$$

          ", "content": {"math_content": "m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

          ", "content": [{"c": "Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

          ", "content": {"math_content": "\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          $$v^3$$ can be determined by the equation we deduced $$v^3-u^3=-\\dfrac{101}{27}$$. Then,

          ", "content": [{"c": "$$v^3$$ can be determined by the equation we deduced $$v^3-u^3=-\\dfrac{101}{27}$$. Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

          ", "content": {"math_content": "\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Now we have,

          ", "content": [{"c": "Now we have,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$ and $$v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$

          ", "content": {"math_content": "u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6} and v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Evaluating the simplest cubic equation $$x^3-A=0$$,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

          ", "content": [{"c": "Evaluating the simplest cubic equation $$x^3-A=0$$,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          If $$\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}$$, then its reciprocal is equal to its conjugate, $$\\dfrac{1}{\u03c9}=\\overline{\u03c9}$$.

          ", "content": [{"c": "If $$\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}$$, then its reciprocal is equal to its conjugate, $$\\dfrac{1}{\u03c9}=\\overline{\u03c9}$$.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}$$

          ", "content": {"math_content": "\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Similary, taking cubic root for $$u^3$$ and $$v^3$$ also gives 3 roots.

          ", "content": [{"c": "Similary, taking cubic root for $$u^3$$ and $$v^3$$ also gives 3 roots.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

          ", "content": {"math_content": "\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          For $$v_2$$ and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and $$u_3$$, which is the same in value.

          ", "content": [{"c": "For $$v_2$$ and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and $$u_3$$, which is the same in value.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

          ", "content": {"math_content": "\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Verification for the redicand in $$v$$.

          ", "content": [{"c": "Verification for the redicand in $$v$$.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          ", "content": {"math_content": "\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Since $$x=u-v$$, combining the real and imaginary parts gives\n 3 results for $$t$$

          ", "content": [{"c": "Since $$x=u-v$$, combining the real and imaginary parts gives\n 3 results for $$t$$", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          ", "content": {"math_content": "\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

          $$\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          ", "content": {"math_content": "\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

          $$\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          ", "content": {"math_content": "\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          3. Vieta's Substitution

          ", "content": {"title_content": "3. Vieta's Substitution", "level": "2"}}, {"type": "paragraph", "raw_content": "

          In Cardano' solution, $$t$$ is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$. This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

          ", "content": [{"c": "In Cardano' solution, $$t$$ is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$. This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$t=u-\\dfrac{p}{3u}$$

          ", "content": {"math_content": "t=u-\\dfrac{p}{3u}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Substitute the expression $$t=u-\\dfrac{20}{9u}$$ to the cubic equation

          ", "content": [{"c": "Substitute the expression $$t=u-\\dfrac{20}{9u}$$ to the cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0$$

          ", "content": {"math_content": "\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Expand brackets and cancel the like terms

          ", "content": [{"c": "Expand brackets and cancel the like terms", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0$$

          ", "content": {"math_content": "u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Then we get the same equation as (2)

          ", "content": [{"c": "Then we get the same equation as (2)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$

          ", "content": {"math_content": "u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          The rest of the steps will be the same as those of Cardano's solution

          ", "content": [{"c": "The rest of the steps will be the same as those of Cardano's solution", "t": "text"}]}, {"type": "title", "raw_content": "

          4. Euler's Solution

          ", "content": {"title_content": "4. Euler's Solution", "level": "2"}}, {"type": "title", "raw_content": "

          $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

          ", "content": {"title_content": "$$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

          ", "content": [{"c": "Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          $$t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} $$3

          ", "content": [{"c": "$$t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} $$3", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Let the root of the cubic equation be the sum of two cubic roots

          ", "content": [{"c": "Let the root of the cubic equation be the sum of two cubic roots", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          $$t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} $$4

          ", "content": [{"c": "$$t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} $$4", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          in which $$r_1$$ and $$r_2$$ are two roots of a quadratic equation

          ", "content": [{"c": "in which $$r_1$$ and $$r_2$$ are two roots of a quadratic equation", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          $$z^2-\\alpha z+ \u03b2=0 $$5

          ", "content": [{"c": "$$z^2-\\alpha z+ \u03b2=0 $$5", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Using Vieta's Formula, the following equations are established.

          ", "content": [{"c": "Using Vieta's Formula, the following equations are established.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2 $$

          ", "content": {"math_content": "r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          To determine $$\\alpha$$, $$\u03b2$$, cube both sides of the equation (4)

          ", "content": [{"c": "To determine $$\\alpha$$, $$\u03b2$$, cube both sides of the equation (4)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2 $$

          ", "content": {"math_content": "t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Substituting, the equation is simplified to

          ", "content": [{"c": "Substituting, the equation is simplified to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$t^3=3\\sqrt[3]{\u03b2}t+\\alpha $$

          ", "content": {"math_content": "t^3=3\\sqrt[3]{\u03b2}t+\\alpha", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Compare the cubic equation with (3), the following equations are established

          ", "content": [{"c": "Compare the cubic equation with (3), the following equations are established", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}$$

          ", "content": {"math_content": "\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Solving for $$\u03b2$$ gives

          ", "content": [{"c": "Solving for $$\u03b2$$ gives", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\u03b2=-\\dfrac{8000}{729} $$

          ", "content": {"math_content": "\u03b2=-\\dfrac{8000}{729}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          So the quadratic equation (5) is determined as

          ", "content": [{"c": "So the quadratic equation (5) is determined as", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          $$z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0$$6

          ", "content": [{"c": "$$z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0$$6", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solving the quadratic equation yields

          ", "content": [{"c": "Solving the quadratic equation yields", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}$$

          ", "content": {"math_content": "\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Therefore, one of the roots of the cubic equation could be obtained from (4).

          ", "content": [{"c": "Therefore, one of the roots of the cubic equation could be obtained from (4).", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          ", "content": {"math_content": "t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          in decimals,

          ", "content": [{"c": "in decimals,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$t_1=0.53778143658824 $$

          ", "content": {"math_content": "t_1=0.53778143658824", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          However, since the cube root of a quantity has triple values,

          ", "content": [{"c": "However, since the cube root of a quantity has triple values,", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          The other two roots could be determined as,

          ", "content": [{"c": "The other two roots could be determined as,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          ", "content": {"math_content": "t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

          $$t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          ", "content": {"math_content": "t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

          ", "content": [{"c": "Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          For the equation $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}$$, we have $$p=\\dfrac{20}{3}$$ and $$q = -\\dfrac{101}{27}$$

          ", "content": [{"c": "For the equation $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}$$, we have $$p=\\dfrac{20}{3}$$ and $$q = -\\dfrac{101}{27}$$", "t": "text"}]}, {"type": "title", "raw_content": "

          Calculate the discriminant

          ", "content": {"title_content": "Calculate the discriminant", "level": "3"}}, {"type": "paragraph", "raw_content": "

          The nature of the roots are determined by the sign of the discriminant.

          ", "content": [{"c": "The nature of the roots are determined by the sign of the discriminant.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}$$

          ", "content": {"math_content": "\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          4.1 Use the root formula directly

          ", "content": {"title_content": "4.1 Use the root formula directly", "level": "3"}}, {"type": "paragraph", "raw_content": "

          If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

          ", "content": [{"c": "If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}$$

          ", "content": {"math_content": "t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          in which, $$ \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2} $$ and $$ \\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}$$

          ", "content": [{"c": "in which, $$ \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2} $$ and $$ \\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}$$", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Substitute the values of $$p, q$$ and $$\\Delta$$ which we have calculated. Then,

          ", "content": [{"c": "Substitute the values of $$p, q$$ and $$\\Delta$$ which we have calculated. Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          ", "content": {"math_content": "\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          If we denote

          ", "content": [{"c": "If we denote", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

          ", "content": {"math_content": "R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

          $$\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

          ", "content": {"math_content": "\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          then,

          ", "content": [{"c": "then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}$$, $$\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}$$

          ", "content": {"math_content": "\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}, \\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

          $$\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          ", "content": {"math_content": "\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

          $$\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          ", "content": {"math_content": "\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          Roots of the general cubic equation

          ", "content": {"title_content": "Roots of the general cubic equation", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Since $$x = t - \\dfrac{b}{3a}$$, substituting the values of $$t$$, $$a$$ and $$b$$ gives

          ", "content": [{"c": "Since $$x = t - \\dfrac{b}{3a}$$, substituting the values of $$t$$, $$a$$ and $$b$$ gives", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$x_1 = t_1-\\dfrac{2}{3}$$

          ", "content": {"math_content": "x_1 = t_1-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

          $$x_2 = t_2-\\dfrac{2}{3}$$

          ", "content": {"math_content": "x_2 = t_2-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

          $$x_3 = t_3-\\dfrac{2}{3}$$

          ", "content": {"math_content": "x_3 = t_3-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          5. Summary

          ", "content": {"title_content": "5. Summary", "level": "2"}}, {"type": "paragraph", "raw_content": "

          In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is found to have one real root and two complex roots. Exact values and approximations are given below.

          ", "content": [{"c": "In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is found to have one real root and two complex roots. Exact values and approximations are given below.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

          ", "content": {"math_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          in decimal notation,

          ", "content": [{"c": "in decimal notation,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

          ", "content": {"math_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$

          ", "content": {"title_content": "6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Since the discriminat is greater than zero, the curve of the cubic function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$ has one intersection point with the x-axis.

          ", "content": [{"c": "Since the discriminat is greater than zero, the curve of the cubic function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$ has one intersection point with the x-axis.", "t": "text"}]}, {"type": "title", "raw_content": "

          More cubic equations

          ", "content": {"title_content": "More cubic equations", "level": "2"}}]], "main_html": "

          Solve the cubic equation:

          $$x^3+2x^2+8x+1=0 $$

          Quick Answer

          Since the discriminant $$\\Delta >0$$, the cubic equation has one real root and two conjugate complex roots.

          $$ \\Delta=14.472222222222$$

          $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

          In decimals,

          $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

          Detailed Steps on Solution

          1. Convert to depressed cubic equation

          The idea is to convert general form of cubic equation

          $$ax^3+bx^2+cx+d = 0$$

          to the form without quadratic term.

          $$t^3+pt+q = 0$$

          By substituting $$x$$ with $$t - \\dfrac{b}{3a}$$, the general cubic equation could be transformed to

          $$t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0 $$

          Compare with the depressed cubic equation. Then,

          $$p = \\dfrac{3ac-b^2}{3a^2}$$

          $$q = \\dfrac{2b^3-9abc+27a^2d}{27a^3} $$

          Substitute the values of coefficients, $$p, q$$ is obtained as

          $$p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}$$

          $$q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}$$

          Use the substitution to transform

          Let $$p$$ and $$q$$ being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

          $$t^3 +pt+q=0$$

          Let $$x=t-\\dfrac{2}{3}$$

          The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is transformed to

          $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

          2. Cardano's solution

          Let $$t=u-v$$

          Cube both sides and extract common factor from two middle terms after expanding the bracket.

          $$\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}$$

          Since $$u-v=t$$, substitution gives a linear term for the equation.\n Rearrange terms.

          $$x^3+3uvx-u^3+v^3=0$$

          Compare the cubic equation with the original one (1)

          $$\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}$$

          $$v=\\dfrac{20}{9u}$$ gives relationship between the two variables. Substitute the value of $$v$$ to the second equation

          $$\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}$$

          Simplifying gives,

          $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$2

          Let $$m=u^3$$, then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by $$v^3=-\\dfrac{101}{27}+u^3$$.

          $$m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0$$

          Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

          $$\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

          $$v^3$$ can be determined by the equation we deduced $$v^3-u^3=-\\dfrac{101}{27}$$. Then,

          $$\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

          Now we have,

          $$u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$ and $$v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$

          Evaluating the simplest cubic equation $$x^3-A=0$$,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

          If $$\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}$$, then its reciprocal is equal to its conjugate, $$\\dfrac{1}{\u03c9}=\\overline{\u03c9}$$.

          $$\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}$$

          Similary, taking cubic root for $$u^3$$ and $$v^3$$ also gives 3 roots.

          $$\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

          For $$v_2$$ and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and $$u_3$$, which is the same in value.

          $$\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

          Verification for the redicand in $$v$$.

          $$\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          Since $$x=u-v$$, combining the real and imaginary parts gives\n 3 results for $$t$$

          $$\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          $$\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          $$\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          3. Vieta's Substitution

          In Cardano' solution, $$t$$ is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$. This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

          $$t=u-\\dfrac{p}{3u}$$

          Substitute the expression $$t=u-\\dfrac{20}{9u}$$ to the cubic equation

          $$\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0$$

          Expand brackets and cancel the like terms

          $$u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0$$

          Then we get the same equation as (2)

          $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$

          The rest of the steps will be the same as those of Cardano's solution

          4. Euler's Solution

          $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

          Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

          $$t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} $$3

          Let the root of the cubic equation be the sum of two cubic roots

          $$t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} $$4

          in which $$r_1$$ and $$r_2$$ are two roots of a quadratic equation

          $$z^2-\\alpha z+ \u03b2=0 $$5

          Using Vieta's Formula, the following equations are established.

          $$r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2 $$

          To determine $$\\alpha$$, $$\u03b2$$, cube both sides of the equation (4)

          $$t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2 $$

          Substituting, the equation is simplified to

          $$t^3=3\\sqrt[3]{\u03b2}t+\\alpha $$

          Compare the cubic equation with (3), the following equations are established

          $$\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}$$

          Solving for $$\u03b2$$ gives

          $$\u03b2=-\\dfrac{8000}{729} $$

          So the quadratic equation (5) is determined as

          $$z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0$$6

          Solving the quadratic equation yields

          $$\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}$$

          Therefore, one of the roots of the cubic equation could be obtained from (4).

          $$t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          in decimals,

          $$t_1=0.53778143658824 $$

          However, since the cube root of a quantity has triple values,

          The other two roots could be determined as,

          $$t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          $$t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

          For the equation $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}$$, we have $$p=\\dfrac{20}{3}$$ and $$q = -\\dfrac{101}{27}$$

          Calculate the discriminant

          The nature of the roots are determined by the sign of the discriminant.

          $$\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}$$

          4.1 Use the root formula directly

          If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

          $$t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}$$

          in which, $$ \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2} $$ and $$ \\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}$$

          Substitute the values of $$p, q$$ and $$\\Delta$$ which we have calculated. Then,

          $$\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          If we denote

          $$R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

          $$\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

          then,

          $$\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}$$, $$\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}$$

          $$\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          $$\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          Roots of the general cubic equation

          Since $$x = t - \\dfrac{b}{3a}$$, substituting the values of $$t$$, $$a$$ and $$b$$ gives

          $$x_1 = t_1-\\dfrac{2}{3}$$

          $$x_2 = t_2-\\dfrac{2}{3}$$

          $$x_3 = t_3-\\dfrac{2}{3}$$

          5. Summary

          In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is found to have one real root and two complex roots. Exact values and approximations are given below.

          $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

          in decimal notation,

          $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

          6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$

          Since the discriminat is greater than zero, the curve of the cubic function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$ has one intersection point with the x-axis.

          More cubic equations

          ", "statics": {"title": 14, "paragraph": 64, "paragraph.text": 64, "equation-interline": 54}, "url": "https://uniteasy.com/solver/cubicequation/x%5E3%2B2x%5E2%2B8x%2B1%3D0/", "content": "# Solve the cubic equation:\n\n## $$x^3+2x^2+8x+1=0 $$\n\nQuick Answer\n\nSince the discriminant $$\\Delta >0$$, the cubic equation has one real root and two conjugate complex roots.\n\n$$\n\\Delta=14.472222222222\n$$\n\n$$\n\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}\n$$\n\nIn decimals,\n\n$$\n\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}\n$$\n\nDetailed Steps on Solution\n\n## 1. Convert to depressed cubic equation\n\nThe idea is to convert general form of cubic equation\n\n$$\nax^3+bx^2+cx+d = 0\n$$\n\nto the form without quadratic term.\n\n$$\nt^3+pt+q = 0\n$$\n\nBy substituting $$x$$ with $$t - \\dfrac{b}{3a}$$, the general cubic equation could be transformed to\n\n$$\nt^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0\n$$\n\nCompare with the depressed cubic equation. Then,\n\n$$\np = \\dfrac{3ac-b^2}{3a^2}\n$$\n\n$$\nq = \\dfrac{2b^3-9abc+27a^2d}{27a^3}\n$$\n\nSubstitute the values of coefficients, $$p, q$$ is obtained as\n\n$$\np = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}\n$$\n\n$$\nq = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}\n$$\n\n### Use the substitution to transform\n\nLet $$p$$ and $$q$$ being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.\n\n$$\nt^3 +pt+q=0\n$$\n\nLet $$x=t-\\dfrac{2}{3}$$\n\nThe cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is transformed to\n\n$$\nt^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0\n$$\n\n## 2. Cardano's solution\n\nLet $$t=u-v$$\n\nCube both sides and extract common factor from two middle terms after expanding the bracket.\n\n$$\n\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}\n$$\n\nSince $$u-v=t$$, substitution gives a linear term for the equation.\n Rearrange terms.\n\n$$\nx^3+3uvx-u^3+v^3=0\n$$\n\nCompare the cubic equation with the original one (1)\n\n$$\n\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}\n$$\n\n$$v=\\dfrac{20}{9u}$$ gives relationship between the two variables. Substitute the value of $$v$$ to the second equation\n\n$$\n\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}\n$$\n\nSimplifying gives,\n\n$$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$2\n\nLet $$m=u^3$$, then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by $$v^3=-\\dfrac{101}{27}+u^3$$.\n\n$$\nm^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0\n$$\n\nSovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.\n\n$$\n\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}\n$$\n\n$$v^3$$ can be determined by the equation we deduced $$v^3-u^3=-\\dfrac{101}{27}$$. Then,\n\n$$\n\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}\n$$\n\nNow we have,\n\n$$\nu^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6} and v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\n$$\n\nEvaluating the simplest cubic equation $$x^3-A=0$$,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.\n\nIf $$\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}$$, then its reciprocal is equal to its conjugate, $$\\dfrac{1}{\u03c9}=\\overline{\u03c9}$$.\n\n$$\n\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}\n$$\n\nSimilary, taking cubic root for $$u^3$$ and $$v^3$$ also gives 3 roots.\n\n$$\n\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}\n$$\n\nFor $$v_2$$ and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and $$u_3$$, which is the same in value.\n\n$$\n\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}\n$$\n\nVerification for the redicand in $$v$$.\n\n$$\n\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\nSince $$x=u-v$$, combining the real and imaginary parts gives\n 3 results for $$t$$\n\n$$\n\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n## 3. Vieta's Substitution\n\nIn Cardano' solution, $$t$$ is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$. This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.\n\n$$\nt=u-\\dfrac{p}{3u}\n$$\n\nSubstitute the expression $$t=u-\\dfrac{20}{9u}$$ to the cubic equation\n\n$$\n\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0\n$$\n\nExpand brackets and cancel the like terms\n\n$$\nu^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0\n$$\n\nThen we get the same equation as (2)\n\n$$\nu^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0\n$$\n\nThe rest of the steps will be the same as those of Cardano's solution\n\n## 4. Euler's Solution\n\n## $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$\n\nMove the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.\n\n$$t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} $$3\n\nLet the root of the cubic equation be the sum of two cubic roots\n\n$$t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} $$4\n\nin which $$r_1$$ and $$r_2$$ are two roots of a quadratic equation\n\n$$z^2-\\alpha z+ \u03b2=0 $$5\n\nUsing Vieta's Formula, the following equations are established.\n\n$$\nr_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2\n$$\n\nTo determine $$\\alpha$$, $$\u03b2$$, cube both sides of the equation (4)\n\n$$\nt^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2\n$$\n\nSubstituting, the equation is simplified to\n\n$$\nt^3=3\\sqrt[3]{\u03b2}t+\\alpha\n$$\n\nCompare the cubic equation with (3), the following equations are established\n\n$$\n\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}\n$$\n\nSolving for $$\u03b2$$ gives\n\n$$\n\u03b2=-\\dfrac{8000}{729}\n$$\n\nSo the quadratic equation (5) is determined as\n\n$$z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0$$6\n\nSolving the quadratic equation yields\n\n$$\n\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}\n$$\n\nTherefore, one of the roots of the cubic equation could be obtained from (4).\n\n$$\nt_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\nin decimals,\n\n$$\nt_1=0.53778143658824\n$$\n\nHowever, since the cube root of a quantity has triple values,\n\nThe other two roots could be determined as,\n\n$$\nt_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\n$$\nt_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\nCombining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.\n\nFor the equation $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}$$, we have $$p=\\dfrac{20}{3}$$ and $$q = -\\dfrac{101}{27}$$\n\n### Calculate the discriminant\n\nThe nature of the roots are determined by the sign of the discriminant.\n\n$$\n\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}\n$$\n\n### 4.1 Use the root formula directly\n\nIf the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.\n\n$$\nt_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}\n$$\n\nin which, $$ \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2} $$ and $$ \\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}$$\n\nSubstitute the values of $$p, q$$ and $$\\Delta$$ which we have calculated. Then,\n\n$$\n\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\nIf we denote\n\n$$\nR = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }\n$$\n\n$$\n\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }\n$$\n\nthen,\n\n$$\n\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}, \\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\n$$\n\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n## Roots of the general cubic equation\n\nSince $$x = t - \\dfrac{b}{3a}$$, substituting the values of $$t$$, $$a$$ and $$b$$ gives\n\n$$\nx_1 = t_1-\\dfrac{2}{3}\n$$\n\n$$\nx_2 = t_2-\\dfrac{2}{3}\n$$\n\n$$\nx_3 = t_3-\\dfrac{2}{3}\n$$\n\n## 5. Summary\n\nIn summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is found to have one real root and two complex roots. Exact values and approximations are given below.\n\n$$\n\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}\n$$\n\nin decimal notation,\n\n$$\n\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}\n$$\n\n## 6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$\n\nSince the discriminat is greater than zero, the curve of the cubic function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$ has one intersection point with the x-axis.\n\n## More cubic equations\n", "html": "\n\n\n\n\n\nSolve x^3+2x^2+8x+1=0 | Uniteasy.com\n\n\n\n\n\n\n\n\n\n\n\n \n
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          Solve the cubic equation:

          \n

          $$x^3+2x^2+8x+1=0 $$

          \n

          Quick Answer

          \n

          Since the discriminant $$\\Delta >0$$, the cubic equation has one real root and two conjugate complex roots.

          $$ \\Delta=14.472222222222$$

          $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

          In decimals,

          $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

          Detailed Steps on Solution

          1. Convert to depressed cubic equation

          The idea is to convert general form of cubic equation

          $$ax^3+bx^2+cx+d = 0$$

          to the form without quadratic term.

          $$t^3+pt+q = 0$$

          By substituting $$x$$ with $$t - \\dfrac{b}{3a}$$, the general cubic equation could be transformed to

          $$t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0 $$

          Compare with the depressed cubic equation. Then,

          $$p = \\dfrac{3ac-b^2}{3a^2}$$

          $$q = \\dfrac{2b^3-9abc+27a^2d}{27a^3} $$

          Substitute the values of coefficients, $$p, q$$ is obtained as

          $$p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}$$

          $$q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}$$

          Use the substitution to transform

          Let $$p$$ and $$q$$ being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

          $$t^3 +pt+q=0$$

          Let $$x=t-\\dfrac{2}{3}$$

          The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is transformed to

          $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

          2. Cardano's solution

          Let $$t=u-v$$

          Cube both sides and extract common factor from two middle terms after expanding the bracket.

          $$\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}$$

          Since $$u-v=t$$, substitution gives a linear term for the equation.\n Rearrange terms.

          $$x^3+3uvx-u^3+v^3=0$$

          Compare the cubic equation with the original one (1)

          $$\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}$$

          $$v=\\dfrac{20}{9u}$$ gives relationship between the two variables. Substitute the value of $$v$$ to the second equation

          $$\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}$$

          Simplifying gives,

          $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$2

          Let $$m=u^3$$, then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by $$v^3=-\\dfrac{101}{27}+u^3$$.

          $$m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0$$

          Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

          $$\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

          $$v^3$$ can be determined by the equation we deduced $$v^3-u^3=-\\dfrac{101}{27}$$. Then,

          $$\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

          Now we have,

          $$u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$ and $$v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$

          Evaluating the simplest cubic equation $$x^3-A=0$$,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

          If $$\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}$$, then its reciprocal is equal to its conjugate, $$\\dfrac{1}{\u03c9}=\\overline{\u03c9}$$.

          $$\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}$$

          Similary, taking cubic root for $$u^3$$ and $$v^3$$ also gives 3 roots.

          $$\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

          For $$v_2$$ and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and $$u_3$$, which is the same in value.

          $$\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

          Verification for the redicand in $$v$$.

          $$\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          Since $$x=u-v$$, combining the real and imaginary parts gives\n 3 results for $$t$$

          $$\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          $$\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          $$\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          3. Vieta's Substitution

          In Cardano' solution, $$t$$ is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$. This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

          $$t=u-\\dfrac{p}{3u}$$

          Substitute the expression $$t=u-\\dfrac{20}{9u}$$ to the cubic equation

          $$\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0$$

          Expand brackets and cancel the like terms

          $$u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0$$

          Then we get the same equation as (2)

          $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$

          The rest of the steps will be the same as those of Cardano's solution

          4. Euler's Solution

          $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

          Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

          $$t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} $$3

          Let the root of the cubic equation be the sum of two cubic roots

          $$t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} $$4

          in which $$r_1$$ and $$r_2$$ are two roots of a quadratic equation

          $$z^2-\\alpha z+ \u03b2=0 $$5

          Using Vieta's Formula, the following equations are established.

          $$r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2 $$

          To determine $$\\alpha$$, $$\u03b2$$, cube both sides of the equation (4)

          $$t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2 $$

          Substituting, the equation is simplified to

          $$t^3=3\\sqrt[3]{\u03b2}t+\\alpha $$

          Compare the cubic equation with (3), the following equations are established

          $$\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}$$

          Solving for $$\u03b2$$ gives

          $$\u03b2=-\\dfrac{8000}{729} $$

          So the quadratic equation (5) is determined as

          $$z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0$$6

          Solving the quadratic equation yields

          $$\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}$$

          Therefore, one of the roots of the cubic equation could be obtained from (4).

          $$t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          in decimals,

          $$t_1=0.53778143658824 $$

          However, since the cube root of a quantity has triple values,

          The other two roots could be determined as,

          $$t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          $$t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

          For the equation $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}$$, we have $$p=\\dfrac{20}{3}$$ and $$q = -\\dfrac{101}{27}$$

          Calculate the discriminant

          The nature of the roots are determined by the sign of the discriminant.

          $$\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}$$

          4.1 Use the root formula directly

          If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

          $$t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}$$

          in which, $$ \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2} $$ and $$ \\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}$$

          Substitute the values of $$p, q$$ and $$\\Delta$$ which we have calculated. Then,

          $$\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          If we denote

          $$R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

          $$\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

          then,

          $$\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}$$, $$\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}$$

          $$\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          $$\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          Roots of the general cubic equation

          Since $$x = t - \\dfrac{b}{3a}$$, substituting the values of $$t$$, $$a$$ and $$b$$ gives

          $$x_1 = t_1-\\dfrac{2}{3}$$

          $$x_2 = t_2-\\dfrac{2}{3}$$

          $$x_3 = t_3-\\dfrac{2}{3}$$

          5. Summary

          In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is found to have one real root and two complex roots. Exact values and approximations are given below.

          $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

          in decimal notation,

          $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

          6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$

          Since the discriminat is greater than zero, the curve of the cubic function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$ has one intersection point with the x-axis.

          \n\n\n\n\n\n\n
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          \n\n\n"} +{"content_list": [[{"type": "title", "raw_content": "

          Solve the cubic equation:

          ", "content": {"title_content": "Solve the cubic equation:", "level": "1"}}, {"type": "title", "raw_content": "

          $$x^3+2x^2+8x+1=0 $$

          ", "content": {"title_content": "$$x^3+2x^2+8x+1=0 $$", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Quick Answer

          ", "content": [{"c": "Quick Answer", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Since the discriminant

          ", "content": [{"c": "Since the discriminant", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\Delta >0", "content": {"math_content": "\\Delta >0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          , the cubic equation has one real root and two conjugate complex roots.

          ", "content": [{"c": ", the cubic equation has one real root and two conjugate complex roots.", "t": "text"}]}, {"type": "equation-interline", "raw_content": " \\Delta=14.472222222222", "content": {"math_content": "\\Delta=14.472222222222", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "content": {"math_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          In decimals,

          ", "content": [{"c": "In decimals,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "content": {"math_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Detailed Steps on Solution

          ", "content": [{"c": "Detailed Steps on Solution", "t": "text"}]}, {"type": "title", "raw_content": "

          1. Convert to depressed cubic equation

          ", "content": {"title_content": "1. Convert to depressed cubic equation", "level": "2"}}, {"type": "paragraph", "raw_content": "

          The idea is to convert general form of cubic equation

          ", "content": [{"c": "The idea is to convert general form of cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "ax^3+bx^2+cx+d = 0", "content": {"math_content": "ax^3+bx^2+cx+d = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          to the form without quadratic term.

          ", "content": [{"c": "to the form without quadratic term.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3+pt+q = 0", "content": {"math_content": "t^3+pt+q = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          By substituting

          ", "content": [{"c": "By substituting", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x", "content": {"math_content": "x", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          with

          ", "content": [{"c": "with", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t - \\dfrac{b}{3a}", "content": {"math_content": "t - \\dfrac{b}{3a}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          , the general cubic equation could be transformed to

          ", "content": [{"c": ", the general cubic equation could be transformed to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0 ", "content": {"math_content": "t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Compare with the depressed cubic equation. Then,

          ", "content": [{"c": "Compare with the depressed cubic equation. Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "p = \\dfrac{3ac-b^2}{3a^2}", "content": {"math_content": "p = \\dfrac{3ac-b^2}{3a^2}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "q = \\dfrac{2b^3-9abc+27a^2d}{27a^3} ", "content": {"math_content": "q = \\dfrac{2b^3-9abc+27a^2d}{27a^3}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Substitute the values of coefficients,

          ", "content": [{"c": "Substitute the values of coefficients,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "p, q", "content": {"math_content": "p, q", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          is obtained as

          ", "content": [{"c": "is obtained as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}", "content": {"math_content": "p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}", "content": {"math_content": "q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          Use the substitution to transform

          ", "content": {"title_content": "Use the substitution to transform", "level": "3"}}, {"type": "paragraph", "raw_content": "

          Let

          ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "p", "content": {"math_content": "p", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          and

          ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "q", "content": {"math_content": "q", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

          ", "content": [{"c": "being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3 +pt+q=0", "content": {"math_content": "t^3 +pt+q=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Let

          ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x=t-\\dfrac{2}{3}", "content": {"math_content": "x=t-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          The cubic equation

          ", "content": [{"c": "The cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x\u00b3 + 2x\u00b2 + 8x + 1=0", "content": {"math_content": "x\u00b3 + 2x\u00b2 + 8x + 1=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          is transformed to

          ", "content": [{"c": "is transformed to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "content": {"math_content": "t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          2. Cardano's solution

          ", "content": {"title_content": "2. Cardano's solution", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Let

          ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t=u-v", "content": {"math_content": "t=u-v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Cube both sides and extract common factor from two middle terms after expanding the bracket.

          ", "content": [{"c": "Cube both sides and extract common factor from two middle terms after expanding the bracket.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Since

          ", "content": [{"c": "Since", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u-v=t", "content": {"math_content": "u-v=t", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          , substitution gives a linear term for the equation.\n Rearrange terms.

          ", "content": [{"c": ", substitution gives a linear term for the equation.\n Rearrange terms.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x^3+3uvx-u^3+v^3=0", "content": {"math_content": "x^3+3uvx-u^3+v^3=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Compare the cubic equation with the original one (1)

          ", "content": [{"c": "Compare the cubic equation with the original one (1)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "v=\\dfrac{20}{9u}", "content": {"math_content": "v=\\dfrac{20}{9u}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          gives relationship between the two variables. Substitute the value of

          ", "content": [{"c": "gives relationship between the two variables. Substitute the value of", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v", "content": {"math_content": "v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          to the second equation

          ", "content": [{"c": "to the second equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}", "content": {"math_content": "\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Simplifying gives,

          ", "content": [{"c": "Simplifying gives,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0", "content": {"math_content": "u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          2

          ", "content": [{"c": "2", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Let

          ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "m=u^3", "content": {"math_content": "m=u^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          , then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by

          ", "content": [{"c": ", then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v^3=-\\dfrac{101}{27}+u^3", "content": {"math_content": "v^3=-\\dfrac{101}{27}+u^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          .

          ", "content": [{"c": ".", "t": "text"}]}, {"type": "equation-interline", "raw_content": "m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0", "content": {"math_content": "m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

          ", "content": [{"c": "Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "v^3", "content": {"math_content": "v^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          can be determined by the equation we deduced

          ", "content": [{"c": "can be determined by the equation we deduced", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v^3-u^3=-\\dfrac{101}{27}", "content": {"math_content": "v^3-u^3=-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          . Then,

          ", "content": [{"c": ". Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Now we have,

          ", "content": [{"c": "Now we have,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}", "content": {"math_content": "u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          and

          ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}", "content": {"math_content": "v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Evaluating the simplest cubic equation

          ", "content": [{"c": "Evaluating the simplest cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x^3-A=0", "content": {"math_content": "x^3-A=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          ,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

          ", "content": [{"c": ",\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          If

          ", "content": [{"c": "If", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}", "content": {"math_content": "\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          , then its reciprocal is equal to its conjugate,

          ", "content": [{"c": ", then its reciprocal is equal to its conjugate,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\dfrac{1}{\u03c9}=\\overline{\u03c9}", "content": {"math_content": "\\dfrac{1}{\u03c9}=\\overline{\u03c9}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          .

          ", "content": [{"c": ".", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Similary, taking cubic root for

          ", "content": [{"c": "Similary, taking cubic root for", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u^3", "content": {"math_content": "u^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          and

          ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v^3", "content": {"math_content": "v^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          also gives 3 roots.

          ", "content": [{"c": "also gives 3 roots.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          For

          ", "content": [{"c": "For", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v_2", "content": {"math_content": "v_2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and

          ", "content": [{"c": "and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u_3", "content": {"math_content": "u_3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          , which is the same in value.

          ", "content": [{"c": ", which is the same in value.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Verification for the redicand in

          ", "content": [{"c": "Verification for the redicand in", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v", "content": {"math_content": "v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          .

          ", "content": [{"c": ".", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Since

          ", "content": [{"c": "Since", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x=u-v", "content": {"math_content": "x=u-v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          , combining the real and imaginary parts gives\n 3 results for

          ", "content": [{"c": ", combining the real and imaginary parts gives\n 3 results for", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t", "content": {"math_content": "t", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          3. Vieta's Substitution

          ", "content": {"title_content": "3. Vieta's Substitution", "level": "2"}}, {"type": "paragraph", "raw_content": "

          In Cardano' solution,

          ", "content": [{"c": "In Cardano' solution,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t", "content": {"math_content": "t", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n

          ", "content": [{"c": "is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "content": {"math_content": "t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          . This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

          ", "content": [{"c": ". This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t=u-\\dfrac{p}{3u}", "content": {"math_content": "t=u-\\dfrac{p}{3u}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Substitute the expression

          ", "content": [{"c": "Substitute the expression", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t=u-\\dfrac{20}{9u}", "content": {"math_content": "t=u-\\dfrac{20}{9u}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          to the cubic equation

          ", "content": [{"c": "to the cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0", "content": {"math_content": "\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Expand brackets and cancel the like terms

          ", "content": [{"c": "Expand brackets and cancel the like terms", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0", "content": {"math_content": "u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Then we get the same equation as (2)

          ", "content": [{"c": "Then we get the same equation as (2)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0", "content": {"math_content": "u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          The rest of the steps will be the same as those of Cardano's solution

          ", "content": [{"c": "The rest of the steps will be the same as those of Cardano's solution", "t": "text"}]}, {"type": "title", "raw_content": "

          4. Euler's Solution

          ", "content": {"title_content": "4. Euler's Solution", "level": "2"}}, {"type": "title", "raw_content": "

          $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

          ", "content": {"title_content": "$$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

          ", "content": [{"c": "Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} ", "content": {"math_content": "t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          3

          ", "content": [{"c": "3", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Let the root of the cubic equation be the sum of two cubic roots

          ", "content": [{"c": "Let the root of the cubic equation be the sum of two cubic roots", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} ", "content": {"math_content": "t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          4

          ", "content": [{"c": "4", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          in which

          ", "content": [{"c": "in which", "t": "text"}]}, {"type": "equation-interline", "raw_content": "r_1", "content": {"math_content": "r_1", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          and

          ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "r_2", "content": {"math_content": "r_2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          are two roots of a quadratic equation

          ", "content": [{"c": "are two roots of a quadratic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "z^2-\\alpha z+ \u03b2=0 ", "content": {"math_content": "z^2-\\alpha z+ \u03b2=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          5

          ", "content": [{"c": "5", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Using Vieta's Formula, the following equations are established.

          ", "content": [{"c": "Using Vieta's Formula, the following equations are established.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2 ", "content": {"math_content": "r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          To determine

          ", "content": [{"c": "To determine", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\alpha", "content": {"math_content": "\\alpha", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          ,

          ", "content": [{"c": ",", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\u03b2", "content": {"math_content": "\u03b2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          , cube both sides of the equation (4)

          ", "content": [{"c": ", cube both sides of the equation (4)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2 ", "content": {"math_content": "t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Substituting, the equation is simplified to

          ", "content": [{"c": "Substituting, the equation is simplified to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3=3\\sqrt[3]{\u03b2}t+\\alpha ", "content": {"math_content": "t^3=3\\sqrt[3]{\u03b2}t+\\alpha", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Compare the cubic equation with (3), the following equations are established

          ", "content": [{"c": "Compare the cubic equation with (3), the following equations are established", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Solving for

          ", "content": [{"c": "Solving for", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\u03b2", "content": {"math_content": "\u03b2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          gives

          ", "content": [{"c": "gives", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\u03b2=-\\dfrac{8000}{729} ", "content": {"math_content": "\u03b2=-\\dfrac{8000}{729}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          So the quadratic equation (5) is determined as

          ", "content": [{"c": "So the quadratic equation (5) is determined as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0", "content": {"math_content": "z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          6

          ", "content": [{"c": "6", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solving the quadratic equation yields

          ", "content": [{"c": "Solving the quadratic equation yields", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Therefore, one of the roots of the cubic equation could be obtained from (4).

          ", "content": [{"c": "Therefore, one of the roots of the cubic equation could be obtained from (4).", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} ", "content": {"math_content": "t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          in decimals,

          ", "content": [{"c": "in decimals,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t_1=0.53778143658824 ", "content": {"math_content": "t_1=0.53778143658824", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          However, since the cube root of a quantity has triple values,

          ", "content": [{"c": "However, since the cube root of a quantity has triple values,", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          The other two roots could be determined as,

          ", "content": [{"c": "The other two roots could be determined as,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} ", "content": {"math_content": "t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} ", "content": {"math_content": "t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

          ", "content": [{"c": "Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          For the equation

          ", "content": [{"c": "For the equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}", "content": {"math_content": "t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          , we have $$p=\\dfrac{20}{3}$$ and

          ", "content": [{"c": ", we have $$p=\\dfrac{20}{3}$$ and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "q = -\\dfrac{101}{27}", "content": {"math_content": "q = -\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          Calculate the discriminant

          ", "content": {"title_content": "Calculate the discriminant", "level": "3"}}, {"type": "paragraph", "raw_content": "

          The nature of the roots are determined by the sign of the discriminant.

          ", "content": [{"c": "The nature of the roots are determined by the sign of the discriminant.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          4.1 Use the root formula directly

          ", "content": {"title_content": "4.1 Use the root formula directly", "level": "3"}}, {"type": "paragraph", "raw_content": "

          If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

          ", "content": [{"c": "If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}", "content": {"math_content": "t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          in which,

          ", "content": [{"c": "in which,", "t": "text"}]}, {"type": "equation-interline", "raw_content": " \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2} ", "content": {"math_content": "\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          and

          ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": " \\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}", "content": {"math_content": "\\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          Substitute the values of

          ", "content": [{"c": "Substitute the values of", "t": "text"}]}, {"type": "equation-interline", "raw_content": "p, q", "content": {"math_content": "p, q", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          and

          ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\Delta", "content": {"math_content": "\\Delta", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          which we have calculated. Then,

          ", "content": [{"c": "which we have calculated. Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          If we denote

          ", "content": [{"c": "If we denote", "t": "text"}]}, {"type": "equation-interline", "raw_content": "R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "content": {"math_content": "R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "content": {"math_content": "\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          then,

          ", "content": [{"c": "then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}", "content": {"math_content": "\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          ,

          ", "content": [{"c": ",", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "content": {"math_content": "\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          Roots of the general cubic equation

          ", "content": {"title_content": "Roots of the general cubic equation", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Since

          ", "content": [{"c": "Since", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x = t - \\dfrac{b}{3a}", "content": {"math_content": "x = t - \\dfrac{b}{3a}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          , substituting the values of $$t$$, $$a$$ and

          ", "content": [{"c": ", substituting the values of $$t$$, $$a$$ and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "b", "content": {"math_content": "b", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          gives

          ", "content": [{"c": "gives", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x_1 = t_1-\\dfrac{2}{3}", "content": {"math_content": "x_1 = t_1-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "x_2 = t_2-\\dfrac{2}{3}", "content": {"math_content": "x_2 = t_2-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "x_3 = t_3-\\dfrac{2}{3}", "content": {"math_content": "x_3 = t_3-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          5. Summary

          ", "content": {"title_content": "5. Summary", "level": "2"}}, {"type": "paragraph", "raw_content": "

          In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation

          ", "content": [{"c": "In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x\u00b3 + 2x\u00b2 + 8x + 1=0", "content": {"math_content": "x\u00b3 + 2x\u00b2 + 8x + 1=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          is found to have one real root and two complex roots. Exact values and approximations are given below.

          ", "content": [{"c": "is found to have one real root and two complex roots. Exact values and approximations are given below.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "content": {"math_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          in decimal notation,

          ", "content": [{"c": "in decimal notation,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "content": {"math_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

          6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$

          ", "content": {"title_content": "6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Since the discriminat is greater than zero, the curve of the cubic function

          ", "content": [{"c": "Since the discriminat is greater than zero, the curve of the cubic function", "t": "text"}]}, {"type": "equation-interline", "raw_content": "f(x) = x\u00b3 + 2x\u00b2 + 8x + 1", "content": {"math_content": "f(x) = x\u00b3 + 2x\u00b2 + 8x + 1", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

          has one intersection point with the x-axis.

          ", "content": [{"c": "has one intersection point with the x-axis.", "t": "text"}]}, {"type": "title", "raw_content": "

          More cubic equations

          ", "content": {"title_content": "More cubic equations", "level": "2"}}]], "main_html": "

          Solve the cubic equation:

          $$x^3+2x^2+8x+1=0 $$

          Quick Answer

          Since the discriminant

          \\Delta >0

          , the cubic equation has one real root and two conjugate complex roots.

          \\Delta=14.472222222222\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}

          In decimals,

          \\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}

          Detailed Steps on Solution

          1. Convert to depressed cubic equation

          The idea is to convert general form of cubic equation

          ax^3+bx^2+cx+d = 0

          to the form without quadratic term.

          t^3+pt+q = 0

          By substituting

          x

          with

          t - \\dfrac{b}{3a}

          , the general cubic equation could be transformed to

          t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0

          Compare with the depressed cubic equation. Then,

          p = \\dfrac{3ac-b^2}{3a^2}q = \\dfrac{2b^3-9abc+27a^2d}{27a^3}

          Substitute the values of coefficients,

          p, q

          is obtained as

          p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}

          Use the substitution to transform

          Let

          p

          and

          q

          being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

          t^3 +pt+q=0

          Let

          x=t-\\dfrac{2}{3}

          The cubic equation

          x\u00b3 + 2x\u00b2 + 8x + 1=0

          is transformed to

          t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0

          2. Cardano's solution

          Let

          t=u-v

          Cube both sides and extract common factor from two middle terms after expanding the bracket.

          \\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}

          Since

          u-v=t

          , substitution gives a linear term for the equation.\n Rearrange terms.

          x^3+3uvx-u^3+v^3=0

          Compare the cubic equation with the original one (1)

          \\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}v=\\dfrac{20}{9u}

          gives relationship between the two variables. Substitute the value of

          v

          to the second equation

          \\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}

          Simplifying gives,

          u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0

          2

          Let

          m=u^3

          , then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by

          v^3=-\\dfrac{101}{27}+u^3

          .

          m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0

          Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

          \\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}v^3

          can be determined by the equation we deduced

          v^3-u^3=-\\dfrac{101}{27}

          . Then,

          \\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}

          Now we have,

          u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}

          and

          v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}

          Evaluating the simplest cubic equation

          x^3-A=0

          ,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

          If

          \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}

          , then its reciprocal is equal to its conjugate,

          \\dfrac{1}{\u03c9}=\\overline{\u03c9}

          .

          \\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}

          Similary, taking cubic root for

          u^3

          and

          v^3

          also gives 3 roots.

          \\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}

          For

          v_2

          and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and

          u_3

          , which is the same in value.

          \\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}

          Verification for the redicand in

          v

          .

          \\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}

          Since

          x=u-v

          , combining the real and imaginary parts gives\n 3 results for

          t\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}

          3. Vieta's Substitution

          In Cardano' solution,

          t

          is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n

          t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0

          . This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

          t=u-\\dfrac{p}{3u}

          Substitute the expression

          t=u-\\dfrac{20}{9u}

          to the cubic equation

          \\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0

          Expand brackets and cancel the like terms

          u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0

          Then we get the same equation as (2)

          u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0

          The rest of the steps will be the same as those of Cardano's solution

          4. Euler's Solution

          $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

          Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

          t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27}

          3

          Let the root of the cubic equation be the sum of two cubic roots

          t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2}

          4

          in which

          r_1

          and

          r_2

          are two roots of a quadratic equation

          z^2-\\alpha z+ \u03b2=0

          5

          Using Vieta's Formula, the following equations are established.

          r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2

          To determine

          \\alpha

          ,

          \u03b2

          , cube both sides of the equation (4)

          t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2

          Substituting, the equation is simplified to

          t^3=3\\sqrt[3]{\u03b2}t+\\alpha

          Compare the cubic equation with (3), the following equations are established

          \\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}

          Solving for

          \u03b2

          gives

          \u03b2=-\\dfrac{8000}{729}

          So the quadratic equation (5) is determined as

          z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0

          6

          Solving the quadratic equation yields

          \\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}

          Therefore, one of the roots of the cubic equation could be obtained from (4).

          t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}

          in decimals,

          t_1=0.53778143658824

          However, since the cube root of a quantity has triple values,

          The other two roots could be determined as,

          t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}

          Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

          For the equation

          t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}

          , we have $$p=\\dfrac{20}{3}$$ and

          q = -\\dfrac{101}{27}

          Calculate the discriminant

          The nature of the roots are determined by the sign of the discriminant.

          \\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}

          4.1 Use the root formula directly

          If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

          t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}

          in which,

          \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}

          and

          \\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}

          Substitute the values of

          p, q

          and

          \\Delta

          which we have calculated. Then,

          \\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}

          If we denote

          R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }

          then,

          \\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}

          ,

          \\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}

          Roots of the general cubic equation

          Since

          x = t - \\dfrac{b}{3a}

          , substituting the values of $$t$$, $$a$$ and

          b

          gives

          x_1 = t_1-\\dfrac{2}{3}x_2 = t_2-\\dfrac{2}{3}x_3 = t_3-\\dfrac{2}{3}

          5. Summary

          In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation

          x\u00b3 + 2x\u00b2 + 8x + 1=0

          is found to have one real root and two complex roots. Exact values and approximations are given below.

          \\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}

          in decimal notation,

          \\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}

          6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$

          Since the discriminat is greater than zero, the curve of the cubic function

          f(x) = x\u00b3 + 2x\u00b2 + 8x + 1

          has one intersection point with the x-axis.

          More cubic equations

          ", "statics": {"title": 14, "paragraph": 103, "paragraph.text": 103, "equation-interline": 105}, "url": "https://uniteasy.com/solver/cubicequation/x%5E3%2B2x%5E2%2B8x%2B1%3D0/", "content": "# Solve the cubic equation:\n\n## $$x^3+2x^2+8x+1=0 $$\n\nQuick Answer\n\nSince the discriminant\n\n$$\n\\Delta >0\n$$\n\n, the cubic equation has one real root and two conjugate complex roots.\n\n$$\n\\Delta=14.472222222222\n$$\n\n$$\n\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}\n$$\n\nIn decimals,\n\n$$\n\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}\n$$\n\nDetailed Steps on Solution\n\n## 1. Convert to depressed cubic equation\n\nThe idea is to convert general form of cubic equation\n\n$$\nax^3+bx^2+cx+d = 0\n$$\n\nto the form without quadratic term.\n\n$$\nt^3+pt+q = 0\n$$\n\nBy substituting\n\n$$\nx\n$$\n\nwith\n\n$$\nt - \\dfrac{b}{3a}\n$$\n\n, the general cubic equation could be transformed to\n\n$$\nt^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0\n$$\n\nCompare with the depressed cubic equation. Then,\n\n$$\np = \\dfrac{3ac-b^2}{3a^2}\n$$\n\n$$\nq = \\dfrac{2b^3-9abc+27a^2d}{27a^3}\n$$\n\nSubstitute the values of coefficients,\n\n$$\np, q\n$$\n\nis obtained as\n\n$$\np = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}\n$$\n\n$$\nq = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}\n$$\n\n### Use the substitution to transform\n\nLet\n\n$$\np\n$$\n\nand\n\n$$\nq\n$$\n\nbeing the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.\n\n$$\nt^3 +pt+q=0\n$$\n\nLet\n\n$$\nx=t-\\dfrac{2}{3}\n$$\n\nThe cubic equation\n\n$$\nx\u00b3 + 2x\u00b2 + 8x + 1=0\n$$\n\nis transformed to\n\n$$\nt^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0\n$$\n\n## 2. Cardano's solution\n\nLet\n\n$$\nt=u-v\n$$\n\nCube both sides and extract common factor from two middle terms after expanding the bracket.\n\n$$\n\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}\n$$\n\nSince\n\n$$\nu-v=t\n$$\n\n, substitution gives a linear term for the equation.\n Rearrange terms.\n\n$$\nx^3+3uvx-u^3+v^3=0\n$$\n\nCompare the cubic equation with the original one (1)\n\n$$\n\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}\n$$\n\n$$\nv=\\dfrac{20}{9u}\n$$\n\ngives relationship between the two variables. Substitute the value of\n\n$$\nv\n$$\n\nto the second equation\n\n$$\n\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}\n$$\n\nSimplifying gives,\n\n$$\nu^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0\n$$\n\n2\n\nLet\n\n$$\nm=u^3\n$$\n\n, then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by\n\n$$\nv^3=-\\dfrac{101}{27}+u^3\n$$\n\n.\n\n$$\nm^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0\n$$\n\nSovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.\n\n$$\n\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}\n$$\n\n$$\nv^3\n$$\n\ncan be determined by the equation we deduced\n\n$$\nv^3-u^3=-\\dfrac{101}{27}\n$$\n\n. Then,\n\n$$\n\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}\n$$\n\nNow we have,\n\n$$\nu^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\n$$\n\nand\n\n$$\nv^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\n$$\n\nEvaluating the simplest cubic equation\n\n$$\nx^3-A=0\n$$\n\n,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.\n\nIf\n\n$$\n\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}\n$$\n\n, then its reciprocal is equal to its conjugate,\n\n$$\n\\dfrac{1}{\u03c9}=\\overline{\u03c9}\n$$\n\n.\n\n$$\n\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}\n$$\n\nSimilary, taking cubic root for\n\n$$\nu^3\n$$\n\nand\n\n$$\nv^3\n$$\n\nalso gives 3 roots.\n\n$$\n\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}\n$$\n\nFor\n\n$$\nv_2\n$$\n\nand $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and\n\n$$\nu_3\n$$\n\n, which is the same in value.\n\n$$\n\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}\n$$\n\nVerification for the redicand in\n\n$$\nv\n$$\n\n.\n\n$$\n\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\nSince\n\n$$\nx=u-v\n$$\n\n, combining the real and imaginary parts gives\n 3 results for\n\n$$\nt\n$$\n\n$$\n\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n## 3. Vieta's Substitution\n\nIn Cardano' solution,\n\n$$\nt\n$$\n\nis defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n\n$$\nt^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0\n$$\n\n. This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.\n\n$$\nt=u-\\dfrac{p}{3u}\n$$\n\nSubstitute the expression\n\n$$\nt=u-\\dfrac{20}{9u}\n$$\n\nto the cubic equation\n\n$$\n\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0\n$$\n\nExpand brackets and cancel the like terms\n\n$$\nu^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0\n$$\n\nThen we get the same equation as (2)\n\n$$\nu^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0\n$$\n\nThe rest of the steps will be the same as those of Cardano's solution\n\n## 4. Euler's Solution\n\n## $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$\n\nMove the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.\n\n$$\nt^3=-\\dfrac{20}{3}t+\\dfrac{101}{27}\n$$\n\n3\n\nLet the root of the cubic equation be the sum of two cubic roots\n\n$$\nt=\\sqrt[3]{r_1}+\\sqrt[3]{r_2}\n$$\n\n4\n\nin which\n\n$$\nr_1\n$$\n\nand\n\n$$\nr_2\n$$\n\nare two roots of a quadratic equation\n\n$$\nz^2-\\alpha z+ \u03b2=0\n$$\n\n5\n\nUsing Vieta's Formula, the following equations are established.\n\n$$\nr_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2\n$$\n\nTo determine\n\n$$\n\\alpha\n$$\n\n,\n\n$$\n\u03b2\n$$\n\n, cube both sides of the equation (4)\n\n$$\nt^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2\n$$\n\nSubstituting, the equation is simplified to\n\n$$\nt^3=3\\sqrt[3]{\u03b2}t+\\alpha\n$$\n\nCompare the cubic equation with (3), the following equations are established\n\n$$\n\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}\n$$\n\nSolving for\n\n$$\n\u03b2\n$$\n\ngives\n\n$$\n\u03b2=-\\dfrac{8000}{729}\n$$\n\nSo the quadratic equation (5) is determined as\n\n$$\nz^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0\n$$\n\n6\n\nSolving the quadratic equation yields\n\n$$\n\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}\n$$\n\nTherefore, one of the roots of the cubic equation could be obtained from (4).\n\n$$\nt_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\nin decimals,\n\n$$\nt_1=0.53778143658824\n$$\n\nHowever, since the cube root of a quantity has triple values,\n\nThe other two roots could be determined as,\n\n$$\nt_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\n$$\nt_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\nCombining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.\n\nFor the equation\n\n$$\nt^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}\n$$\n\n, we have $$p=\\dfrac{20}{3}$$ and\n\n$$\nq = -\\dfrac{101}{27}\n$$\n\n### Calculate the discriminant\n\nThe nature of the roots are determined by the sign of the discriminant.\n\n$$\n\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}\n$$\n\n### 4.1 Use the root formula directly\n\nIf the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.\n\n$$\nt_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}\n$$\n\nin which,\n\n$$\n\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}\n$$\n\nand\n\n$$\n\\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}\n$$\n\nSubstitute the values of\n\n$$\np, q\n$$\n\nand\n\n$$\n\\Delta\n$$\n\nwhich we have calculated. Then,\n\n$$\n\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\nIf we denote\n\n$$\nR = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }\n$$\n\n$$\n\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }\n$$\n\nthen,\n\n$$\n\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\n$$\n\n,\n\n$$\n\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\n$$\n\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n## Roots of the general cubic equation\n\nSince\n\n$$\nx = t - \\dfrac{b}{3a}\n$$\n\n, substituting the values of $$t$$, $$a$$ and\n\n$$\nb\n$$\n\ngives\n\n$$\nx_1 = t_1-\\dfrac{2}{3}\n$$\n\n$$\nx_2 = t_2-\\dfrac{2}{3}\n$$\n\n$$\nx_3 = t_3-\\dfrac{2}{3}\n$$\n\n## 5. Summary\n\nIn summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation\n\n$$\nx\u00b3 + 2x\u00b2 + 8x + 1=0\n$$\n\nis found to have one real root and two complex roots. Exact values and approximations are given below.\n\n$$\n\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}\n$$\n\nin decimal notation,\n\n$$\n\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}\n$$\n\n## 6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$\n\nSince the discriminat is greater than zero, the curve of the cubic function\n\n$$\nf(x) = x\u00b3 + 2x\u00b2 + 8x + 1\n$$\n\nhas one intersection point with the x-axis.\n\n## More cubic equations\n", "html": "\n\n\n\n\n\nSolve x^3+2x^2+8x+1=0 | Uniteasy.com\n\n\n\n\n\n\n\n\n\n\n\n \n
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          \n

          Solve the cubic equation:

          \n

          $$x^3+2x^2+8x+1=0 $$

          \n

          Quick Answer

          \n

          Since the discriminant $$\\Delta >0$$, the cubic equation has one real root and two conjugate complex roots.

          $$ \\Delta=14.472222222222$$

          $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

          In decimals,

          $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

          Detailed Steps on Solution

          1. Convert to depressed cubic equation

          The idea is to convert general form of cubic equation

          $$ax^3+bx^2+cx+d = 0$$

          to the form without quadratic term.

          $$t^3+pt+q = 0$$

          By substituting $$x$$ with $$t - \\dfrac{b}{3a}$$, the general cubic equation could be transformed to

          $$t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0 $$

          Compare with the depressed cubic equation. Then,

          $$p = \\dfrac{3ac-b^2}{3a^2}$$

          $$q = \\dfrac{2b^3-9abc+27a^2d}{27a^3} $$

          Substitute the values of coefficients, $$p, q$$ is obtained as

          $$p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}$$

          $$q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}$$

          Use the substitution to transform

          Let $$p$$ and $$q$$ being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

          $$t^3 +pt+q=0$$

          Let $$x=t-\\dfrac{2}{3}$$

          The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is transformed to

          $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

          2. Cardano's solution

          Let $$t=u-v$$

          Cube both sides and extract common factor from two middle terms after expanding the bracket.

          $$\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}$$

          Since $$u-v=t$$, substitution gives a linear term for the equation.\n Rearrange terms.

          $$x^3+3uvx-u^3+v^3=0$$

          Compare the cubic equation with the original one (1)

          $$\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}$$

          $$v=\\dfrac{20}{9u}$$ gives relationship between the two variables. Substitute the value of $$v$$ to the second equation

          $$\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}$$

          Simplifying gives,

          $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$2

          Let $$m=u^3$$, then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by $$v^3=-\\dfrac{101}{27}+u^3$$.

          $$m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0$$

          Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

          $$\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

          $$v^3$$ can be determined by the equation we deduced $$v^3-u^3=-\\dfrac{101}{27}$$. Then,

          $$\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

          Now we have,

          $$u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$ and $$v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$

          Evaluating the simplest cubic equation $$x^3-A=0$$,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

          If $$\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}$$, then its reciprocal is equal to its conjugate, $$\\dfrac{1}{\u03c9}=\\overline{\u03c9}$$.

          $$\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}$$

          Similary, taking cubic root for $$u^3$$ and $$v^3$$ also gives 3 roots.

          $$\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

          For $$v_2$$ and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and $$u_3$$, which is the same in value.

          $$\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

          Verification for the redicand in $$v$$.

          $$\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          Since $$x=u-v$$, combining the real and imaginary parts gives\n 3 results for $$t$$

          $$\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          $$\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          $$\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          3. Vieta's Substitution

          In Cardano' solution, $$t$$ is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$. This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

          $$t=u-\\dfrac{p}{3u}$$

          Substitute the expression $$t=u-\\dfrac{20}{9u}$$ to the cubic equation

          $$\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0$$

          Expand brackets and cancel the like terms

          $$u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0$$

          Then we get the same equation as (2)

          $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$

          The rest of the steps will be the same as those of Cardano's solution

          4. Euler's Solution

          $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

          Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

          $$t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} $$3

          Let the root of the cubic equation be the sum of two cubic roots

          $$t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} $$4

          in which $$r_1$$ and $$r_2$$ are two roots of a quadratic equation

          $$z^2-\\alpha z+ \u03b2=0 $$5

          Using Vieta's Formula, the following equations are established.

          $$r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2 $$

          To determine $$\\alpha$$, $$\u03b2$$, cube both sides of the equation (4)

          $$t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2 $$

          Substituting, the equation is simplified to

          $$t^3=3\\sqrt[3]{\u03b2}t+\\alpha $$

          Compare the cubic equation with (3), the following equations are established

          $$\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}$$

          Solving for $$\u03b2$$ gives

          $$\u03b2=-\\dfrac{8000}{729} $$

          So the quadratic equation (5) is determined as

          $$z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0$$6

          Solving the quadratic equation yields

          $$\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}$$

          Therefore, one of the roots of the cubic equation could be obtained from (4).

          $$t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          in decimals,

          $$t_1=0.53778143658824 $$

          However, since the cube root of a quantity has triple values,

          The other two roots could be determined as,

          $$t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          $$t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

          Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

          For the equation $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}$$, we have $$p=\\dfrac{20}{3}$$ and $$q = -\\dfrac{101}{27}$$

          Calculate the discriminant

          The nature of the roots are determined by the sign of the discriminant.

          $$\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}$$

          4.1 Use the root formula directly

          If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

          $$t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}$$

          in which, $$ \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2} $$ and $$ \\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}$$

          Substitute the values of $$p, q$$ and $$\\Delta$$ which we have calculated. Then,

          $$\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

          If we denote

          $$R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

          $$\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

          then,

          $$\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}$$, $$\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}$$

          $$\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          $$\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

          Roots of the general cubic equation

          Since $$x = t - \\dfrac{b}{3a}$$, substituting the values of $$t$$, $$a$$ and $$b$$ gives

          $$x_1 = t_1-\\dfrac{2}{3}$$

          $$x_2 = t_2-\\dfrac{2}{3}$$

          $$x_3 = t_3-\\dfrac{2}{3}$$

          5. Summary

          In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is found to have one real root and two complex roots. Exact values and approximations are given below.

          $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

          in decimal notation,

          $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

          6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$

          Since the discriminat is greater than zero, the curve of the cubic function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$ has one intersection point with the x-axis.

          \n\n\n\n\n\n\n
          \n
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          More cubic equations

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          \n\n\n"} diff --git a/bench/data/groundtruth/math_katex_latex_3.jsonl b/bench/data/groundtruth/math_katex_latex_3.jsonl index 5ebdfd92..ac465409 100644 --- a/bench/data/groundtruth/math_katex_latex_3.jsonl +++ b/bench/data/groundtruth/math_katex_latex_3.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "paragraph", "raw_content": "
          \n Show commands:\n Magma\n / PariGP\n / SageMath
          ", "content": [{"c": "Show commands: Magma/ PariGP/ SageMath", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          [N,k,chi] = [3332,1,Mod(667,3332)]
          mf = mfinit([N,k,chi],0)
          lf = mfeigenbasis(mf)
          ", "content": [{"c": "[N,k,chi] = [3332,1,Mod(667,3332)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          from sage.modular.dirichlet import DirichletCharacter
          H = DirichletGroup(3332, base_ring=CyclotomicField(12))
          chi = DirichletCharacter(H, H._module([6, 4, 9]))
          N = Newforms(chi, 1, names=\"a\")
          ", "content": [{"c": "from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3332, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 4, 9])) N = Newforms(chi, 1, names=\"a\")", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
          chi := DirichletCharacter(\"3332.667\");
          S:= CuspForms(chi, 1);
          N := Newforms(S);
          ", "content": [{"c": "//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter(\"3332.667\"); S:= CuspForms(chi, 1); N := Newforms(S);", "t": "text"}]}, {"type": "table", "raw_content": "
          Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
          Weight: \\( k \\) \\(=\\)\\( 1 \\)
          Character orbit: \\([\\chi]\\) \\(=\\)3332.bc (of order \\(12\\), degree \\(4\\), not minimal)
          ", "content": {"html": "
          Level<br>:\\( N \\)\\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
          Weight<br>:\\( k \\)\\(=\\)\\( 1 \\)
          Character orbit<br>:\\([\\chi]\\)\\(=\\)3332.bc<br>(of<br>order<br>\\(12\\),<br>degree<br>\\(4\\), not<br>minimal<br>)
          ", "is_complex": false}}, {"type": "title", "raw_content": "

          Newform invariants

          ", "content": {"title_content": "Newform invariants", "level": "2"}}, {"type": "paragraph", "raw_content": "
          sage:\u00a0f = N[0] # Warning: the index may be different
          ", "content": [{"c": "sage:\u00a0f = N[0] # Warning: the index may be different", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          gp:\u00a0f = lf[1] \\\\ Warning: the index may be different
          ", "content": [{"c": "gp:\u00a0f = lf[1] \\\\ Warning: the index may be different", "t": "text"}]}, {"type": "table", "raw_content": "
          Self dual: no
          Analytic conductor: \\(1.66288462209\\)
          Analytic rank: \\(0\\)
          Dimension: \\(4\\)
          Coefficient field: \\(\\Q(\\zeta_{12})\\)
          gp:\u00a0f.mod \\\\ as an extension of the character field
          Defining polynomial: \\( x^{4} - x^{2} + 1 \\)\"Copy\"Toggle
          Coefficient ring: \\(\\Z[a_1, a_2]\\)
          Coefficient ring index: \\( 1 \\)
          Twist minimal: no (minimal twist has level 68)
          Projective image:\\(D_{4}\\)
          Projective field:Galois closure of 4.2.19652.1
          Artin image:$C_4\\wr C_2\\times C_6$
          Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
          ", "content": {"html": "
          Self dual<br>:no
          Analytic conductor<br>:\\(1.66288462209\\)
          Analytic rank<br>:\\(0\\)
          Dimension<br>:\\(4\\)
          Coefficient field<br>:\\(\\Q(\\zeta_{12})\\)
          gp:\u00a0f.mod \\\\ as an extension of the character field
          Defining polynomial<br>:\\( x^{4} - x^{2} + 1 \\)
          Coefficient ring<br>:\\(\\Z[a_1, a_2]\\)
          Coefficient ring index<br>:\\( 1 \\)
          Twist minimal<br>:no (minimal twist has level 68)
          Projective image<br>:\\(D_{4}\\)
          Projective field<br>:Galois closure of <br>4.2.19652.1
          Artin image<br>:$C_4\\wr C_2\\times C_6$
          Artin field<br>:Galois closure of <br>\\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
          ", "is_complex": false}}, {"type": "title", "raw_content": "

          Embedding invariants

          ", "content": {"title_content": "Embedding invariants", "level": "2"}}, {"type": "table", "raw_content": "
          Embedding label 2027.1
          Root\\(0.866025 - 0.500000i\\) of defining polynomial
          Character\\(\\chi\\)\\(=\\)3332.2027
          Dual form3332.1.bc.b.863.1
          ", "content": {"html": "
          Embedding label2027.1
          Root\\(0.866025 - 0.500000i\\) of defining polynomial
          Character\\(\\chi\\)\\(=\\)3332.2027
          Dual form3332.1.bc.b.863.1
          ", "is_complex": false}}, {"type": "paragraph", "raw_content": "
          sage:\u00a0f.q_expansion() # note that sage often uses an isomorphic number field
          ", "content": [{"c": "sage:\u00a0f.q_expansion() # note that sage often uses an isomorphic number field", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          gp:\u00a0mfcoefs(f, 20)
          ", "content": [{"c": "gp:\u00a0mfcoefs(f, 20)", "t": "text"}]}, {"type": "table", "raw_content": "
          \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
          \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\"Copy\"Toggle
          ", "content": {"html": "
          \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
          \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)
          ", "is_complex": false}}, {"type": "title", "raw_content": "

          Character values

          ", "content": {"title_content": "Character values", "level": "2"}}, {"type": "paragraph", "raw_content": "

          We give the values of \\(\\chi\\) on generators for \\(\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times\\).

          ", "content": [{"c": "We give the values of \\(\\chi\\) on generators for \\(\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times\\).", "t": "text"}]}, {"type": "table", "raw_content": "
          \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
          \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
          ", "content": {"html": "
          \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
          \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
          ", "is_complex": false}}, {"type": "title", "raw_content": "

          Coefficient data

          ", "content": {"title_content": "Coefficient data", "level": "2"}}, {"type": "paragraph", "raw_content": "

          For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the\nSatake parameters \\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).

          ", "content": [{"c": "For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the Satake parameters\\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "table", "raw_content": "
          \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
          \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
          \n \\(2\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(3\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(4\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(5\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(6\\)\n \n 0\n \n 0\n
          \n \\(7\\)\n \n 0\n \n 0\n
          \n \\(8\\)\n \n 1.00000i\n 1.00000i
          \n \\(9\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(10\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(11\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(12\\)\n \n 0\n \n 0\n
          \n \\(13\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(14\\)\n \n 0\n \n 0\n
          \n \\(15\\)\n \n 0\n \n 0\n
          \n \\(16\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(17\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(18\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(19\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(20\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(21\\)\n \n 0\n \n 0\n
          \n \\(22\\)\n \n 0\n \n 0\n
          \n \\(23\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(24\\)\n \n 0\n \n 0\n
          \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(26\\)\n \n 0\n \n 0\n
          \n \\(27\\)\n \n 0\n \n 0\n
          \n \\(28\\)\n \n 0\n \n 0\n
          \n \\(29\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(30\\)\n \n 0\n \n 0\n
          \n \\(31\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(33\\)\n \n 0\n \n 0\n
          \n \\(34\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(35\\)\n \n 0\n \n 0\n
          \n \\(36\\)\n \n 1.00000i\n 1.00000i
          \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(38\\)\n \n 0\n \n 0\n
          \n \\(39\\)\n \n 0\n \n 0\n
          \n \\(40\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(41\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(42\\)\n \n 0\n \n 0\n
          \n \\(43\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(44\\)\n \n 0\n \n 0\n
          \n \\(45\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(46\\)\n \n 0\n \n 0\n
          \n \\(47\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(48\\)\n \n 0\n \n 0\n
          \n \\(49\\)\n \n 0\n \n 0\n
          \n \\(50\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(51\\)\n \n 0\n \n 0\n
          \n \\(52\\)\n \n 0\n \n 0\n
          \n \\(53\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(54\\)\n \n 0\n \n 0\n
          \n \\(55\\)\n \n 0\n \n 0\n
          \n \\(56\\)\n \n 0\n \n 0\n
          \n \\(57\\)\n \n 0\n \n 0\n
          \n \\(58\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(59\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(60\\)\n \n 0\n \n 0\n
          \n \\(61\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(62\\)\n \n 0\n \n 0\n
          \n \\(63\\)\n \n 0\n \n 0\n
          \n \\(64\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(65\\)\n \n 0\n \n 0\n
          \n \\(66\\)\n \n 0\n \n 0\n
          \n \\(67\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(69\\)\n \n 0\n \n 0\n
          \n \\(70\\)\n \n 0\n \n 0\n
          \n \\(71\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(72\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(73\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(74\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(75\\)\n \n 0\n \n 0\n
          \n \\(76\\)\n \n 0\n \n 0\n
          \n \\(77\\)\n \n 0\n \n 0\n
          \n \\(78\\)\n \n 0\n \n 0\n
          \n \\(79\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(81\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(83\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(84\\)\n \n 0\n \n 0\n
          \n \\(85\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(86\\)\n \n 0\n \n 0\n
          \n \\(87\\)\n \n 0\n \n 0\n
          \n \\(88\\)\n \n 0\n \n 0\n
          \n \\(89\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(90\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(91\\)\n \n 0\n \n 0\n
          \n \\(92\\)\n \n 0\n \n 0\n
          \n \\(93\\)\n \n 0\n \n 0\n
          \n \\(94\\)\n \n 0\n \n 0\n
          \n \\(95\\)\n \n 0\n \n 0\n
          \n \\(96\\)\n \n 0\n \n 0\n
          \n \\(97\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(98\\)\n \n 0\n \n 0\n
          \n \\(99\\)\n \n 0\n \n 0\n
          \n \\(100\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(101\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(102\\)\n \n 0\n \n 0\n
          \n \\(103\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(104\\)\n \n 0\n \n 0\n
          \n \\(105\\)\n \n 0\n \n 0\n
          \n \\(106\\)\n \n 0\n \n 0\n
          \n \\(107\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(108\\)\n \n 0\n \n 0\n
          \n \\(109\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(110\\)\n \n 0\n \n 0\n
          \n \\(111\\)\n \n 0\n \n 0\n
          \n \\(112\\)\n \n 0\n \n 0\n
          \n \\(113\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(114\\)\n \n 0\n \n 0\n
          \n \\(115\\)\n \n 0\n \n 0\n
          \n \\(116\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(117\\)\n \n 0\n \n 0\n
          \n \\(118\\)\n \n 0\n \n 0\n
          \n \\(119\\)\n \n 0\n \n 0\n
          \n \\(120\\)\n \n 0\n \n 0\n
          \n \\(121\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(123\\)\n \n 0\n \n 0\n
          \n \\(124\\)\n \n 0\n \n 0\n
          \n \\(125\\)\n \n 0\n \n 0\n
          \n \\(126\\)\n \n 0\n \n 0\n
          \n \\(127\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(128\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(129\\)\n \n 0\n \n 0\n
          \n \\(130\\)\n \n 0\n \n 0\n
          \n \\(131\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(132\\)\n \n 0\n \n 0\n
          \n \\(133\\)\n \n 0\n \n 0\n
          \n \\(134\\)\n \n 0\n \n 0\n
          \n \\(135\\)\n \n 0\n \n 0\n
          \n \\(136\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(137\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(138\\)\n \n 0\n \n 0\n
          \n \\(139\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(140\\)\n \n 0\n \n 0\n
          \n \\(141\\)\n \n 0\n \n 0\n
          \n \\(142\\)\n \n 0\n \n 0\n
          \n \\(143\\)\n \n 0\n \n 0\n
          \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(145\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(147\\)\n \n 0\n \n 0\n
          \n \\(148\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(149\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \\(150\\)\n \n 0\n \n 0\n
          \n \\(151\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(152\\)\n \n 0\n \n 0\n
          \n \\(153\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(154\\)\n \n 0\n \n 0\n
          \n \\(155\\)\n \n 0\n \n 0\n
          \n \\(156\\)\n \n 0\n \n 0\n
          \n \\(157\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(158\\)\n \n 0\n \n 0\n
          \n \\(159\\)\n \n 0\n \n 0\n
          \n \\(160\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(161\\)\n \n 0\n \n 0\n
          \n \\(162\\)\n \n 1.00000i\n 1.00000i
          \n \\(163\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(164\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(165\\)\n \n 0\n \n 0\n
          \n \\(166\\)\n \n 0\n \n 0\n
          \n \\(167\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(168\\)\n \n 0\n \n 0\n
          \n \\(169\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(170\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(171\\)\n \n 0\n \n 0\n
          \n \\(172\\)\n \n 0\n \n 0\n
          \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(174\\)\n \n 0\n \n 0\n
          \n \\(175\\)\n \n 0\n \n 0\n
          \n \\(176\\)\n \n 0\n \n 0\n
          \n \\(177\\)\n \n 0\n \n 0\n
          \n \\(178\\)\n \n 0\n \n 0\n
          \n \\(179\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(180\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(182\\)\n \n 0\n \n 0\n
          \n \\(183\\)\n \n 0\n \n 0\n
          \n \\(184\\)\n \n 0\n \n 0\n
          \n \\(185\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(186\\)\n \n 0\n \n 0\n
          \n \\(187\\)\n \n 0\n \n 0\n
          \n \\(188\\)\n \n 0\n \n 0\n
          \n \\(189\\)\n \n 0\n \n 0\n
          \n \\(190\\)\n \n 0\n \n 0\n
          \n \\(191\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(192\\)\n \n 0\n \n 0\n
          \n \\(193\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(194\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(195\\)\n \n 0\n \n 0\n
          \n \\(196\\)\n \n 0\n \n 0\n
          \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(198\\)\n \n 0\n \n 0\n
          \n \\(199\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(200\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(201\\)\n \n 0\n \n 0\n
          \n \\(202\\)\n \n 0\n \n 0\n
          \n \\(203\\)\n \n 0\n \n 0\n
          \n \\(204\\)\n \n 0\n \n 0\n
          \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
          \n \\(206\\)\n \n 0\n \n 0\n
          \n \\(207\\)\n \n 0\n \n 0\n
          \n \\(208\\)\n \n 0\n \n 0\n
          \n \\(209\\)\n \n 0\n \n 0\n
          \n \\(210\\)\n \n 0\n \n 0\n
          \n \\(211\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(212\\)\n \n 0\n \n 0\n
          \n \\(213\\)\n \n 0\n \n 0\n
          \n \\(214\\)\n \n 0\n \n 0\n
          \n \\(215\\)\n \n 0\n \n 0\n
          \n \\(216\\)\n \n 0\n \n 0\n
          \n \\(217\\)\n \n 0\n \n 0\n
          \n \\(218\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(219\\)\n \n 0\n \n 0\n
          \n \\(220\\)\n \n 0\n \n 0\n
          \n \\(221\\)\n \n 0\n \n 0\n
          \n \\(222\\)\n \n 0\n \n 0\n
          \n \\(223\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(224\\)\n \n 0\n \n 0\n
          \n \\(225\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(227\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(228\\)\n \n 0\n \n 0\n
          \n \\(229\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(230\\)\n \n 0\n \n 0\n
          \n \\(231\\)\n \n 0\n \n 0\n
          \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(234\\)\n \n 0\n \n 0\n
          \n \\(235\\)\n \n 0\n \n 0\n
          \n \\(236\\)\n \n 0\n \n 0\n
          \n \\(237\\)\n \n 0\n \n 0\n
          \n \\(238\\)\n \n 0\n \n 0\n
          \n \\(239\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(240\\)\n \n 0\n \n 0\n
          \n \\(241\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(242\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(243\\)\n \n 0\n \n 0\n
          \n \\(244\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(245\\)\n \n 0\n \n 0\n
          \n \\(246\\)\n \n 0\n \n 0\n
          \n \\(247\\)\n \n 0\n \n 0\n
          \n \\(248\\)\n \n 0\n \n 0\n
          \n \\(249\\)\n \n 0\n \n 0\n
          \n \\(250\\)\n \n 0\n \n 0\n
          \n \\(251\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(252\\)\n \n 0\n \n 0\n
          \n \\(253\\)\n \n 0\n \n 0\n
          \n \\(254\\)\n \n 0\n \n 0\n
          \n \\(255\\)\n \n 0\n \n 0\n
          \n \\(256\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(257\\)\n \n 1.73205\n \n \u2212\n \n 1.00000i\n 1.73205\n \n \u2212\n \n 1.00000i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(258\\)\n \n 0\n \n 0\n
          \n \\(259\\)\n \n 0\n \n 0\n
          \n \\(260\\)\n \n 0\n \n 0\n
          \n \\(261\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(262\\)\n \n 0\n \n 0\n
          \n \\(263\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(264\\)\n \n 0\n \n 0\n
          \n \\(265\\)\n \n 0\n \n 0\n
          \n \\(266\\)\n \n 0\n \n 0\n
          \n \\(267\\)\n \n 0\n \n 0\n
          \n \\(268\\)\n \n 0\n \n 0\n
          \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(270\\)\n \n 0\n \n 0\n
          \n \\(271\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(272\\)\n \n 1.00000\n \n 1.00000\n
          \n \\(273\\)\n \n 0\n \n 0\n
          \n \\(274\\)\n \n 0\n \n 0\n
          \n \\(275\\)\n \n 0\n \n 0\n
          \n \\(276\\)\n \n 0\n \n 0\n
          \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(278\\)\n \n 0\n \n 0\n
          \n \\(279\\)\n \n 0\n \n 0\n
          \n \\(280\\)\n \n 0\n \n 0\n
          \n \\(281\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(282\\)\n \n 0\n \n 0\n
          \n \\(283\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(284\\)\n \n 0\n \n 0\n
          \n \\(285\\)\n \n 0\n \n 0\n
          \n \\(286\\)\n \n 0\n \n 0\n
          \n \\(287\\)\n \n 0\n \n 0\n
          \n \\(288\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(289\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(290\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(291\\)\n \n 0\n \n 0\n
          \n \\(292\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(293\\)\n \n 2.00000\n \n 2.00000\n \n 1.00000\n \n \\(0\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(294\\)\n \n 0\n \n 0\n
          \n \\(295\\)\n \n 0\n \n 0\n
          \n \\(296\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(297\\)\n \n 0\n \n 0\n
          \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n 1.73205\n \n +\n \n 1.00000i
          \n \\(299\\)\n \n 0\n \n 0\n
          \n \\(300\\)\n \n 0\n \n 0\n
          \n \\(301\\)\n \n 0\n \n 0\n
          \n \\(302\\)\n \n 0\n \n 0\n
          \n \\(303\\)\n \n 0\n \n 0\n
          \n \\(304\\)\n \n 0\n \n 0\n
          \n \\(305\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
          \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(307\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(308\\)\n \n 0\n \n 0\n
          \n \\(309\\)\n \n 0\n \n 0\n
          \n \\(310\\)\n \n 0\n \n 0\n
          \n \\(311\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(312\\)\n \n 0\n \n 0\n
          \n \\(313\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(314\\)\n \n 2.00000i\n 2.00000i
          \n \\(315\\)\n \n 0\n \n 0\n
          \n \\(316\\)\n \n 0\n \n 0\n
          \n \\(317\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(318\\)\n \n 0\n \n 0\n
          \n \\(319\\)\n \n 0\n \n 0\n
          \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(321\\)\n \n 0\n \n 0\n
          \n \\(322\\)\n \n 0\n \n 0\n
          \n \\(323\\)\n \n 0\n \n 0\n
          \n \\(324\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(325\\)\n \n 0\n \n 0\n
          \n \\(326\\)\n \n 0\n \n 0\n
          \n \\(327\\)\n \n 0\n \n 0\n
          \n \\(328\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(329\\)\n \n 0\n \n 0\n
          \n \\(330\\)\n \n 0\n \n 0\n
          \n \\(331\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(332\\)\n \n 0\n \n 0\n
          \n \\(333\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(334\\)\n \n 0\n \n 0\n
          \n \\(335\\)\n \n 0\n \n 0\n
          \n \\(336\\)\n \n 0\n \n 0\n
          \n \\(337\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(338\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(339\\)\n \n 0\n \n 0\n
          \n \\(340\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(341\\)\n \n 0\n \n 0\n
          \n \\(342\\)\n \n 0\n \n 0\n
          \n \\(343\\)\n \n 0\n \n 0\n
          \n \\(344\\)\n \n 0\n \n 0\n
          \n \\(345\\)\n \n 0\n \n 0\n
          \n \\(346\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(347\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(348\\)\n \n 0\n \n 0\n
          \n \\(349\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(350\\)\n \n 0\n \n 0\n
          \n \\(351\\)\n \n 0\n \n 0\n
          \n \\(352\\)\n \n 0\n \n 0\n
          \n \\(353\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(354\\)\n \n 0\n \n 0\n
          \n \\(355\\)\n \n 0\n \n 0\n
          \n \\(356\\)\n \n 0\n \n 0\n
          \n \\(357\\)\n \n 0\n \n 0\n
          \n \\(358\\)\n \n 0\n \n 0\n
          \n \\(359\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(361\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(362\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(363\\)\n \n 0\n \n 0\n
          \n \\(364\\)\n \n 0\n \n 0\n
          \n \\(365\\)\n \n 2.00000i\n 2.00000i
          \n \\(366\\)\n \n 0\n \n 0\n
          \n \\(367\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(368\\)\n \n 0\n \n 0\n
          \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(370\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(371\\)\n \n 0\n \n 0\n
          \n \\(372\\)\n \n 0\n \n 0\n
          \n \\(373\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(374\\)\n \n 0\n \n 0\n
          \n \\(375\\)\n \n 0\n \n 0\n
          \n \\(376\\)\n \n 0\n \n 0\n
          \n \\(377\\)\n \n 0\n \n 0\n
          \n \\(378\\)\n \n 0\n \n 0\n
          \n \\(379\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(380\\)\n \n 0\n \n 0\n
          \n \\(381\\)\n \n 0\n \n 0\n
          \n \\(382\\)\n \n 0\n \n 0\n
          \n \\(383\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(384\\)\n \n 0\n \n 0\n
          \n \\(385\\)\n \n 0\n \n 0\n
          \n \\(386\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(387\\)\n \n 0\n \n 0\n
          \n \\(388\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(389\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \\(390\\)\n \n 0\n \n 0\n
          \n \\(391\\)\n \n 0\n \n 0\n
          \n \\(392\\)\n \n 0\n \n 0\n
          \n \\(393\\)\n \n 0\n \n 0\n
          \n \\(394\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(395\\)\n \n 0\n \n 0\n
          \n \\(396\\)\n \n 0\n \n 0\n
          \n \\(397\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(398\\)\n \n 0\n \n 0\n
          \n \\(399\\)\n \n 0\n \n 0\n
          \n \\(400\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(401\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(402\\)\n \n 0\n \n 0\n
          \n \\(403\\)\n \n 0\n \n 0\n
          \n \\(404\\)\n \n 0\n \n 0\n
          \n \\(405\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(406\\)\n \n 0\n \n 0\n
          \n \\(407\\)\n \n 0\n \n 0\n
          \n \\(408\\)\n \n 0\n \n 0\n
          \n \\(409\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(410\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(411\\)\n \n 0\n \n 0\n
          \n \\(412\\)\n \n 0\n \n 0\n
          \n \\(413\\)\n \n 0\n \n 0\n
          \n \\(414\\)\n \n 0\n \n 0\n
          \n \\(415\\)\n \n 0\n \n 0\n
          \n \\(416\\)\n \n 0\n \n 0\n
          \n \\(417\\)\n \n 0\n \n 0\n
          \n \\(418\\)\n \n 0\n \n 0\n
          \n \\(419\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(420\\)\n \n 0\n \n 0\n
          \n \\(421\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(422\\)\n \n 0\n \n 0\n
          \n \\(423\\)\n \n 0\n \n 0\n
          \n \\(424\\)\n \n 0\n \n 0\n
          \n \\(425\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(426\\)\n \n 0\n \n 0\n
          \n \\(427\\)\n \n 0\n \n 0\n
          \n \\(428\\)\n \n 0\n \n 0\n
          \n \\(429\\)\n \n 0\n \n 0\n
          \n \\(430\\)\n \n 0\n \n 0\n
          \n \\(431\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(432\\)\n \n 0\n \n 0\n
          \n \\(433\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(434\\)\n \n 0\n \n 0\n
          \n \\(435\\)\n \n 0\n \n 0\n
          \n \\(436\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(437\\)\n \n 0\n \n 0\n
          \n \\(438\\)\n \n 0\n \n 0\n
          \n \\(439\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(440\\)\n \n 0\n \n 0\n
          \n \\(441\\)\n \n 0\n \n 0\n
          \n \\(442\\)\n \n 0\n \n 0\n
          \n \\(443\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(444\\)\n \n 0\n \n 0\n
          \n \\(445\\)\n \n 0\n \n 0\n
          \n \\(446\\)\n \n 0\n \n 0\n
          \n \\(447\\)\n \n 0\n \n 0\n
          \n \\(448\\)\n \n 0\n \n 0\n
          \n \\(449\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(450\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(451\\)\n \n 0\n \n 0\n
          \n \\(452\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(453\\)\n \n 0\n \n 0\n
          \n \\(454\\)\n \n 0\n \n 0\n
          \n \\(455\\)\n \n 0\n \n 0\n
          \n \\(456\\)\n \n 0\n \n 0\n
          \n \\(457\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(458\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(459\\)\n \n 0\n \n 0\n
          \n \\(460\\)\n \n 0\n \n 0\n
          \n \\(461\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(462\\)\n \n 0\n \n 0\n
          \n \\(463\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(464\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(465\\)\n \n 0\n \n 0\n
          \n \\(466\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(467\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(468\\)\n \n 0\n \n 0\n
          \n \\(469\\)\n \n 0\n \n 0\n
          \n \\(470\\)\n \n 0\n \n 0\n
          \n \\(471\\)\n \n 0\n \n 0\n
          \n \\(472\\)\n \n 0\n \n 0\n
          \n \\(473\\)\n \n 0\n \n 0\n
          \n \\(474\\)\n \n 0\n \n 0\n
          \n \\(475\\)\n \n 0\n \n 0\n
          \n \\(476\\)\n \n 0\n \n 0\n
          \n \\(477\\)\n \n 0\n \n 0\n
          \n \\(478\\)\n \n 0\n \n 0\n
          \n \\(479\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(480\\)\n \n 0\n \n 0\n
          \n \\(481\\)\n \n 0\n \n 0\n
          \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(483\\)\n \n 0\n \n 0\n
          \n \\(484\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(485\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(486\\)\n \n 0\n \n 0\n
          \n \\(487\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(489\\)\n \n 0\n \n 0\n
          \n \\(490\\)\n \n 0\n \n 0\n
          \n \\(491\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(492\\)\n \n 0\n \n 0\n
          \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(494\\)\n \n 0\n \n 0\n
          \n \\(495\\)\n \n 0\n \n 0\n
          \n \\(496\\)\n \n 0\n \n 0\n
          \n \\(497\\)\n \n 0\n \n 0\n
          \n \\(498\\)\n \n 0\n \n 0\n
          \n \\(499\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(500\\)\n \n 0\n \n 0\n
          \n \\(501\\)\n \n 0\n \n 0\n
          \n \\(502\\)\n \n 0\n \n 0\n
          \n \\(503\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(504\\)\n \n 0\n \n 0\n
          \n \\(505\\)\n \n 0\n \n 0\n
          \n \\(506\\)\n \n 0\n \n 0\n
          \n \\(507\\)\n \n 0\n \n 0\n
          \n \\(508\\)\n \n 0\n \n 0\n
          \n \\(509\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \\(510\\)\n \n 0\n \n 0\n
          \n \\(511\\)\n \n 0\n \n 0\n
          \n \\(512\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(513\\)\n \n 0\n \n 0\n
          \n \\(514\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
          \n \\(515\\)\n \n 0\n \n 0\n
          \n \\(516\\)\n \n 0\n \n 0\n
          \n \\(517\\)\n \n 0\n \n 0\n
          \n \\(518\\)\n \n 0\n \n 0\n
          \n \\(519\\)\n \n 0\n \n 0\n
          \n \\(520\\)\n \n 0\n \n 0\n
          \n \\(521\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(522\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(523\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(524\\)\n \n 0\n \n 0\n
          \n \\(525\\)\n \n 0\n \n 0\n
          \n \\(526\\)\n \n 0\n \n 0\n
          \n \\(527\\)\n \n 0\n \n 0\n
          \n \\(528\\)\n \n 0\n \n 0\n
          \n \\(529\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(530\\)\n \n 0\n \n 0\n
          \n \\(531\\)\n \n 0\n \n 0\n
          \n \\(532\\)\n \n 0\n \n 0\n
          \n \\(533\\)\n \n 0\n \n 0\n
          \n \\(534\\)\n \n 0\n \n 0\n
          \n \\(535\\)\n \n 0\n \n 0\n
          \n \\(536\\)\n \n 0\n \n 0\n
          \n \\(537\\)\n \n 0\n \n 0\n
          \n \\(538\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(539\\)\n \n 0\n \n 0\n
          \n \\(540\\)\n \n 0\n \n 0\n
          \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(542\\)\n \n 0\n \n 0\n
          \n \\(543\\)\n \n 0\n \n 0\n
          \n \\(544\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(545\\)\n \n \u22122.00000\n \n \u22122.00000\n
          \n \\(546\\)\n \n 0\n \n 0\n
          \n \\(547\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(548\\)\n \n 0\n \n 0\n
          \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(550\\)\n \n 0\n \n 0\n
          \n \\(551\\)\n \n 0\n \n 0\n
          \n \\(552\\)\n \n 0\n \n 0\n
          \n \\(553\\)\n \n 0\n \n 0\n
          \n \\(554\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(555\\)\n \n 0\n \n 0\n
          \n \\(556\\)\n \n 0\n \n 0\n
          \n \\(557\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(558\\)\n \n 0\n \n 0\n
          \n \\(559\\)\n \n 0\n \n 0\n
          \n \\(560\\)\n \n 0\n \n 0\n
          \n \\(561\\)\n \n 0\n \n 0\n
          \n \\(562\\)\n \n 0\n \n 0\n
          \n \\(563\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(564\\)\n \n 0\n \n 0\n
          \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
          \n \\(566\\)\n \n 0\n \n 0\n
          \n \\(567\\)\n \n 0\n \n 0\n
          \n \\(568\\)\n \n 0\n \n 0\n
          \n \\(569\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(570\\)\n \n 0\n \n 0\n
          \n \\(571\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(572\\)\n \n 0\n \n 0\n
          \n \\(573\\)\n \n 0\n \n 0\n
          \n \\(574\\)\n \n 0\n \n 0\n
          \n \\(575\\)\n \n 0\n \n 0\n
          \n \\(576\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(577\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(579\\)\n \n 0\n \n 0\n
          \n \\(580\\)\n \n 2.00000i\n 2.00000i
          \n \\(581\\)\n \n 0\n \n 0\n
          \n \\(582\\)\n \n 0\n \n 0\n
          \n \\(583\\)\n \n 0\n \n 0\n
          \n \\(584\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(585\\)\n \n 0\n \n 0\n
          \n \\(586\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(587\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(588\\)\n \n 0\n \n 0\n
          \n \\(589\\)\n \n 0\n \n 0\n
          \n \\(590\\)\n \n 0\n \n 0\n
          \n \\(591\\)\n \n 0\n \n 0\n
          \n \\(592\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(593\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(594\\)\n \n 0\n \n 0\n
          \n \\(595\\)\n \n 0\n \n 0\n
          \n \\(596\\)\n \n \u22122.00000\n \n \u22122.00000\n
          \n \\(597\\)\n \n 0\n \n 0\n
          \n \\(598\\)\n \n 0\n \n 0\n
          \n \\(599\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(600\\)\n \n 0\n \n 0\n
          \n \\(601\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(602\\)\n \n 0\n \n 0\n
          \n \\(603\\)\n \n 0\n \n 0\n
          \n \\(604\\)\n \n 0\n \n 0\n
          \n \\(605\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(606\\)\n \n 0\n \n 0\n
          \n \\(607\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(608\\)\n \n 0\n \n 0\n
          \n \\(609\\)\n \n 0\n \n 0\n
          \n \\(610\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(611\\)\n \n 0\n \n 0\n
          \n \\(612\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(613\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(614\\)\n \n 0\n \n 0\n
          \n \\(615\\)\n \n 0\n \n 0\n
          \n \\(616\\)\n \n 0\n \n 0\n
          \n \\(617\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(618\\)\n \n 0\n \n 0\n
          \n \\(619\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(620\\)\n \n 0\n \n 0\n
          \n \\(621\\)\n \n 0\n \n 0\n
          \n \\(622\\)\n \n 0\n \n 0\n
          \n \\(623\\)\n \n 0\n \n 0\n
          \n \\(624\\)\n \n 0\n \n 0\n
          \n \\(625\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(626\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(627\\)\n \n 0\n \n 0\n
          \n \\(628\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(629\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(630\\)\n \n 0\n \n 0\n
          \n \\(631\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(632\\)\n \n 0\n \n 0\n
          \n \\(633\\)\n \n 0\n \n 0\n
          \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(635\\)\n \n 0\n \n 0\n
          \n \\(636\\)\n \n 0\n \n 0\n
          \n \\(637\\)\n \n 0\n \n 0\n
          \n \\(638\\)\n \n 0\n \n 0\n
          \n \\(639\\)\n \n 0\n \n 0\n
          \n \\(640\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(641\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(642\\)\n \n 0\n \n 0\n
          \n \\(643\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(644\\)\n \n 0\n \n 0\n
          \n \\(645\\)\n \n 0\n \n 0\n
          \n \\(646\\)\n \n 0\n \n 0\n
          \n \\(647\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(649\\)\n \n 0\n \n 0\n
          \n \\(650\\)\n \n 0\n \n 0\n
          \n \\(651\\)\n \n 0\n \n 0\n
          \n \\(652\\)\n \n 0\n \n 0\n
          \n \\(653\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(654\\)\n \n 0\n \n 0\n
          \n \\(655\\)\n \n 0\n \n 0\n
          \n \\(656\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(658\\)\n \n 0\n \n 0\n
          \n \\(659\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(660\\)\n \n 0\n \n 0\n
          \n \\(661\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(662\\)\n \n 0\n \n 0\n
          \n \\(663\\)\n \n 0\n \n 0\n
          \n \\(664\\)\n \n 0\n \n 0\n
          \n \\(665\\)\n \n 0\n \n 0\n
          \n \\(666\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(667\\)\n \n 0\n \n 0\n
          \n \\(668\\)\n \n 0\n \n 0\n
          \n \\(669\\)\n \n 0\n \n 0\n
          \n \\(670\\)\n \n 0\n \n 0\n
          \n \\(671\\)\n \n 0\n \n 0\n
          \n \\(672\\)\n \n 0\n \n 0\n
          \n \\(673\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(674\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(675\\)\n \n 0\n \n 0\n
          \n \\(676\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(678\\)\n \n 0\n \n 0\n
          \n \\(679\\)\n \n 0\n \n 0\n
          \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(681\\)\n \n 0\n \n 0\n
          \n \\(682\\)\n \n 0\n \n 0\n
          \n \\(683\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(684\\)\n \n 0\n \n 0\n
          \n \\(685\\)\n \n 0\n \n 0\n
          \n \\(686\\)\n \n 0\n \n 0\n
          \n \\(687\\)\n \n 0\n \n 0\n
          \n \\(688\\)\n \n 0\n \n 0\n
          \n \\(689\\)\n \n 0\n \n 0\n
          \n \\(690\\)\n \n 0\n \n 0\n
          \n \\(691\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(692\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(693\\)\n \n 0\n \n 0\n
          \n \\(694\\)\n \n 0\n \n 0\n
          \n \\(695\\)\n \n 0\n \n 0\n
          \n \\(696\\)\n \n 0\n \n 0\n
          \n \\(697\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(698\\)\n \n 0\n \n 0\n
          \n \\(699\\)\n \n 0\n \n 0\n
          \n \\(700\\)\n \n 0\n \n 0\n
          \n \\(701\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(702\\)\n \n 0\n \n 0\n
          \n \\(703\\)\n \n 0\n \n 0\n
          \n \\(704\\)\n \n 0\n \n 0\n
          \n \\(705\\)\n \n 0\n \n 0\n
          \n \\(706\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(707\\)\n \n 0\n \n 0\n
          \n \\(708\\)\n \n 0\n \n 0\n
          \n \\(709\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(710\\)\n \n 0\n \n 0\n
          \n \\(711\\)\n \n 0\n \n 0\n
          \n \\(712\\)\n \n 0\n \n 0\n
          \n \\(713\\)\n \n 0\n \n 0\n
          \n \\(714\\)\n \n 0\n \n 0\n
          \n \\(715\\)\n \n 0\n \n 0\n
          \n \\(716\\)\n \n 0\n \n 0\n
          \n \\(717\\)\n \n 0\n \n 0\n
          \n \\(718\\)\n \n 0\n \n 0\n
          \n \\(719\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(720\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(721\\)\n \n 0\n \n 0\n
          \n \\(722\\)\n \n 1.00000i\n 1.00000i
          \n \\(723\\)\n \n 0\n \n 0\n
          \n \\(724\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(725\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(726\\)\n \n 0\n \n 0\n
          \n \\(727\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(728\\)\n \n 0\n \n 0\n
          \n \\(729\\)\n \n 1.00000i\n 1.00000i
          \n \\(730\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(731\\)\n \n 0\n \n 0\n
          \n \\(732\\)\n \n 0\n \n 0\n
          \n \\(733\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(734\\)\n \n 0\n \n 0\n
          \n \\(735\\)\n \n 0\n \n 0\n
          \n \\(736\\)\n \n 0\n \n 0\n
          \n \\(737\\)\n \n 0\n \n 0\n
          \n \\(738\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(739\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(740\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(741\\)\n \n 0\n \n 0\n
          \n \\(742\\)\n \n 0\n \n 0\n
          \n \\(743\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(744\\)\n \n 0\n \n 0\n
          \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n 0.732051\n \n +\n \n 2.73205i
          \n \\(746\\)\n \n 0\n \n 0\n
          \n \\(747\\)\n \n 0\n \n 0\n
          \n \\(748\\)\n \n 0\n \n 0\n
          \n \\(749\\)\n \n 0\n \n 0\n
          \n \\(750\\)\n \n 0\n \n 0\n
          \n \\(751\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(752\\)\n \n 0\n \n 0\n
          \n \\(753\\)\n \n 0\n \n 0\n
          \n \\(754\\)\n \n 0\n \n 0\n
          \n \\(755\\)\n \n 0\n \n 0\n
          \n \\(756\\)\n \n 0\n \n 0\n
          \n \\(757\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(758\\)\n \n 0\n \n 0\n
          \n \\(759\\)\n \n 0\n \n 0\n
          \n \\(760\\)\n \n 0\n \n 0\n
          \n \\(761\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(762\\)\n \n 0\n \n 0\n
          \n \\(763\\)\n \n 0\n \n 0\n
          \n \\(764\\)\n \n 0\n \n 0\n
          \n \\(765\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(766\\)\n \n 0\n \n 0\n
          \n \\(767\\)\n \n 0\n \n 0\n
          \n \\(768\\)\n \n 0\n \n 0\n
          \n \\(769\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(770\\)\n \n 0\n \n 0\n
          \n \\(771\\)\n \n 0\n \n 0\n
          \n \\(772\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(773\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \\(774\\)\n \n 0\n \n 0\n
          \n \\(775\\)\n \n 0\n \n 0\n
          \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(777\\)\n \n 0\n \n 0\n
          \n \\(778\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(779\\)\n \n 0\n \n 0\n
          \n \\(780\\)\n \n 0\n \n 0\n
          \n \\(781\\)\n \n 0\n \n 0\n
          \n \\(782\\)\n \n 0\n \n 0\n
          \n \\(783\\)\n \n 0\n \n 0\n
          \n \\(784\\)\n \n 0\n \n 0\n
          \n \\(785\\)\n \n \u22122.00000\n \n +\n \n 2.00000i\n \u22122.00000\n \n +\n \n 2.00000i
          \n \\(786\\)\n \n 0\n \n 0\n
          \n \\(787\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(788\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(789\\)\n \n 0\n \n 0\n
          \n \\(790\\)\n \n 0\n \n 0\n
          \n \\(791\\)\n \n 0\n \n 0\n
          \n \\(792\\)\n \n 0\n \n 0\n
          \n \\(793\\)\n \n 0\n \n 0\n
          \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(795\\)\n \n 0\n \n 0\n
          \n \\(796\\)\n \n 0\n \n 0\n
          \n \\(797\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(798\\)\n \n 0\n \n 0\n
          \n \\(799\\)\n \n 0\n \n 0\n
          \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(801\\)\n \n 0\n \n 0\n
          \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(803\\)\n \n 0\n \n 0\n
          \n \\(804\\)\n \n 0\n \n 0\n
          \n \\(805\\)\n \n 0\n \n 0\n
          \n \\(806\\)\n \n 0\n \n 0\n
          \n \\(807\\)\n \n 0\n \n 0\n
          \n \\(808\\)\n \n 0\n \n 0\n
          \n \\(809\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(810\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(811\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(812\\)\n \n 0\n \n 0\n
          \n \\(813\\)\n \n 0\n \n 0\n
          \n \\(814\\)\n \n 0\n \n 0\n
          \n \\(815\\)\n \n 0\n \n 0\n
          \n \\(816\\)\n \n 0\n \n 0\n
          \n \\(817\\)\n \n 0\n \n 0\n
          \n \\(818\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(819\\)\n \n 0\n \n 0\n
          \n \\(820\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(821\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(822\\)\n \n 0\n \n 0\n
          \n \\(823\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(824\\)\n \n 0\n \n 0\n
          \n \\(825\\)\n \n 0\n \n 0\n
          \n \\(826\\)\n \n 0\n \n 0\n
          \n \\(827\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(828\\)\n \n 0\n \n 0\n
          \n \\(829\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(830\\)\n \n 0\n \n 0\n
          \n \\(831\\)\n \n 0\n \n 0\n
          \n \\(832\\)\n \n 0\n \n 0\n
          \n \\(833\\)\n \n 0\n \n 0\n
          \n \\(834\\)\n \n 0\n \n 0\n
          \n \\(835\\)\n \n 0\n \n 0\n
          \n \\(836\\)\n \n 0\n \n 0\n
          \n \\(837\\)\n \n 0\n \n 0\n
          \n \\(838\\)\n \n 0\n \n 0\n
          \n \\(839\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(840\\)\n \n 0\n \n 0\n
          \n \\(841\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(842\\)\n \n 0\n \n 0\n
          \n \\(843\\)\n \n 0\n \n 0\n
          \n \\(844\\)\n \n 0\n \n 0\n
          \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(846\\)\n \n 0\n \n 0\n
          \n \\(847\\)\n \n 0\n \n 0\n
          \n \\(848\\)\n \n 0\n \n 0\n
          \n \\(849\\)\n \n 0\n \n 0\n
          \n \\(850\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(851\\)\n \n 0\n \n 0\n
          \n \\(852\\)\n \n 0\n \n 0\n
          \n \\(853\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(854\\)\n \n 0\n \n 0\n
          \n \\(855\\)\n \n 0\n \n 0\n
          \n \\(856\\)\n \n 0\n \n 0\n
          \n \\(857\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(858\\)\n \n 0\n \n 0\n
          \n \\(859\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(860\\)\n \n 0\n \n 0\n
          \n \\(861\\)\n \n 0\n \n 0\n
          \n \\(862\\)\n \n 0\n \n 0\n
          \n \\(863\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(864\\)\n \n 0\n \n 0\n
          \n \\(865\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(866\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(867\\)\n \n 0\n \n 0\n
          \n \\(868\\)\n \n 0\n \n 0\n
          \n \\(869\\)\n \n 0\n \n 0\n
          \n \\(870\\)\n \n 0\n \n 0\n
          \n \\(871\\)\n \n 0\n \n 0\n
          \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(873\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(874\\)\n \n 0\n \n 0\n
          \n \\(875\\)\n \n 0\n \n 0\n
          \n \\(876\\)\n \n 0\n \n 0\n
          \n \\(877\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(878\\)\n \n 0\n \n 0\n
          \n \\(879\\)\n \n 0\n \n 0\n
          \n \\(880\\)\n \n 0\n \n 0\n
          \n \\(881\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(882\\)\n \n 0\n \n 0\n
          \n \\(883\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(884\\)\n \n 0\n \n 0\n
          \n \\(885\\)\n \n 0\n \n 0\n
          \n \\(886\\)\n \n 0\n \n 0\n
          \n \\(887\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(888\\)\n \n 0\n \n 0\n
          \n \\(889\\)\n \n 0\n \n 0\n
          \n \\(890\\)\n \n 0\n \n 0\n
          \n \\(891\\)\n \n 0\n \n 0\n
          \n \\(892\\)\n \n 0\n \n 0\n
          \n \\(893\\)\n \n 0\n \n 0\n
          \n \\(894\\)\n \n 0\n \n 0\n
          \n \\(895\\)\n \n 0\n \n 0\n
          \n \\(896\\)\n \n 0\n \n 0\n
          \n \\(897\\)\n \n 0\n \n 0\n
          \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(899\\)\n \n 0\n \n 0\n
          \n \\(900\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(901\\)\n \n 0\n \n 0\n
          \n \\(902\\)\n \n 0\n \n 0\n
          \n \\(903\\)\n \n 0\n \n 0\n
          \n \\(904\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(905\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(906\\)\n \n 0\n \n 0\n
          \n \\(907\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(908\\)\n \n 0\n \n 0\n
          \n \\(909\\)\n \n 0\n \n 0\n
          \n \\(910\\)\n \n 0\n \n 0\n
          \n \\(911\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(912\\)\n \n 0\n \n 0\n
          \n \\(913\\)\n \n 0\n \n 0\n
          \n \\(914\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(915\\)\n \n 0\n \n 0\n
          \n \\(916\\)\n \n 2.00000i\n 2.00000i
          \n \\(917\\)\n \n 0\n \n 0\n
          \n \\(918\\)\n \n 0\n \n 0\n
          \n \\(919\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(920\\)\n \n 0\n \n 0\n
          \n \\(921\\)\n \n 0\n \n 0\n
          \n \\(922\\)\n \n 0\n \n 0\n
          \n \\(923\\)\n \n 0\n \n 0\n
          \n \\(924\\)\n \n 0\n \n 0\n
          \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(926\\)\n \n 0\n \n 0\n
          \n \\(927\\)\n \n 0\n \n 0\n
          \n \\(928\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(929\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(930\\)\n \n 0\n \n 0\n
          \n \\(931\\)\n \n 0\n \n 0\n
          \n \\(932\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(933\\)\n \n 0\n \n 0\n
          \n \\(934\\)\n \n 0\n \n 0\n
          \n \\(935\\)\n \n 0\n \n 0\n
          \n \\(936\\)\n \n 0\n \n 0\n
          \n \\(937\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(938\\)\n \n 0\n \n 0\n
          \n \\(939\\)\n \n 0\n \n 0\n
          \n \\(940\\)\n \n 0\n \n 0\n
          \n \\(941\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(942\\)\n \n 0\n \n 0\n
          \n \\(943\\)\n \n 0\n \n 0\n
          \n \\(944\\)\n \n 0\n \n 0\n
          \n \\(945\\)\n \n 0\n \n 0\n
          \n \\(946\\)\n \n 0\n \n 0\n
          \n \\(947\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(948\\)\n \n 0\n \n 0\n
          \n \\(949\\)\n \n 0\n \n 0\n
          \n \\(950\\)\n \n 0\n \n 0\n
          \n \\(951\\)\n \n 0\n \n 0\n
          \n \\(952\\)\n \n 0\n \n 0\n
          \n \\(953\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(954\\)\n \n 0\n \n 0\n
          \n \\(955\\)\n \n 0\n \n 0\n
          \n \\(956\\)\n \n 0\n \n 0\n
          \n \\(957\\)\n \n 0\n \n 0\n
          \n \\(958\\)\n \n 0\n \n 0\n
          \n \\(959\\)\n \n 0\n \n 0\n
          \n \\(960\\)\n \n 0\n \n 0\n
          \n \\(961\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(962\\)\n \n 0\n \n 0\n
          \n \\(963\\)\n \n 0\n \n 0\n
          \n \\(964\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(965\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(966\\)\n \n 0\n \n 0\n
          \n \\(967\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(969\\)\n \n 0\n \n 0\n
          \n \\(970\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(971\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(972\\)\n \n 0\n \n 0\n
          \n \\(973\\)\n \n 0\n \n 0\n
          \n \\(974\\)\n \n 0\n \n 0\n
          \n \\(975\\)\n \n 0\n \n 0\n
          \n \\(976\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(977\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(978\\)\n \n 0\n \n 0\n
          \n \\(979\\)\n \n 0\n \n 0\n
          \n \\(980\\)\n \n 0\n \n 0\n
          \n \\(981\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(982\\)\n \n 0\n \n 0\n
          \n \\(983\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(984\\)\n \n 0\n \n 0\n
          \n \\(985\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(986\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(987\\)\n \n 0\n \n 0\n
          \n \\(988\\)\n \n 0\n \n 0\n
          \n \\(989\\)\n \n 0\n \n 0\n
          \n \\(990\\)\n \n 0\n \n 0\n
          \n \\(991\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(992\\)\n \n 0\n \n 0\n
          \n \\(993\\)\n \n 0\n \n 0\n
          \n \\(994\\)\n \n 0\n \n 0\n
          \n \\(995\\)\n \n 0\n \n 0\n
          \n \\(996\\)\n \n 0\n \n 0\n
          \n \\(997\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(998\\)\n \n 0\n \n 0\n
          \n \\(999\\)\n \n 0\n \n 0\n
          ", "content": {"html": "
          \\(n\\)\\(a_n\\)\\(a_n / n^{(k-1)/2}\\)\\( \\alpha_n \\)\\( \\theta_n \\)
          \\(p\\)\\(a_p\\)\\(a_p / p^{(k-1)/2}\\)\\( \\alpha_p\\)\\( \\theta_p \\)
          \\(2\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(3\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(4\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(5\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(6\\)00
          \\(7\\)00
          \\(8\\)1.00000<br>i1.00000<br>i
          \\(9\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(10\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(11\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(12\\)00
          \\(13\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(14\\)00
          \\(15\\)00
          \\(16\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(17\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(18\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(19\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(20\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(21\\)00
          \\(22\\)00
          \\(23\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(24\\)00
          \\(25\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(26\\)00
          \\(27\\)00
          \\(28\\)00
          \\(29\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(30\\)00
          \\(31\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(32\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(33\\)00
          \\(34\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(35\\)00
          \\(36\\)1.00000<br>i1.00000<br>i
          \\(37\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(38\\)00
          \\(39\\)00
          \\(40\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(41\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(42\\)00
          \\(43\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(44\\)00
          \\(45\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(46\\)00
          \\(47\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(48\\)00
          \\(49\\)00
          \\(50\\)\u22121.00000\u22121.00000
          \\(51\\)00
          \\(52\\)00
          \\(53\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(54\\)00
          \\(55\\)00
          \\(56\\)00
          \\(57\\)00
          \\(58\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(59\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(60\\)00
          \\(61\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(62\\)00
          \\(63\\)00
          \\(64\\)\u22121.00000\u22121.00000
          \\(65\\)00
          \\(66\\)00
          \\(67\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(68\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(69\\)00
          \\(70\\)00
          \\(71\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(72\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(73\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(74\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(75\\)00
          \\(76\\)00
          \\(77\\)00
          \\(78\\)00
          \\(79\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(80\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(81\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(82\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(83\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(84\\)00
          \\(85\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(86\\)00
          \\(87\\)00
          \\(88\\)00
          \\(89\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(90\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(91\\)00
          \\(92\\)00
          \\(93\\)00
          \\(94\\)00
          \\(95\\)00
          \\(96\\)00
          \\(97\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(98\\)00
          \\(99\\)00
          \\(100\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(101\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(102\\)00
          \\(103\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(104\\)00
          \\(105\\)00
          \\(106\\)00
          \\(107\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(108\\)00
          \\(109\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(110\\)00
          \\(111\\)00
          \\(112\\)00
          \\(113\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(114\\)00
          \\(115\\)00
          \\(116\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(117\\)00
          \\(118\\)00
          \\(119\\)00
          \\(120\\)00
          \\(121\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(122\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(123\\)00
          \\(124\\)00
          \\(125\\)00
          \\(126\\)00
          \\(127\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(128\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(129\\)00
          \\(130\\)00
          \\(131\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(132\\)00
          \\(133\\)00
          \\(134\\)00
          \\(135\\)00
          \\(136\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(137\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(138\\)00
          \\(139\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(140\\)00
          \\(141\\)00
          \\(142\\)00
          \\(143\\)00
          \\(144\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(145\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(146\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(147\\)00
          \\(148\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(149\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \\(150\\)00
          \\(151\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(152\\)00
          \\(153\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(154\\)00
          \\(155\\)00
          \\(156\\)00
          \\(157\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(158\\)00
          \\(159\\)00
          \\(160\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(161\\)00
          \\(162\\)1.00000<br>i1.00000<br>i
          \\(163\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(164\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(165\\)00
          \\(166\\)00
          \\(167\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(168\\)00
          \\(169\\)\u22121.00000\u22121.00000
          \\(170\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(171\\)00
          \\(172\\)00
          \\(173\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(174\\)00
          \\(175\\)00
          \\(176\\)00
          \\(177\\)00
          \\(178\\)00
          \\(179\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(180\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(181\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(182\\)00
          \\(183\\)00
          \\(184\\)00
          \\(185\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(186\\)00
          \\(187\\)00
          \\(188\\)00
          \\(189\\)00
          \\(190\\)00
          \\(191\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(192\\)00
          \\(193\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(194\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(195\\)00
          \\(196\\)00
          \\(197\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(198\\)00
          \\(199\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(200\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(201\\)00
          \\(202\\)00
          \\(203\\)00
          \\(204\\)00
          \\(205\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
          \\(206\\)00
          \\(207\\)00
          \\(208\\)00
          \\(209\\)00
          \\(210\\)00
          \\(211\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(212\\)00
          \\(213\\)00
          \\(214\\)00
          \\(215\\)00
          \\(216\\)00
          \\(217\\)00
          \\(218\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(219\\)00
          \\(220\\)00
          \\(221\\)00
          \\(222\\)00
          \\(223\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(224\\)00
          \\(225\\)\u22121.00000\u22121.00000
          \\(226\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(227\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(228\\)00
          \\(229\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(230\\)00
          \\(231\\)00
          \\(232\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(233\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(234\\)00
          \\(235\\)00
          \\(236\\)00
          \\(237\\)00
          \\(238\\)00
          \\(239\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(240\\)00
          \\(241\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(242\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(243\\)00
          \\(244\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(245\\)00
          \\(246\\)00
          \\(247\\)00
          \\(248\\)00
          \\(249\\)00
          \\(250\\)00
          \\(251\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(252\\)00
          \\(253\\)00
          \\(254\\)00
          \\(255\\)00
          \\(256\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(257\\)1.73205\u22121.00000<br>i1.73205\u22121.00000<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(258\\)00
          \\(259\\)00
          \\(260\\)00
          \\(261\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(262\\)00
          \\(263\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(264\\)00
          \\(265\\)00
          \\(266\\)00
          \\(267\\)00
          \\(268\\)00
          \\(269\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(270\\)00
          \\(271\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(272\\)1.000001.00000
          \\(273\\)00
          \\(274\\)00
          \\(275\\)00
          \\(276\\)00
          \\(277\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(278\\)00
          \\(279\\)00
          \\(280\\)00
          \\(281\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(282\\)00
          \\(283\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(284\\)00
          \\(285\\)00
          \\(286\\)00
          \\(287\\)00
          \\(288\\)\u22121.00000\u22121.00000
          \\(289\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(290\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(291\\)00
          \\(292\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(293\\)2.000002.000001.00000\\(0\\)
          1.00000\\(0\\)
          \\(294\\)00
          \\(295\\)00
          \\(296\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(297\\)00
          \\(298\\)1.73205+1.00000<br>i1.73205+1.00000<br>i
          \\(299\\)00
          \\(300\\)00
          \\(301\\)00
          \\(302\\)00
          \\(303\\)00
          \\(304\\)00
          \\(305\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
          \\(306\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(307\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(308\\)00
          \\(309\\)00
          \\(310\\)00
          \\(311\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(312\\)00
          \\(313\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(314\\)2.00000<br>i2.00000<br>i
          \\(315\\)00
          \\(316\\)00
          \\(317\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(318\\)00
          \\(319\\)00
          \\(320\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(321\\)00
          \\(322\\)00
          \\(323\\)00
          \\(324\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(325\\)00
          \\(326\\)00
          \\(327\\)00
          \\(328\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(329\\)00
          \\(330\\)00
          \\(331\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(332\\)00
          \\(333\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(334\\)00
          \\(335\\)00
          \\(336\\)00
          \\(337\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(338\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(339\\)00
          \\(340\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(341\\)00
          \\(342\\)00
          \\(343\\)00
          \\(344\\)00
          \\(345\\)00
          \\(346\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(347\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(348\\)00
          \\(349\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(350\\)00
          \\(351\\)00
          \\(352\\)00
          \\(353\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(354\\)00
          \\(355\\)00
          \\(356\\)00
          \\(357\\)00
          \\(358\\)00
          \\(359\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(360\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(361\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(362\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(363\\)00
          \\(364\\)00
          \\(365\\)2.00000<br>i2.00000<br>i
          \\(366\\)00
          \\(367\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(368\\)00
          \\(369\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(370\\)2.000002.00000
          \\(371\\)00
          \\(372\\)00
          \\(373\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(374\\)00
          \\(375\\)00
          \\(376\\)00
          \\(377\\)00
          \\(378\\)00
          \\(379\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(380\\)00
          \\(381\\)00
          \\(382\\)00
          \\(383\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(384\\)00
          \\(385\\)00
          \\(386\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(387\\)00
          \\(388\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(389\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \\(390\\)00
          \\(391\\)00
          \\(392\\)00
          \\(393\\)00
          \\(394\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(395\\)00
          \\(396\\)00
          \\(397\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(398\\)00
          \\(399\\)00
          \\(400\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(401\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(402\\)00
          \\(403\\)00
          \\(404\\)00
          \\(405\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(406\\)00
          \\(407\\)00
          \\(408\\)00
          \\(409\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(410\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(411\\)00
          \\(412\\)00
          \\(413\\)00
          \\(414\\)00
          \\(415\\)00
          \\(416\\)00
          \\(417\\)00
          \\(418\\)00
          \\(419\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(420\\)00
          \\(421\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(422\\)00
          \\(423\\)00
          \\(424\\)00
          \\(425\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(426\\)00
          \\(427\\)00
          \\(428\\)00
          \\(429\\)00
          \\(430\\)00
          \\(431\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(432\\)00
          \\(433\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(434\\)00
          \\(435\\)00
          \\(436\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(437\\)00
          \\(438\\)00
          \\(439\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(440\\)00
          \\(441\\)00
          \\(442\\)00
          \\(443\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(444\\)00
          \\(445\\)00
          \\(446\\)00
          \\(447\\)00
          \\(448\\)00
          \\(449\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(450\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(451\\)00
          \\(452\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(453\\)00
          \\(454\\)00
          \\(455\\)00
          \\(456\\)00
          \\(457\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(458\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(459\\)00
          \\(460\\)00
          \\(461\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(462\\)00
          \\(463\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(464\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(465\\)00
          \\(466\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(467\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(468\\)00
          \\(469\\)00
          \\(470\\)00
          \\(471\\)00
          \\(472\\)00
          \\(473\\)00
          \\(474\\)00
          \\(475\\)00
          \\(476\\)00
          \\(477\\)00
          \\(478\\)00
          \\(479\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(480\\)00
          \\(481\\)00
          \\(482\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(483\\)00
          \\(484\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(485\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(486\\)00
          \\(487\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(488\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(489\\)00
          \\(490\\)00
          \\(491\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(492\\)00
          \\(493\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(494\\)00
          \\(495\\)00
          \\(496\\)00
          \\(497\\)00
          \\(498\\)00
          \\(499\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(500\\)00
          \\(501\\)00
          \\(502\\)00
          \\(503\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(504\\)00
          \\(505\\)00
          \\(506\\)00
          \\(507\\)00
          \\(508\\)00
          \\(509\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \\(510\\)00
          \\(511\\)00
          \\(512\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(513\\)00
          \\(514\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
          \\(515\\)00
          \\(516\\)00
          \\(517\\)00
          \\(518\\)00
          \\(519\\)00
          \\(520\\)00
          \\(521\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(522\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(523\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(524\\)00
          \\(525\\)00
          \\(526\\)00
          \\(527\\)00
          \\(528\\)00
          \\(529\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(530\\)00
          \\(531\\)00
          \\(532\\)00
          \\(533\\)00
          \\(534\\)00
          \\(535\\)00
          \\(536\\)00
          \\(537\\)00
          \\(538\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(539\\)00
          \\(540\\)00
          \\(541\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(542\\)00
          \\(543\\)00
          \\(544\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(545\\)\u22122.00000\u22122.00000
          \\(546\\)00
          \\(547\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(548\\)00
          \\(549\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(550\\)00
          \\(551\\)00
          \\(552\\)00
          \\(553\\)00
          \\(554\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(555\\)00
          \\(556\\)00
          \\(557\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(558\\)00
          \\(559\\)00
          \\(560\\)00
          \\(561\\)00
          \\(562\\)00
          \\(563\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(564\\)00
          \\(565\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
          \\(566\\)00
          \\(567\\)00
          \\(568\\)00
          \\(569\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(570\\)00
          \\(571\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(572\\)00
          \\(573\\)00
          \\(574\\)00
          \\(575\\)00
          \\(576\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(577\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(578\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(579\\)00
          \\(580\\)2.00000<br>i2.00000<br>i
          \\(581\\)00
          \\(582\\)00
          \\(583\\)00
          \\(584\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(585\\)00
          \\(586\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(587\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(588\\)00
          \\(589\\)00
          \\(590\\)00
          \\(591\\)00
          \\(592\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(593\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(594\\)00
          \\(595\\)00
          \\(596\\)\u22122.00000\u22122.00000
          \\(597\\)00
          \\(598\\)00
          \\(599\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(600\\)00
          \\(601\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(602\\)00
          \\(603\\)00
          \\(604\\)00
          \\(605\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(606\\)00
          \\(607\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(608\\)00
          \\(609\\)00
          \\(610\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(611\\)00
          \\(612\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(613\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(614\\)00
          \\(615\\)00
          \\(616\\)00
          \\(617\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(618\\)00
          \\(619\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(620\\)00
          \\(621\\)00
          \\(622\\)00
          \\(623\\)00
          \\(624\\)00
          \\(625\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(626\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(627\\)00
          \\(628\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(629\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(630\\)00
          \\(631\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(632\\)00
          \\(633\\)00
          \\(634\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(635\\)00
          \\(636\\)00
          \\(637\\)00
          \\(638\\)00
          \\(639\\)00
          \\(640\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(641\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(642\\)00
          \\(643\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(644\\)00
          \\(645\\)00
          \\(646\\)00
          \\(647\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(648\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(649\\)00
          \\(650\\)00
          \\(651\\)00
          \\(652\\)00
          \\(653\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(654\\)00
          \\(655\\)00
          \\(656\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(657\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(658\\)00
          \\(659\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(660\\)00
          \\(661\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(662\\)00
          \\(663\\)00
          \\(664\\)00
          \\(665\\)00
          \\(666\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(667\\)00
          \\(668\\)00
          \\(669\\)00
          \\(670\\)00
          \\(671\\)00
          \\(672\\)00
          \\(673\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(674\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(675\\)00
          \\(676\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(677\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(678\\)00
          \\(679\\)00
          \\(680\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(681\\)00
          \\(682\\)00
          \\(683\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(684\\)00
          \\(685\\)00
          \\(686\\)00
          \\(687\\)00
          \\(688\\)00
          \\(689\\)00
          \\(690\\)00
          \\(691\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(692\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(693\\)00
          \\(694\\)00
          \\(695\\)00
          \\(696\\)00
          \\(697\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(698\\)00
          \\(699\\)00
          \\(700\\)00
          \\(701\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(702\\)00
          \\(703\\)00
          \\(704\\)00
          \\(705\\)00
          \\(706\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(707\\)00
          \\(708\\)00
          \\(709\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(710\\)00
          \\(711\\)00
          \\(712\\)00
          \\(713\\)00
          \\(714\\)00
          \\(715\\)00
          \\(716\\)00
          \\(717\\)00
          \\(718\\)00
          \\(719\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(720\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(721\\)00
          \\(722\\)1.00000<br>i1.00000<br>i
          \\(723\\)00
          \\(724\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(725\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(726\\)00
          \\(727\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(728\\)00
          \\(729\\)1.00000<br>i1.00000<br>i
          \\(730\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(731\\)00
          \\(732\\)00
          \\(733\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(734\\)00
          \\(735\\)00
          \\(736\\)00
          \\(737\\)00
          \\(738\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(739\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(740\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(741\\)00
          \\(742\\)00
          \\(743\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(744\\)00
          \\(745\\)0.732051+2.73205<br>i0.732051+2.73205<br>i
          \\(746\\)00
          \\(747\\)00
          \\(748\\)00
          \\(749\\)00
          \\(750\\)00
          \\(751\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(752\\)00
          \\(753\\)00
          \\(754\\)00
          \\(755\\)00
          \\(756\\)00
          \\(757\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(758\\)00
          \\(759\\)00
          \\(760\\)00
          \\(761\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(762\\)00
          \\(763\\)00
          \\(764\\)00
          \\(765\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(766\\)00
          \\(767\\)00
          \\(768\\)00
          \\(769\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(770\\)00
          \\(771\\)00
          \\(772\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(773\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \\(774\\)00
          \\(775\\)00
          \\(776\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(777\\)00
          \\(778\\)2.000002.00000
          \\(779\\)00
          \\(780\\)00
          \\(781\\)00
          \\(782\\)00
          \\(783\\)00
          \\(784\\)00
          \\(785\\)\u22122.00000+2.00000<br>i\u22122.00000+2.00000<br>i
          \\(786\\)00
          \\(787\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(788\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(789\\)00
          \\(790\\)00
          \\(791\\)00
          \\(792\\)00
          \\(793\\)00
          \\(794\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(795\\)00
          \\(796\\)00
          \\(797\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(798\\)00
          \\(799\\)00
          \\(800\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(801\\)00
          \\(802\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(803\\)00
          \\(804\\)00
          \\(805\\)00
          \\(806\\)00
          \\(807\\)00
          \\(808\\)00
          \\(809\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(810\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(811\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(812\\)00
          \\(813\\)00
          \\(814\\)00
          \\(815\\)00
          \\(816\\)00
          \\(817\\)00
          \\(818\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(819\\)00
          \\(820\\)2.000002.00000
          \\(821\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(822\\)00
          \\(823\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(824\\)00
          \\(825\\)00
          \\(826\\)00
          \\(827\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(828\\)00
          \\(829\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(830\\)00
          \\(831\\)00
          \\(832\\)00
          \\(833\\)00
          \\(834\\)00
          \\(835\\)00
          \\(836\\)00
          \\(837\\)00
          \\(838\\)00
          \\(839\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(840\\)00
          \\(841\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(842\\)00
          \\(843\\)00
          \\(844\\)00
          \\(845\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(846\\)00
          \\(847\\)00
          \\(848\\)00
          \\(849\\)00
          \\(850\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(851\\)00
          \\(852\\)00
          \\(853\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(854\\)00
          \\(855\\)00
          \\(856\\)00
          \\(857\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(858\\)00
          \\(859\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(860\\)00
          \\(861\\)00
          \\(862\\)00
          \\(863\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(864\\)00
          \\(865\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(866\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(867\\)00
          \\(868\\)00
          \\(869\\)00
          \\(870\\)00
          \\(871\\)00
          \\(872\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(873\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(874\\)00
          \\(875\\)00
          \\(876\\)00
          \\(877\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(878\\)00
          \\(879\\)00
          \\(880\\)00
          \\(881\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(882\\)00
          \\(883\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(884\\)00
          \\(885\\)00
          \\(886\\)00
          \\(887\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(888\\)00
          \\(889\\)00
          \\(890\\)00
          \\(891\\)00
          \\(892\\)00
          \\(893\\)00
          \\(894\\)00
          \\(895\\)00
          \\(896\\)00
          \\(897\\)00
          \\(898\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(899\\)00
          \\(900\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(901\\)00
          \\(902\\)00
          \\(903\\)00
          \\(904\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(905\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(906\\)00
          \\(907\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(908\\)00
          \\(909\\)00
          \\(910\\)00
          \\(911\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(912\\)00
          \\(913\\)00
          \\(914\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(915\\)00
          \\(916\\)2.00000<br>i2.00000<br>i
          \\(917\\)00
          \\(918\\)00
          \\(919\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(920\\)00
          \\(921\\)00
          \\(922\\)00
          \\(923\\)00
          \\(924\\)00
          \\(925\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(926\\)00
          \\(927\\)00
          \\(928\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(929\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(930\\)00
          \\(931\\)00
          \\(932\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(933\\)00
          \\(934\\)00
          \\(935\\)00
          \\(936\\)00
          \\(937\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(938\\)00
          \\(939\\)00
          \\(940\\)00
          \\(941\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(942\\)00
          \\(943\\)00
          \\(944\\)00
          \\(945\\)00
          \\(946\\)00
          \\(947\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(948\\)00
          \\(949\\)00
          \\(950\\)00
          \\(951\\)00
          \\(952\\)00
          \\(953\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(954\\)00
          \\(955\\)00
          \\(956\\)00
          \\(957\\)00
          \\(958\\)00
          \\(959\\)00
          \\(960\\)00
          \\(961\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(962\\)00
          \\(963\\)00
          \\(964\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(965\\)2.000002.00000
          \\(966\\)00
          \\(967\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(968\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(969\\)00
          \\(970\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(971\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(972\\)00
          \\(973\\)00
          \\(974\\)00
          \\(975\\)00
          \\(976\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(977\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(978\\)00
          \\(979\\)00
          \\(980\\)00
          \\(981\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(982\\)00
          \\(983\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(984\\)00
          \\(985\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(986\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(987\\)00
          \\(988\\)00
          \\(989\\)00
          \\(990\\)00
          \\(991\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(992\\)00
          \\(993\\)00
          \\(994\\)00
          \\(995\\)00
          \\(996\\)00
          \\(997\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(998\\)00
          \\(999\\)00
          ", "is_complex": true}}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "table", "raw_content": "
          \u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisting character
          CharParityOrdTypeTwistMinDim
          1.1even1trivial3332.1.bc.b.2027.14
          4.3odd2CM3332.1.bc.b.2027.14
          7.2even3inner3332.1.bc.b.667.14
          7.3odd668.1.f.a.55.1yes2
          7.4even33332.1.m.b.2843.12
          7.5odd63332.1.bc.c.667.14
          7.6odd23332.1.bc.c.2027.14
          17.13even4inner3332.1.bc.b.2223.14
          21.17even6612.1.l.a.55.12
          28.3even668.1.f.a.55.1yes2
          28.11odd63332.1.m.b.2843.12
          28.19even63332.1.bc.c.667.14
          28.23odd6inner3332.1.bc.b.667.14
          28.27even23332.1.bc.c.2027.14
          35.3even121700.1.n.a.599.12
          35.17even121700.1.n.b.599.12
          35.24odd61700.1.p.a.1551.12
          56.3even61088.1.p.a.191.12
          56.45odd61088.1.p.a.191.12
          68.47odd4inner3332.1.bc.b.2223.14
          84.59odd6612.1.l.a.55.12
          119.3even481156.1.g.b.155.18
          119.10even481156.1.g.b.179.18
          119.13odd43332.1.bc.c.2223.14
          119.24even481156.1.g.b.179.28
          119.30even12inner3332.1.bc.b.863.14
          119.31even481156.1.g.b.155.28
          119.38odd121156.1.f.b.251.12
          119.45even481156.1.g.b.399.18
          119.47odd123332.1.bc.c.863.14
          119.59odd241156.1.c.b.579.12
          119.66odd241156.1.d.a.1155.22
          119.73even481156.1.g.b.423.28
          119.80even481156.1.g.b.423.18
          119.81even123332.1.m.b.3039.12
          119.87odd241156.1.d.a.1155.12
          119.94odd241156.1.c.b.579.22
          119.101odd61156.1.f.b.327.12
          119.108even481156.1.g.b.399.28
          119.115odd1268.1.f.a.47.1&check;2
          140.3odd121700.1.n.a.599.12
          140.59even61700.1.p.a.1551.12
          140.87odd121700.1.n.b.599.12
          357.353even12612.1.l.a.523.12
          476.3odd481156.1.g.b.155.18
          476.31odd481156.1.g.b.155.28
          476.47even123332.1.bc.c.863.14
          476.59even241156.1.c.b.579.12
          476.87even241156.1.d.a.1155.12
          476.115even1268.1.f.a.47.1&check;2
          476.143odd481156.1.g.b.179.28
          476.199odd481156.1.g.b.423.18
          476.227odd481156.1.g.b.399.28
          476.251even43332.1.bc.c.2223.14
          476.283odd481156.1.g.b.399.18
          476.311odd481156.1.g.b.423.28
          476.319odd123332.1.m.b.3039.12
          476.339even61156.1.f.b.327.12
          476.367odd481156.1.g.b.179.18
          476.387odd12inner3332.1.bc.b.863.14
          476.395even121156.1.f.b.251.12
          476.423even241156.1.d.a.1155.22
          476.451even241156.1.c.b.579.22
          595.234odd121700.1.p.a.251.12
          595.353even121700.1.n.b.999.12
          595.472even121700.1.n.a.999.12
          952.115even121088.1.p.a.319.12
          952.829odd121088.1.p.a.319.12
          1428.1067odd12612.1.l.a.523.12
          2380.1067odd121700.1.n.a.999.12
          2380.1543odd121700.1.n.b.999.12
          2380.2019even121700.1.p.a.251.12
          ", "content": {"html": "
          By <br>twisting character
          CharParityOrdTypeTwistMinDim
          1.1even1trivial3332.1.bc.b.2027.14
          4.3odd2CM3332.1.bc.b.2027.14
          7.2even3inner3332.1.bc.b.667.14
          7.3odd668.1.f.a.55.1yes2
          7.4even33332.1.m.b.2843.12
          7.5odd63332.1.bc.c.667.14
          7.6odd23332.1.bc.c.2027.14
          17.13even4inner3332.1.bc.b.2223.14
          21.17even6612.1.l.a.55.12
          28.3even668.1.f.a.55.1yes2
          28.11odd63332.1.m.b.2843.12
          28.19even63332.1.bc.c.667.14
          28.23odd6inner3332.1.bc.b.667.14
          28.27even23332.1.bc.c.2027.14
          35.3even121700.1.n.a.599.12
          35.17even121700.1.n.b.599.12
          35.24odd61700.1.p.a.1551.12
          56.3even61088.1.p.a.191.12
          56.45odd61088.1.p.a.191.12
          68.47odd4inner3332.1.bc.b.2223.14
          84.59odd6612.1.l.a.55.12
          119.3even481156.1.g.b.155.18
          119.10even481156.1.g.b.179.18
          119.13odd43332.1.bc.c.2223.14
          119.24even481156.1.g.b.179.28
          119.30even12inner3332.1.bc.b.863.14
          119.31even481156.1.g.b.155.28
          119.38odd121156.1.f.b.251.12
          119.45even481156.1.g.b.399.18
          119.47odd123332.1.bc.c.863.14
          119.59odd241156.1.c.b.579.12
          119.66odd241156.1.d.a.1155.22
          119.73even481156.1.g.b.423.28
          119.80even481156.1.g.b.423.18
          119.81even123332.1.m.b.3039.12
          119.87odd241156.1.d.a.1155.12
          119.94odd241156.1.c.b.579.22
          119.101odd61156.1.f.b.327.12
          119.108even481156.1.g.b.399.28
          119.115odd1268.1.f.a.47.1&check;2
          140.3odd121700.1.n.a.599.12
          140.59even61700.1.p.a.1551.12
          140.87odd121700.1.n.b.599.12
          357.353even12612.1.l.a.523.12
          476.3odd481156.1.g.b.155.18
          476.31odd481156.1.g.b.155.28
          476.47even123332.1.bc.c.863.14
          476.59even241156.1.c.b.579.12
          476.87even241156.1.d.a.1155.12
          476.115even1268.1.f.a.47.1&check;2
          476.143odd481156.1.g.b.179.28
          476.199odd481156.1.g.b.423.18
          476.227odd481156.1.g.b.399.28
          476.251even43332.1.bc.c.2223.14
          476.283odd481156.1.g.b.399.18
          476.311odd481156.1.g.b.423.28
          476.319odd123332.1.m.b.3039.12
          476.339even61156.1.f.b.327.12
          476.367odd481156.1.g.b.179.18
          476.387odd12inner3332.1.bc.b.863.14
          476.395even121156.1.f.b.251.12
          476.423even241156.1.d.a.1155.22
          476.451even241156.1.c.b.579.22
          595.234odd121700.1.p.a.251.12
          595.353even121700.1.n.b.999.12
          595.472even121700.1.n.a.999.12
          952.115even121088.1.p.a.319.12
          952.829odd121088.1.p.a.319.12
          1428.1067odd12612.1.l.a.523.12
          2380.1067odd121700.1.n.a.999.12
          2380.1543odd121700.1.n.b.999.12
          2380.2019even121700.1.p.a.251.12
          ", "is_complex": false}}, {"type": "table", "raw_content": "
          \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisted newform
          TwistMinDimCharParityOrdType
          68.1.f.a.47.1&check;2119.115odd12
          68.1.f.a.47.1&check;2476.115even12
          68.1.f.a.55.1yes27.3odd6
          68.1.f.a.55.1yes228.3even6
          612.1.l.a.55.1221.17even6
          612.1.l.a.55.1284.59odd6
          612.1.l.a.523.12357.353even12
          612.1.l.a.523.121428.1067odd12
          1088.1.p.a.191.1256.3even6
          1088.1.p.a.191.1256.45odd6
          1088.1.p.a.319.12952.115even12
          1088.1.p.a.319.12952.829odd12
          1156.1.c.b.579.12119.59odd24
          1156.1.c.b.579.12476.59even24
          1156.1.c.b.579.22119.94odd24
          1156.1.c.b.579.22476.451even24
          1156.1.d.a.1155.12119.87odd24
          1156.1.d.a.1155.12476.87even24
          1156.1.d.a.1155.22119.66odd24
          1156.1.d.a.1155.22476.423even24
          1156.1.f.b.251.12119.38odd12
          1156.1.f.b.251.12476.395even12
          1156.1.f.b.327.12119.101odd6
          1156.1.f.b.327.12476.339even6
          1156.1.g.b.155.18119.3even48
          1156.1.g.b.155.18476.3odd48
          1156.1.g.b.155.28119.31even48
          1156.1.g.b.155.28476.31odd48
          1156.1.g.b.179.18119.10even48
          1156.1.g.b.179.18476.367odd48
          1156.1.g.b.179.28119.24even48
          1156.1.g.b.179.28476.143odd48
          1156.1.g.b.399.18119.45even48
          1156.1.g.b.399.18476.283odd48
          1156.1.g.b.399.28119.108even48
          1156.1.g.b.399.28476.227odd48
          1156.1.g.b.423.18119.80even48
          1156.1.g.b.423.18476.199odd48
          1156.1.g.b.423.28119.73even48
          1156.1.g.b.423.28476.311odd48
          1700.1.n.a.599.1235.3even12
          1700.1.n.a.599.12140.3odd12
          1700.1.n.a.999.12595.472even12
          1700.1.n.a.999.122380.1067odd12
          1700.1.n.b.599.1235.17even12
          1700.1.n.b.599.12140.87odd12
          1700.1.n.b.999.12595.353even12
          1700.1.n.b.999.122380.1543odd12
          1700.1.p.a.251.12595.234odd12
          1700.1.p.a.251.122380.2019even12
          1700.1.p.a.1551.1235.24odd6
          1700.1.p.a.1551.12140.59even6
          3332.1.m.b.2843.127.4even3
          3332.1.m.b.2843.1228.11odd6
          3332.1.m.b.3039.12119.81even12
          3332.1.m.b.3039.12476.319odd12
          3332.1.bc.b.667.147.2even3inner
          3332.1.bc.b.667.1428.23odd6inner
          3332.1.bc.b.863.14119.30even12inner
          3332.1.bc.b.863.14476.387odd12inner
          3332.1.bc.b.2027.141.1even1trivial
          3332.1.bc.b.2027.144.3odd2CM
          3332.1.bc.b.2223.1417.13even4inner
          3332.1.bc.b.2223.1468.47odd4inner
          3332.1.bc.c.667.147.5odd6
          3332.1.bc.c.667.1428.19even6
          3332.1.bc.c.863.14119.47odd12
          3332.1.bc.c.863.14476.47even12
          3332.1.bc.c.2027.147.6odd2
          3332.1.bc.c.2027.1428.27even2
          3332.1.bc.c.2223.14119.13odd4
          3332.1.bc.c.2223.14476.251even4
          ", "content": {"html": "
          By <br>twisted newform
          TwistMinDimCharParityOrdType
          68.1.f.a.47.1&check;2119.115odd12
          68.1.f.a.47.1&check;2476.115even12
          68.1.f.a.55.1yes27.3odd6
          68.1.f.a.55.1yes228.3even6
          612.1.l.a.55.1221.17even6
          612.1.l.a.55.1284.59odd6
          612.1.l.a.523.12357.353even12
          612.1.l.a.523.121428.1067odd12
          1088.1.p.a.191.1256.3even6
          1088.1.p.a.191.1256.45odd6
          1088.1.p.a.319.12952.115even12
          1088.1.p.a.319.12952.829odd12
          1156.1.c.b.579.12119.59odd24
          1156.1.c.b.579.12476.59even24
          1156.1.c.b.579.22119.94odd24
          1156.1.c.b.579.22476.451even24
          1156.1.d.a.1155.12119.87odd24
          1156.1.d.a.1155.12476.87even24
          1156.1.d.a.1155.22119.66odd24
          1156.1.d.a.1155.22476.423even24
          1156.1.f.b.251.12119.38odd12
          1156.1.f.b.251.12476.395even12
          1156.1.f.b.327.12119.101odd6
          1156.1.f.b.327.12476.339even6
          1156.1.g.b.155.18119.3even48
          1156.1.g.b.155.18476.3odd48
          1156.1.g.b.155.28119.31even48
          1156.1.g.b.155.28476.31odd48
          1156.1.g.b.179.18119.10even48
          1156.1.g.b.179.18476.367odd48
          1156.1.g.b.179.28119.24even48
          1156.1.g.b.179.28476.143odd48
          1156.1.g.b.399.18119.45even48
          1156.1.g.b.399.18476.283odd48
          1156.1.g.b.399.28119.108even48
          1156.1.g.b.399.28476.227odd48
          1156.1.g.b.423.18119.80even48
          1156.1.g.b.423.18476.199odd48
          1156.1.g.b.423.28119.73even48
          1156.1.g.b.423.28476.311odd48
          1700.1.n.a.599.1235.3even12
          1700.1.n.a.599.12140.3odd12
          1700.1.n.a.999.12595.472even12
          1700.1.n.a.999.122380.1067odd12
          1700.1.n.b.599.1235.17even12
          1700.1.n.b.599.12140.87odd12
          1700.1.n.b.999.12595.353even12
          1700.1.n.b.999.122380.1543odd12
          1700.1.p.a.251.12595.234odd12
          1700.1.p.a.251.122380.2019even12
          1700.1.p.a.1551.1235.24odd6
          1700.1.p.a.1551.12140.59even6
          3332.1.m.b.2843.127.4even3
          3332.1.m.b.2843.1228.11odd6
          3332.1.m.b.3039.12119.81even12
          3332.1.m.b.3039.12476.319odd12
          3332.1.bc.b.667.147.2even3inner
          3332.1.bc.b.667.1428.23odd6inner
          3332.1.bc.b.863.14119.30even12inner
          3332.1.bc.b.863.14476.387odd12inner
          3332.1.bc.b.2027.141.1even1trivial
          3332.1.bc.b.2027.144.3odd2CM
          3332.1.bc.b.2223.1417.13even4inner
          3332.1.bc.b.2223.1468.47odd4inner
          3332.1.bc.c.667.147.5odd6
          3332.1.bc.c.667.1428.19even6
          3332.1.bc.c.863.14119.47odd12
          3332.1.bc.c.863.14476.47even12
          3332.1.bc.c.2027.147.6odd2
          3332.1.bc.c.2027.1428.27even2
          3332.1.bc.c.2223.14119.13odd4
          3332.1.bc.c.2223.14476.251even4
          ", "is_complex": false}}]], "main_html": "
          \n Show commands:\n Magma\n / PariGP\n / SageMath
          [N,k,chi] = [3332,1,Mod(667,3332)]
          mf = mfinit([N,k,chi],0)
          lf = mfeigenbasis(mf)
          from sage.modular.dirichlet import DirichletCharacter
          H = DirichletGroup(3332, base_ring=CyclotomicField(12))
          chi = DirichletCharacter(H, H._module([6, 4, 9]))
          N = Newforms(chi, 1, names=\"a\")
          //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
          chi := DirichletCharacter(\"3332.667\");
          S:= CuspForms(chi, 1);
          N := Newforms(S);
          Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
          Weight: \\( k \\) \\(=\\)\\( 1 \\)
          Character orbit: \\([\\chi]\\) \\(=\\)3332.bc (of order \\(12\\), degree \\(4\\), not minimal)

          Newform invariants

          sage:\u00a0f = N[0] # Warning: the index may be different
          gp:\u00a0f = lf[1] \\\\ Warning: the index may be different
          Self dual: no
          Analytic conductor: \\(1.66288462209\\)
          Analytic rank: \\(0\\)
          Dimension: \\(4\\)
          Coefficient field: \\(\\Q(\\zeta_{12})\\)
          gp:\u00a0f.mod \\\\ as an extension of the character field
          Defining polynomial: \\( x^{4} - x^{2} + 1 \\)\"Copy\"Toggle
          Coefficient ring: \\(\\Z[a_1, a_2]\\)
          Coefficient ring index: \\( 1 \\)
          Twist minimal: no (minimal twist has level 68)
          Projective image:\\(D_{4}\\)
          Projective field:Galois closure of 4.2.19652.1
          Artin image:$C_4\\wr C_2\\times C_6$
          Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)

          Embedding invariants

          Embedding label 2027.1
          Root\\(0.866025 - 0.500000i\\) of defining polynomial
          Character\\(\\chi\\)\\(=\\)3332.2027
          Dual form3332.1.bc.b.863.1
          sage:\u00a0f.q_expansion() # note that sage often uses an isomorphic number field
          gp:\u00a0mfcoefs(f, 20)
          \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
          \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\"Copy\"Toggle

          Character values

          We give the values of \\(\\chi\\) on generators for \\(\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times\\).

          \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
          \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)

          Coefficient data

          For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the\nSatake parameters \\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).

          \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
          \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
          \n \\(2\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(3\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(4\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(5\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(6\\)\n \n 0\n \n 0\n
          \n \\(7\\)\n \n 0\n \n 0\n
          \n \\(8\\)\n \n 1.00000i\n 1.00000i
          \n \\(9\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(10\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(11\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(12\\)\n \n 0\n \n 0\n
          \n \\(13\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(14\\)\n \n 0\n \n 0\n
          \n \\(15\\)\n \n 0\n \n 0\n
          \n \\(16\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(17\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(18\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(19\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(20\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(21\\)\n \n 0\n \n 0\n
          \n \\(22\\)\n \n 0\n \n 0\n
          \n \\(23\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(24\\)\n \n 0\n \n 0\n
          \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(26\\)\n \n 0\n \n 0\n
          \n \\(27\\)\n \n 0\n \n 0\n
          \n \\(28\\)\n \n 0\n \n 0\n
          \n \\(29\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(30\\)\n \n 0\n \n 0\n
          \n \\(31\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(33\\)\n \n 0\n \n 0\n
          \n \\(34\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(35\\)\n \n 0\n \n 0\n
          \n \\(36\\)\n \n 1.00000i\n 1.00000i
          \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(38\\)\n \n 0\n \n 0\n
          \n \\(39\\)\n \n 0\n \n 0\n
          \n \\(40\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(41\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(42\\)\n \n 0\n \n 0\n
          \n \\(43\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(44\\)\n \n 0\n \n 0\n
          \n \\(45\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(46\\)\n \n 0\n \n 0\n
          \n \\(47\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(48\\)\n \n 0\n \n 0\n
          \n \\(49\\)\n \n 0\n \n 0\n
          \n \\(50\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(51\\)\n \n 0\n \n 0\n
          \n \\(52\\)\n \n 0\n \n 0\n
          \n \\(53\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(54\\)\n \n 0\n \n 0\n
          \n \\(55\\)\n \n 0\n \n 0\n
          \n \\(56\\)\n \n 0\n \n 0\n
          \n \\(57\\)\n \n 0\n \n 0\n
          \n \\(58\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(59\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(60\\)\n \n 0\n \n 0\n
          \n \\(61\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(62\\)\n \n 0\n \n 0\n
          \n \\(63\\)\n \n 0\n \n 0\n
          \n \\(64\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(65\\)\n \n 0\n \n 0\n
          \n \\(66\\)\n \n 0\n \n 0\n
          \n \\(67\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(69\\)\n \n 0\n \n 0\n
          \n \\(70\\)\n \n 0\n \n 0\n
          \n \\(71\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(72\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(73\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(74\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(75\\)\n \n 0\n \n 0\n
          \n \\(76\\)\n \n 0\n \n 0\n
          \n \\(77\\)\n \n 0\n \n 0\n
          \n \\(78\\)\n \n 0\n \n 0\n
          \n \\(79\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(81\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(83\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(84\\)\n \n 0\n \n 0\n
          \n \\(85\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(86\\)\n \n 0\n \n 0\n
          \n \\(87\\)\n \n 0\n \n 0\n
          \n \\(88\\)\n \n 0\n \n 0\n
          \n \\(89\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(90\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(91\\)\n \n 0\n \n 0\n
          \n \\(92\\)\n \n 0\n \n 0\n
          \n \\(93\\)\n \n 0\n \n 0\n
          \n \\(94\\)\n \n 0\n \n 0\n
          \n \\(95\\)\n \n 0\n \n 0\n
          \n \\(96\\)\n \n 0\n \n 0\n
          \n \\(97\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(98\\)\n \n 0\n \n 0\n
          \n \\(99\\)\n \n 0\n \n 0\n
          \n \\(100\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(101\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(102\\)\n \n 0\n \n 0\n
          \n \\(103\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(104\\)\n \n 0\n \n 0\n
          \n \\(105\\)\n \n 0\n \n 0\n
          \n \\(106\\)\n \n 0\n \n 0\n
          \n \\(107\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(108\\)\n \n 0\n \n 0\n
          \n \\(109\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(110\\)\n \n 0\n \n 0\n
          \n \\(111\\)\n \n 0\n \n 0\n
          \n \\(112\\)\n \n 0\n \n 0\n
          \n \\(113\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(114\\)\n \n 0\n \n 0\n
          \n \\(115\\)\n \n 0\n \n 0\n
          \n \\(116\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(117\\)\n \n 0\n \n 0\n
          \n \\(118\\)\n \n 0\n \n 0\n
          \n \\(119\\)\n \n 0\n \n 0\n
          \n \\(120\\)\n \n 0\n \n 0\n
          \n \\(121\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(123\\)\n \n 0\n \n 0\n
          \n \\(124\\)\n \n 0\n \n 0\n
          \n \\(125\\)\n \n 0\n \n 0\n
          \n \\(126\\)\n \n 0\n \n 0\n
          \n \\(127\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(128\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(129\\)\n \n 0\n \n 0\n
          \n \\(130\\)\n \n 0\n \n 0\n
          \n \\(131\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(132\\)\n \n 0\n \n 0\n
          \n \\(133\\)\n \n 0\n \n 0\n
          \n \\(134\\)\n \n 0\n \n 0\n
          \n \\(135\\)\n \n 0\n \n 0\n
          \n \\(136\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(137\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(138\\)\n \n 0\n \n 0\n
          \n \\(139\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(140\\)\n \n 0\n \n 0\n
          \n \\(141\\)\n \n 0\n \n 0\n
          \n \\(142\\)\n \n 0\n \n 0\n
          \n \\(143\\)\n \n 0\n \n 0\n
          \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(145\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(147\\)\n \n 0\n \n 0\n
          \n \\(148\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(149\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \\(150\\)\n \n 0\n \n 0\n
          \n \\(151\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(152\\)\n \n 0\n \n 0\n
          \n \\(153\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(154\\)\n \n 0\n \n 0\n
          \n \\(155\\)\n \n 0\n \n 0\n
          \n \\(156\\)\n \n 0\n \n 0\n
          \n \\(157\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(158\\)\n \n 0\n \n 0\n
          \n \\(159\\)\n \n 0\n \n 0\n
          \n \\(160\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(161\\)\n \n 0\n \n 0\n
          \n \\(162\\)\n \n 1.00000i\n 1.00000i
          \n \\(163\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(164\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(165\\)\n \n 0\n \n 0\n
          \n \\(166\\)\n \n 0\n \n 0\n
          \n \\(167\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(168\\)\n \n 0\n \n 0\n
          \n \\(169\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(170\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(171\\)\n \n 0\n \n 0\n
          \n \\(172\\)\n \n 0\n \n 0\n
          \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(174\\)\n \n 0\n \n 0\n
          \n \\(175\\)\n \n 0\n \n 0\n
          \n \\(176\\)\n \n 0\n \n 0\n
          \n \\(177\\)\n \n 0\n \n 0\n
          \n \\(178\\)\n \n 0\n \n 0\n
          \n \\(179\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(180\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(182\\)\n \n 0\n \n 0\n
          \n \\(183\\)\n \n 0\n \n 0\n
          \n \\(184\\)\n \n 0\n \n 0\n
          \n \\(185\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(186\\)\n \n 0\n \n 0\n
          \n \\(187\\)\n \n 0\n \n 0\n
          \n \\(188\\)\n \n 0\n \n 0\n
          \n \\(189\\)\n \n 0\n \n 0\n
          \n \\(190\\)\n \n 0\n \n 0\n
          \n \\(191\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(192\\)\n \n 0\n \n 0\n
          \n \\(193\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(194\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(195\\)\n \n 0\n \n 0\n
          \n \\(196\\)\n \n 0\n \n 0\n
          \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(198\\)\n \n 0\n \n 0\n
          \n \\(199\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(200\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(201\\)\n \n 0\n \n 0\n
          \n \\(202\\)\n \n 0\n \n 0\n
          \n \\(203\\)\n \n 0\n \n 0\n
          \n \\(204\\)\n \n 0\n \n 0\n
          \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
          \n \\(206\\)\n \n 0\n \n 0\n
          \n \\(207\\)\n \n 0\n \n 0\n
          \n \\(208\\)\n \n 0\n \n 0\n
          \n \\(209\\)\n \n 0\n \n 0\n
          \n \\(210\\)\n \n 0\n \n 0\n
          \n \\(211\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(212\\)\n \n 0\n \n 0\n
          \n \\(213\\)\n \n 0\n \n 0\n
          \n \\(214\\)\n \n 0\n \n 0\n
          \n \\(215\\)\n \n 0\n \n 0\n
          \n \\(216\\)\n \n 0\n \n 0\n
          \n \\(217\\)\n \n 0\n \n 0\n
          \n \\(218\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(219\\)\n \n 0\n \n 0\n
          \n \\(220\\)\n \n 0\n \n 0\n
          \n \\(221\\)\n \n 0\n \n 0\n
          \n \\(222\\)\n \n 0\n \n 0\n
          \n \\(223\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(224\\)\n \n 0\n \n 0\n
          \n \\(225\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(227\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(228\\)\n \n 0\n \n 0\n
          \n \\(229\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(230\\)\n \n 0\n \n 0\n
          \n \\(231\\)\n \n 0\n \n 0\n
          \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(234\\)\n \n 0\n \n 0\n
          \n \\(235\\)\n \n 0\n \n 0\n
          \n \\(236\\)\n \n 0\n \n 0\n
          \n \\(237\\)\n \n 0\n \n 0\n
          \n \\(238\\)\n \n 0\n \n 0\n
          \n \\(239\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(240\\)\n \n 0\n \n 0\n
          \n \\(241\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(242\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(243\\)\n \n 0\n \n 0\n
          \n \\(244\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(245\\)\n \n 0\n \n 0\n
          \n \\(246\\)\n \n 0\n \n 0\n
          \n \\(247\\)\n \n 0\n \n 0\n
          \n \\(248\\)\n \n 0\n \n 0\n
          \n \\(249\\)\n \n 0\n \n 0\n
          \n \\(250\\)\n \n 0\n \n 0\n
          \n \\(251\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(252\\)\n \n 0\n \n 0\n
          \n \\(253\\)\n \n 0\n \n 0\n
          \n \\(254\\)\n \n 0\n \n 0\n
          \n \\(255\\)\n \n 0\n \n 0\n
          \n \\(256\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(257\\)\n \n 1.73205\n \n \u2212\n \n 1.00000i\n 1.73205\n \n \u2212\n \n 1.00000i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(258\\)\n \n 0\n \n 0\n
          \n \\(259\\)\n \n 0\n \n 0\n
          \n \\(260\\)\n \n 0\n \n 0\n
          \n \\(261\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(262\\)\n \n 0\n \n 0\n
          \n \\(263\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(264\\)\n \n 0\n \n 0\n
          \n \\(265\\)\n \n 0\n \n 0\n
          \n \\(266\\)\n \n 0\n \n 0\n
          \n \\(267\\)\n \n 0\n \n 0\n
          \n \\(268\\)\n \n 0\n \n 0\n
          \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(270\\)\n \n 0\n \n 0\n
          \n \\(271\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(272\\)\n \n 1.00000\n \n 1.00000\n
          \n \\(273\\)\n \n 0\n \n 0\n
          \n \\(274\\)\n \n 0\n \n 0\n
          \n \\(275\\)\n \n 0\n \n 0\n
          \n \\(276\\)\n \n 0\n \n 0\n
          \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(278\\)\n \n 0\n \n 0\n
          \n \\(279\\)\n \n 0\n \n 0\n
          \n \\(280\\)\n \n 0\n \n 0\n
          \n \\(281\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(282\\)\n \n 0\n \n 0\n
          \n \\(283\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(284\\)\n \n 0\n \n 0\n
          \n \\(285\\)\n \n 0\n \n 0\n
          \n \\(286\\)\n \n 0\n \n 0\n
          \n \\(287\\)\n \n 0\n \n 0\n
          \n \\(288\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(289\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(290\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(291\\)\n \n 0\n \n 0\n
          \n \\(292\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(293\\)\n \n 2.00000\n \n 2.00000\n \n 1.00000\n \n \\(0\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(294\\)\n \n 0\n \n 0\n
          \n \\(295\\)\n \n 0\n \n 0\n
          \n \\(296\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(297\\)\n \n 0\n \n 0\n
          \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n 1.73205\n \n +\n \n 1.00000i
          \n \\(299\\)\n \n 0\n \n 0\n
          \n \\(300\\)\n \n 0\n \n 0\n
          \n \\(301\\)\n \n 0\n \n 0\n
          \n \\(302\\)\n \n 0\n \n 0\n
          \n \\(303\\)\n \n 0\n \n 0\n
          \n \\(304\\)\n \n 0\n \n 0\n
          \n \\(305\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
          \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(307\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(308\\)\n \n 0\n \n 0\n
          \n \\(309\\)\n \n 0\n \n 0\n
          \n \\(310\\)\n \n 0\n \n 0\n
          \n \\(311\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(312\\)\n \n 0\n \n 0\n
          \n \\(313\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(314\\)\n \n 2.00000i\n 2.00000i
          \n \\(315\\)\n \n 0\n \n 0\n
          \n \\(316\\)\n \n 0\n \n 0\n
          \n \\(317\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(318\\)\n \n 0\n \n 0\n
          \n \\(319\\)\n \n 0\n \n 0\n
          \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(321\\)\n \n 0\n \n 0\n
          \n \\(322\\)\n \n 0\n \n 0\n
          \n \\(323\\)\n \n 0\n \n 0\n
          \n \\(324\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(325\\)\n \n 0\n \n 0\n
          \n \\(326\\)\n \n 0\n \n 0\n
          \n \\(327\\)\n \n 0\n \n 0\n
          \n \\(328\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(329\\)\n \n 0\n \n 0\n
          \n \\(330\\)\n \n 0\n \n 0\n
          \n \\(331\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(332\\)\n \n 0\n \n 0\n
          \n \\(333\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(334\\)\n \n 0\n \n 0\n
          \n \\(335\\)\n \n 0\n \n 0\n
          \n \\(336\\)\n \n 0\n \n 0\n
          \n \\(337\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(338\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(339\\)\n \n 0\n \n 0\n
          \n \\(340\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(341\\)\n \n 0\n \n 0\n
          \n \\(342\\)\n \n 0\n \n 0\n
          \n \\(343\\)\n \n 0\n \n 0\n
          \n \\(344\\)\n \n 0\n \n 0\n
          \n \\(345\\)\n \n 0\n \n 0\n
          \n \\(346\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(347\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(348\\)\n \n 0\n \n 0\n
          \n \\(349\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(350\\)\n \n 0\n \n 0\n
          \n \\(351\\)\n \n 0\n \n 0\n
          \n \\(352\\)\n \n 0\n \n 0\n
          \n \\(353\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(354\\)\n \n 0\n \n 0\n
          \n \\(355\\)\n \n 0\n \n 0\n
          \n \\(356\\)\n \n 0\n \n 0\n
          \n \\(357\\)\n \n 0\n \n 0\n
          \n \\(358\\)\n \n 0\n \n 0\n
          \n \\(359\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(361\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(362\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(363\\)\n \n 0\n \n 0\n
          \n \\(364\\)\n \n 0\n \n 0\n
          \n \\(365\\)\n \n 2.00000i\n 2.00000i
          \n \\(366\\)\n \n 0\n \n 0\n
          \n \\(367\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(368\\)\n \n 0\n \n 0\n
          \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(370\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(371\\)\n \n 0\n \n 0\n
          \n \\(372\\)\n \n 0\n \n 0\n
          \n \\(373\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(374\\)\n \n 0\n \n 0\n
          \n \\(375\\)\n \n 0\n \n 0\n
          \n \\(376\\)\n \n 0\n \n 0\n
          \n \\(377\\)\n \n 0\n \n 0\n
          \n \\(378\\)\n \n 0\n \n 0\n
          \n \\(379\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(380\\)\n \n 0\n \n 0\n
          \n \\(381\\)\n \n 0\n \n 0\n
          \n \\(382\\)\n \n 0\n \n 0\n
          \n \\(383\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(384\\)\n \n 0\n \n 0\n
          \n \\(385\\)\n \n 0\n \n 0\n
          \n \\(386\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(387\\)\n \n 0\n \n 0\n
          \n \\(388\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(389\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \\(390\\)\n \n 0\n \n 0\n
          \n \\(391\\)\n \n 0\n \n 0\n
          \n \\(392\\)\n \n 0\n \n 0\n
          \n \\(393\\)\n \n 0\n \n 0\n
          \n \\(394\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(395\\)\n \n 0\n \n 0\n
          \n \\(396\\)\n \n 0\n \n 0\n
          \n \\(397\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(398\\)\n \n 0\n \n 0\n
          \n \\(399\\)\n \n 0\n \n 0\n
          \n \\(400\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(401\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(402\\)\n \n 0\n \n 0\n
          \n \\(403\\)\n \n 0\n \n 0\n
          \n \\(404\\)\n \n 0\n \n 0\n
          \n \\(405\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(406\\)\n \n 0\n \n 0\n
          \n \\(407\\)\n \n 0\n \n 0\n
          \n \\(408\\)\n \n 0\n \n 0\n
          \n \\(409\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(410\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(411\\)\n \n 0\n \n 0\n
          \n \\(412\\)\n \n 0\n \n 0\n
          \n \\(413\\)\n \n 0\n \n 0\n
          \n \\(414\\)\n \n 0\n \n 0\n
          \n \\(415\\)\n \n 0\n \n 0\n
          \n \\(416\\)\n \n 0\n \n 0\n
          \n \\(417\\)\n \n 0\n \n 0\n
          \n \\(418\\)\n \n 0\n \n 0\n
          \n \\(419\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(420\\)\n \n 0\n \n 0\n
          \n \\(421\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(422\\)\n \n 0\n \n 0\n
          \n \\(423\\)\n \n 0\n \n 0\n
          \n \\(424\\)\n \n 0\n \n 0\n
          \n \\(425\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(426\\)\n \n 0\n \n 0\n
          \n \\(427\\)\n \n 0\n \n 0\n
          \n \\(428\\)\n \n 0\n \n 0\n
          \n \\(429\\)\n \n 0\n \n 0\n
          \n \\(430\\)\n \n 0\n \n 0\n
          \n \\(431\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(432\\)\n \n 0\n \n 0\n
          \n \\(433\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(434\\)\n \n 0\n \n 0\n
          \n \\(435\\)\n \n 0\n \n 0\n
          \n \\(436\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(437\\)\n \n 0\n \n 0\n
          \n \\(438\\)\n \n 0\n \n 0\n
          \n \\(439\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(440\\)\n \n 0\n \n 0\n
          \n \\(441\\)\n \n 0\n \n 0\n
          \n \\(442\\)\n \n 0\n \n 0\n
          \n \\(443\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(444\\)\n \n 0\n \n 0\n
          \n \\(445\\)\n \n 0\n \n 0\n
          \n \\(446\\)\n \n 0\n \n 0\n
          \n \\(447\\)\n \n 0\n \n 0\n
          \n \\(448\\)\n \n 0\n \n 0\n
          \n \\(449\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(450\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(451\\)\n \n 0\n \n 0\n
          \n \\(452\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(453\\)\n \n 0\n \n 0\n
          \n \\(454\\)\n \n 0\n \n 0\n
          \n \\(455\\)\n \n 0\n \n 0\n
          \n \\(456\\)\n \n 0\n \n 0\n
          \n \\(457\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(458\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(459\\)\n \n 0\n \n 0\n
          \n \\(460\\)\n \n 0\n \n 0\n
          \n \\(461\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(462\\)\n \n 0\n \n 0\n
          \n \\(463\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(464\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(465\\)\n \n 0\n \n 0\n
          \n \\(466\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(467\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(468\\)\n \n 0\n \n 0\n
          \n \\(469\\)\n \n 0\n \n 0\n
          \n \\(470\\)\n \n 0\n \n 0\n
          \n \\(471\\)\n \n 0\n \n 0\n
          \n \\(472\\)\n \n 0\n \n 0\n
          \n \\(473\\)\n \n 0\n \n 0\n
          \n \\(474\\)\n \n 0\n \n 0\n
          \n \\(475\\)\n \n 0\n \n 0\n
          \n \\(476\\)\n \n 0\n \n 0\n
          \n \\(477\\)\n \n 0\n \n 0\n
          \n \\(478\\)\n \n 0\n \n 0\n
          \n \\(479\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(480\\)\n \n 0\n \n 0\n
          \n \\(481\\)\n \n 0\n \n 0\n
          \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(483\\)\n \n 0\n \n 0\n
          \n \\(484\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(485\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(486\\)\n \n 0\n \n 0\n
          \n \\(487\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(489\\)\n \n 0\n \n 0\n
          \n \\(490\\)\n \n 0\n \n 0\n
          \n \\(491\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(492\\)\n \n 0\n \n 0\n
          \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(494\\)\n \n 0\n \n 0\n
          \n \\(495\\)\n \n 0\n \n 0\n
          \n \\(496\\)\n \n 0\n \n 0\n
          \n \\(497\\)\n \n 0\n \n 0\n
          \n \\(498\\)\n \n 0\n \n 0\n
          \n \\(499\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(500\\)\n \n 0\n \n 0\n
          \n \\(501\\)\n \n 0\n \n 0\n
          \n \\(502\\)\n \n 0\n \n 0\n
          \n \\(503\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(504\\)\n \n 0\n \n 0\n
          \n \\(505\\)\n \n 0\n \n 0\n
          \n \\(506\\)\n \n 0\n \n 0\n
          \n \\(507\\)\n \n 0\n \n 0\n
          \n \\(508\\)\n \n 0\n \n 0\n
          \n \\(509\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \\(510\\)\n \n 0\n \n 0\n
          \n \\(511\\)\n \n 0\n \n 0\n
          \n \\(512\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(513\\)\n \n 0\n \n 0\n
          \n \\(514\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
          \n \\(515\\)\n \n 0\n \n 0\n
          \n \\(516\\)\n \n 0\n \n 0\n
          \n \\(517\\)\n \n 0\n \n 0\n
          \n \\(518\\)\n \n 0\n \n 0\n
          \n \\(519\\)\n \n 0\n \n 0\n
          \n \\(520\\)\n \n 0\n \n 0\n
          \n \\(521\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(522\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(523\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(524\\)\n \n 0\n \n 0\n
          \n \\(525\\)\n \n 0\n \n 0\n
          \n \\(526\\)\n \n 0\n \n 0\n
          \n \\(527\\)\n \n 0\n \n 0\n
          \n \\(528\\)\n \n 0\n \n 0\n
          \n \\(529\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(530\\)\n \n 0\n \n 0\n
          \n \\(531\\)\n \n 0\n \n 0\n
          \n \\(532\\)\n \n 0\n \n 0\n
          \n \\(533\\)\n \n 0\n \n 0\n
          \n \\(534\\)\n \n 0\n \n 0\n
          \n \\(535\\)\n \n 0\n \n 0\n
          \n \\(536\\)\n \n 0\n \n 0\n
          \n \\(537\\)\n \n 0\n \n 0\n
          \n \\(538\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(539\\)\n \n 0\n \n 0\n
          \n \\(540\\)\n \n 0\n \n 0\n
          \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(542\\)\n \n 0\n \n 0\n
          \n \\(543\\)\n \n 0\n \n 0\n
          \n \\(544\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(545\\)\n \n \u22122.00000\n \n \u22122.00000\n
          \n \\(546\\)\n \n 0\n \n 0\n
          \n \\(547\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(548\\)\n \n 0\n \n 0\n
          \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(550\\)\n \n 0\n \n 0\n
          \n \\(551\\)\n \n 0\n \n 0\n
          \n \\(552\\)\n \n 0\n \n 0\n
          \n \\(553\\)\n \n 0\n \n 0\n
          \n \\(554\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(555\\)\n \n 0\n \n 0\n
          \n \\(556\\)\n \n 0\n \n 0\n
          \n \\(557\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(558\\)\n \n 0\n \n 0\n
          \n \\(559\\)\n \n 0\n \n 0\n
          \n \\(560\\)\n \n 0\n \n 0\n
          \n \\(561\\)\n \n 0\n \n 0\n
          \n \\(562\\)\n \n 0\n \n 0\n
          \n \\(563\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(564\\)\n \n 0\n \n 0\n
          \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
          \n \\(566\\)\n \n 0\n \n 0\n
          \n \\(567\\)\n \n 0\n \n 0\n
          \n \\(568\\)\n \n 0\n \n 0\n
          \n \\(569\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(570\\)\n \n 0\n \n 0\n
          \n \\(571\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(572\\)\n \n 0\n \n 0\n
          \n \\(573\\)\n \n 0\n \n 0\n
          \n \\(574\\)\n \n 0\n \n 0\n
          \n \\(575\\)\n \n 0\n \n 0\n
          \n \\(576\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(577\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(579\\)\n \n 0\n \n 0\n
          \n \\(580\\)\n \n 2.00000i\n 2.00000i
          \n \\(581\\)\n \n 0\n \n 0\n
          \n \\(582\\)\n \n 0\n \n 0\n
          \n \\(583\\)\n \n 0\n \n 0\n
          \n \\(584\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(585\\)\n \n 0\n \n 0\n
          \n \\(586\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(587\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(588\\)\n \n 0\n \n 0\n
          \n \\(589\\)\n \n 0\n \n 0\n
          \n \\(590\\)\n \n 0\n \n 0\n
          \n \\(591\\)\n \n 0\n \n 0\n
          \n \\(592\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(593\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(594\\)\n \n 0\n \n 0\n
          \n \\(595\\)\n \n 0\n \n 0\n
          \n \\(596\\)\n \n \u22122.00000\n \n \u22122.00000\n
          \n \\(597\\)\n \n 0\n \n 0\n
          \n \\(598\\)\n \n 0\n \n 0\n
          \n \\(599\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(600\\)\n \n 0\n \n 0\n
          \n \\(601\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(602\\)\n \n 0\n \n 0\n
          \n \\(603\\)\n \n 0\n \n 0\n
          \n \\(604\\)\n \n 0\n \n 0\n
          \n \\(605\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(606\\)\n \n 0\n \n 0\n
          \n \\(607\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(608\\)\n \n 0\n \n 0\n
          \n \\(609\\)\n \n 0\n \n 0\n
          \n \\(610\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(611\\)\n \n 0\n \n 0\n
          \n \\(612\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(613\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(614\\)\n \n 0\n \n 0\n
          \n \\(615\\)\n \n 0\n \n 0\n
          \n \\(616\\)\n \n 0\n \n 0\n
          \n \\(617\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(618\\)\n \n 0\n \n 0\n
          \n \\(619\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(620\\)\n \n 0\n \n 0\n
          \n \\(621\\)\n \n 0\n \n 0\n
          \n \\(622\\)\n \n 0\n \n 0\n
          \n \\(623\\)\n \n 0\n \n 0\n
          \n \\(624\\)\n \n 0\n \n 0\n
          \n \\(625\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(626\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(627\\)\n \n 0\n \n 0\n
          \n \\(628\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(629\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(630\\)\n \n 0\n \n 0\n
          \n \\(631\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(632\\)\n \n 0\n \n 0\n
          \n \\(633\\)\n \n 0\n \n 0\n
          \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(635\\)\n \n 0\n \n 0\n
          \n \\(636\\)\n \n 0\n \n 0\n
          \n \\(637\\)\n \n 0\n \n 0\n
          \n \\(638\\)\n \n 0\n \n 0\n
          \n \\(639\\)\n \n 0\n \n 0\n
          \n \\(640\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(641\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(642\\)\n \n 0\n \n 0\n
          \n \\(643\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(644\\)\n \n 0\n \n 0\n
          \n \\(645\\)\n \n 0\n \n 0\n
          \n \\(646\\)\n \n 0\n \n 0\n
          \n \\(647\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(649\\)\n \n 0\n \n 0\n
          \n \\(650\\)\n \n 0\n \n 0\n
          \n \\(651\\)\n \n 0\n \n 0\n
          \n \\(652\\)\n \n 0\n \n 0\n
          \n \\(653\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(654\\)\n \n 0\n \n 0\n
          \n \\(655\\)\n \n 0\n \n 0\n
          \n \\(656\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(658\\)\n \n 0\n \n 0\n
          \n \\(659\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(660\\)\n \n 0\n \n 0\n
          \n \\(661\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(662\\)\n \n 0\n \n 0\n
          \n \\(663\\)\n \n 0\n \n 0\n
          \n \\(664\\)\n \n 0\n \n 0\n
          \n \\(665\\)\n \n 0\n \n 0\n
          \n \\(666\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(667\\)\n \n 0\n \n 0\n
          \n \\(668\\)\n \n 0\n \n 0\n
          \n \\(669\\)\n \n 0\n \n 0\n
          \n \\(670\\)\n \n 0\n \n 0\n
          \n \\(671\\)\n \n 0\n \n 0\n
          \n \\(672\\)\n \n 0\n \n 0\n
          \n \\(673\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(674\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(675\\)\n \n 0\n \n 0\n
          \n \\(676\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(678\\)\n \n 0\n \n 0\n
          \n \\(679\\)\n \n 0\n \n 0\n
          \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(681\\)\n \n 0\n \n 0\n
          \n \\(682\\)\n \n 0\n \n 0\n
          \n \\(683\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(684\\)\n \n 0\n \n 0\n
          \n \\(685\\)\n \n 0\n \n 0\n
          \n \\(686\\)\n \n 0\n \n 0\n
          \n \\(687\\)\n \n 0\n \n 0\n
          \n \\(688\\)\n \n 0\n \n 0\n
          \n \\(689\\)\n \n 0\n \n 0\n
          \n \\(690\\)\n \n 0\n \n 0\n
          \n \\(691\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(692\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(693\\)\n \n 0\n \n 0\n
          \n \\(694\\)\n \n 0\n \n 0\n
          \n \\(695\\)\n \n 0\n \n 0\n
          \n \\(696\\)\n \n 0\n \n 0\n
          \n \\(697\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(698\\)\n \n 0\n \n 0\n
          \n \\(699\\)\n \n 0\n \n 0\n
          \n \\(700\\)\n \n 0\n \n 0\n
          \n \\(701\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(702\\)\n \n 0\n \n 0\n
          \n \\(703\\)\n \n 0\n \n 0\n
          \n \\(704\\)\n \n 0\n \n 0\n
          \n \\(705\\)\n \n 0\n \n 0\n
          \n \\(706\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(707\\)\n \n 0\n \n 0\n
          \n \\(708\\)\n \n 0\n \n 0\n
          \n \\(709\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(710\\)\n \n 0\n \n 0\n
          \n \\(711\\)\n \n 0\n \n 0\n
          \n \\(712\\)\n \n 0\n \n 0\n
          \n \\(713\\)\n \n 0\n \n 0\n
          \n \\(714\\)\n \n 0\n \n 0\n
          \n \\(715\\)\n \n 0\n \n 0\n
          \n \\(716\\)\n \n 0\n \n 0\n
          \n \\(717\\)\n \n 0\n \n 0\n
          \n \\(718\\)\n \n 0\n \n 0\n
          \n \\(719\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(720\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(721\\)\n \n 0\n \n 0\n
          \n \\(722\\)\n \n 1.00000i\n 1.00000i
          \n \\(723\\)\n \n 0\n \n 0\n
          \n \\(724\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(725\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(726\\)\n \n 0\n \n 0\n
          \n \\(727\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(728\\)\n \n 0\n \n 0\n
          \n \\(729\\)\n \n 1.00000i\n 1.00000i
          \n \\(730\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(731\\)\n \n 0\n \n 0\n
          \n \\(732\\)\n \n 0\n \n 0\n
          \n \\(733\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(734\\)\n \n 0\n \n 0\n
          \n \\(735\\)\n \n 0\n \n 0\n
          \n \\(736\\)\n \n 0\n \n 0\n
          \n \\(737\\)\n \n 0\n \n 0\n
          \n \\(738\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(739\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(740\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(741\\)\n \n 0\n \n 0\n
          \n \\(742\\)\n \n 0\n \n 0\n
          \n \\(743\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(744\\)\n \n 0\n \n 0\n
          \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n 0.732051\n \n +\n \n 2.73205i
          \n \\(746\\)\n \n 0\n \n 0\n
          \n \\(747\\)\n \n 0\n \n 0\n
          \n \\(748\\)\n \n 0\n \n 0\n
          \n \\(749\\)\n \n 0\n \n 0\n
          \n \\(750\\)\n \n 0\n \n 0\n
          \n \\(751\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(752\\)\n \n 0\n \n 0\n
          \n \\(753\\)\n \n 0\n \n 0\n
          \n \\(754\\)\n \n 0\n \n 0\n
          \n \\(755\\)\n \n 0\n \n 0\n
          \n \\(756\\)\n \n 0\n \n 0\n
          \n \\(757\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(758\\)\n \n 0\n \n 0\n
          \n \\(759\\)\n \n 0\n \n 0\n
          \n \\(760\\)\n \n 0\n \n 0\n
          \n \\(761\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(762\\)\n \n 0\n \n 0\n
          \n \\(763\\)\n \n 0\n \n 0\n
          \n \\(764\\)\n \n 0\n \n 0\n
          \n \\(765\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(766\\)\n \n 0\n \n 0\n
          \n \\(767\\)\n \n 0\n \n 0\n
          \n \\(768\\)\n \n 0\n \n 0\n
          \n \\(769\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(770\\)\n \n 0\n \n 0\n
          \n \\(771\\)\n \n 0\n \n 0\n
          \n \\(772\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(773\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \\(774\\)\n \n 0\n \n 0\n
          \n \\(775\\)\n \n 0\n \n 0\n
          \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(777\\)\n \n 0\n \n 0\n
          \n \\(778\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(779\\)\n \n 0\n \n 0\n
          \n \\(780\\)\n \n 0\n \n 0\n
          \n \\(781\\)\n \n 0\n \n 0\n
          \n \\(782\\)\n \n 0\n \n 0\n
          \n \\(783\\)\n \n 0\n \n 0\n
          \n \\(784\\)\n \n 0\n \n 0\n
          \n \\(785\\)\n \n \u22122.00000\n \n +\n \n 2.00000i\n \u22122.00000\n \n +\n \n 2.00000i
          \n \\(786\\)\n \n 0\n \n 0\n
          \n \\(787\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(788\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(789\\)\n \n 0\n \n 0\n
          \n \\(790\\)\n \n 0\n \n 0\n
          \n \\(791\\)\n \n 0\n \n 0\n
          \n \\(792\\)\n \n 0\n \n 0\n
          \n \\(793\\)\n \n 0\n \n 0\n
          \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(795\\)\n \n 0\n \n 0\n
          \n \\(796\\)\n \n 0\n \n 0\n
          \n \\(797\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(798\\)\n \n 0\n \n 0\n
          \n \\(799\\)\n \n 0\n \n 0\n
          \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(801\\)\n \n 0\n \n 0\n
          \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(803\\)\n \n 0\n \n 0\n
          \n \\(804\\)\n \n 0\n \n 0\n
          \n \\(805\\)\n \n 0\n \n 0\n
          \n \\(806\\)\n \n 0\n \n 0\n
          \n \\(807\\)\n \n 0\n \n 0\n
          \n \\(808\\)\n \n 0\n \n 0\n
          \n \\(809\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(810\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(811\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(812\\)\n \n 0\n \n 0\n
          \n \\(813\\)\n \n 0\n \n 0\n
          \n \\(814\\)\n \n 0\n \n 0\n
          \n \\(815\\)\n \n 0\n \n 0\n
          \n \\(816\\)\n \n 0\n \n 0\n
          \n \\(817\\)\n \n 0\n \n 0\n
          \n \\(818\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(819\\)\n \n 0\n \n 0\n
          \n \\(820\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(821\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(822\\)\n \n 0\n \n 0\n
          \n \\(823\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(824\\)\n \n 0\n \n 0\n
          \n \\(825\\)\n \n 0\n \n 0\n
          \n \\(826\\)\n \n 0\n \n 0\n
          \n \\(827\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(828\\)\n \n 0\n \n 0\n
          \n \\(829\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(830\\)\n \n 0\n \n 0\n
          \n \\(831\\)\n \n 0\n \n 0\n
          \n \\(832\\)\n \n 0\n \n 0\n
          \n \\(833\\)\n \n 0\n \n 0\n
          \n \\(834\\)\n \n 0\n \n 0\n
          \n \\(835\\)\n \n 0\n \n 0\n
          \n \\(836\\)\n \n 0\n \n 0\n
          \n \\(837\\)\n \n 0\n \n 0\n
          \n \\(838\\)\n \n 0\n \n 0\n
          \n \\(839\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(840\\)\n \n 0\n \n 0\n
          \n \\(841\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(842\\)\n \n 0\n \n 0\n
          \n \\(843\\)\n \n 0\n \n 0\n
          \n \\(844\\)\n \n 0\n \n 0\n
          \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(846\\)\n \n 0\n \n 0\n
          \n \\(847\\)\n \n 0\n \n 0\n
          \n \\(848\\)\n \n 0\n \n 0\n
          \n \\(849\\)\n \n 0\n \n 0\n
          \n \\(850\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(851\\)\n \n 0\n \n 0\n
          \n \\(852\\)\n \n 0\n \n 0\n
          \n \\(853\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(854\\)\n \n 0\n \n 0\n
          \n \\(855\\)\n \n 0\n \n 0\n
          \n \\(856\\)\n \n 0\n \n 0\n
          \n \\(857\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(858\\)\n \n 0\n \n 0\n
          \n \\(859\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(860\\)\n \n 0\n \n 0\n
          \n \\(861\\)\n \n 0\n \n 0\n
          \n \\(862\\)\n \n 0\n \n 0\n
          \n \\(863\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(864\\)\n \n 0\n \n 0\n
          \n \\(865\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(866\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(867\\)\n \n 0\n \n 0\n
          \n \\(868\\)\n \n 0\n \n 0\n
          \n \\(869\\)\n \n 0\n \n 0\n
          \n \\(870\\)\n \n 0\n \n 0\n
          \n \\(871\\)\n \n 0\n \n 0\n
          \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(873\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(874\\)\n \n 0\n \n 0\n
          \n \\(875\\)\n \n 0\n \n 0\n
          \n \\(876\\)\n \n 0\n \n 0\n
          \n \\(877\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(878\\)\n \n 0\n \n 0\n
          \n \\(879\\)\n \n 0\n \n 0\n
          \n \\(880\\)\n \n 0\n \n 0\n
          \n \\(881\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(882\\)\n \n 0\n \n 0\n
          \n \\(883\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(884\\)\n \n 0\n \n 0\n
          \n \\(885\\)\n \n 0\n \n 0\n
          \n \\(886\\)\n \n 0\n \n 0\n
          \n \\(887\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(888\\)\n \n 0\n \n 0\n
          \n \\(889\\)\n \n 0\n \n 0\n
          \n \\(890\\)\n \n 0\n \n 0\n
          \n \\(891\\)\n \n 0\n \n 0\n
          \n \\(892\\)\n \n 0\n \n 0\n
          \n \\(893\\)\n \n 0\n \n 0\n
          \n \\(894\\)\n \n 0\n \n 0\n
          \n \\(895\\)\n \n 0\n \n 0\n
          \n \\(896\\)\n \n 0\n \n 0\n
          \n \\(897\\)\n \n 0\n \n 0\n
          \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(899\\)\n \n 0\n \n 0\n
          \n \\(900\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(901\\)\n \n 0\n \n 0\n
          \n \\(902\\)\n \n 0\n \n 0\n
          \n \\(903\\)\n \n 0\n \n 0\n
          \n \\(904\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(905\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(906\\)\n \n 0\n \n 0\n
          \n \\(907\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(908\\)\n \n 0\n \n 0\n
          \n \\(909\\)\n \n 0\n \n 0\n
          \n \\(910\\)\n \n 0\n \n 0\n
          \n \\(911\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(912\\)\n \n 0\n \n 0\n
          \n \\(913\\)\n \n 0\n \n 0\n
          \n \\(914\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(915\\)\n \n 0\n \n 0\n
          \n \\(916\\)\n \n 2.00000i\n 2.00000i
          \n \\(917\\)\n \n 0\n \n 0\n
          \n \\(918\\)\n \n 0\n \n 0\n
          \n \\(919\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(920\\)\n \n 0\n \n 0\n
          \n \\(921\\)\n \n 0\n \n 0\n
          \n \\(922\\)\n \n 0\n \n 0\n
          \n \\(923\\)\n \n 0\n \n 0\n
          \n \\(924\\)\n \n 0\n \n 0\n
          \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(926\\)\n \n 0\n \n 0\n
          \n \\(927\\)\n \n 0\n \n 0\n
          \n \\(928\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(929\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(930\\)\n \n 0\n \n 0\n
          \n \\(931\\)\n \n 0\n \n 0\n
          \n \\(932\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(933\\)\n \n 0\n \n 0\n
          \n \\(934\\)\n \n 0\n \n 0\n
          \n \\(935\\)\n \n 0\n \n 0\n
          \n \\(936\\)\n \n 0\n \n 0\n
          \n \\(937\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(938\\)\n \n 0\n \n 0\n
          \n \\(939\\)\n \n 0\n \n 0\n
          \n \\(940\\)\n \n 0\n \n 0\n
          \n \\(941\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(942\\)\n \n 0\n \n 0\n
          \n \\(943\\)\n \n 0\n \n 0\n
          \n \\(944\\)\n \n 0\n \n 0\n
          \n \\(945\\)\n \n 0\n \n 0\n
          \n \\(946\\)\n \n 0\n \n 0\n
          \n \\(947\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(948\\)\n \n 0\n \n 0\n
          \n \\(949\\)\n \n 0\n \n 0\n
          \n \\(950\\)\n \n 0\n \n 0\n
          \n \\(951\\)\n \n 0\n \n 0\n
          \n \\(952\\)\n \n 0\n \n 0\n
          \n \\(953\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(954\\)\n \n 0\n \n 0\n
          \n \\(955\\)\n \n 0\n \n 0\n
          \n \\(956\\)\n \n 0\n \n 0\n
          \n \\(957\\)\n \n 0\n \n 0\n
          \n \\(958\\)\n \n 0\n \n 0\n
          \n \\(959\\)\n \n 0\n \n 0\n
          \n \\(960\\)\n \n 0\n \n 0\n
          \n \\(961\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(962\\)\n \n 0\n \n 0\n
          \n \\(963\\)\n \n 0\n \n 0\n
          \n \\(964\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(965\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(966\\)\n \n 0\n \n 0\n
          \n \\(967\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(969\\)\n \n 0\n \n 0\n
          \n \\(970\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(971\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(972\\)\n \n 0\n \n 0\n
          \n \\(973\\)\n \n 0\n \n 0\n
          \n \\(974\\)\n \n 0\n \n 0\n
          \n \\(975\\)\n \n 0\n \n 0\n
          \n \\(976\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(977\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(978\\)\n \n 0\n \n 0\n
          \n \\(979\\)\n \n 0\n \n 0\n
          \n \\(980\\)\n \n 0\n \n 0\n
          \n \\(981\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(982\\)\n \n 0\n \n 0\n
          \n \\(983\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(984\\)\n \n 0\n \n 0\n
          \n \\(985\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(986\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(987\\)\n \n 0\n \n 0\n
          \n \\(988\\)\n \n 0\n \n 0\n
          \n \\(989\\)\n \n 0\n \n 0\n
          \n \\(990\\)\n \n 0\n \n 0\n
          \n \\(991\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(992\\)\n \n 0\n \n 0\n
          \n \\(993\\)\n \n 0\n \n 0\n
          \n \\(994\\)\n \n 0\n \n 0\n
          \n \\(995\\)\n \n 0\n \n 0\n
          \n \\(996\\)\n \n 0\n \n 0\n
          \n \\(997\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(998\\)\n \n 0\n \n 0\n
          \n \\(999\\)\n \n 0\n \n 0\n
          \u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisting character
          CharParityOrdTypeTwistMinDim
          1.1even1trivial3332.1.bc.b.2027.14
          4.3odd2CM3332.1.bc.b.2027.14
          7.2even3inner3332.1.bc.b.667.14
          7.3odd668.1.f.a.55.1yes2
          7.4even33332.1.m.b.2843.12
          7.5odd63332.1.bc.c.667.14
          7.6odd23332.1.bc.c.2027.14
          17.13even4inner3332.1.bc.b.2223.14
          21.17even6612.1.l.a.55.12
          28.3even668.1.f.a.55.1yes2
          28.11odd63332.1.m.b.2843.12
          28.19even63332.1.bc.c.667.14
          28.23odd6inner3332.1.bc.b.667.14
          28.27even23332.1.bc.c.2027.14
          35.3even121700.1.n.a.599.12
          35.17even121700.1.n.b.599.12
          35.24odd61700.1.p.a.1551.12
          56.3even61088.1.p.a.191.12
          56.45odd61088.1.p.a.191.12
          68.47odd4inner3332.1.bc.b.2223.14
          84.59odd6612.1.l.a.55.12
          119.3even481156.1.g.b.155.18
          119.10even481156.1.g.b.179.18
          119.13odd43332.1.bc.c.2223.14
          119.24even481156.1.g.b.179.28
          119.30even12inner3332.1.bc.b.863.14
          119.31even481156.1.g.b.155.28
          119.38odd121156.1.f.b.251.12
          119.45even481156.1.g.b.399.18
          119.47odd123332.1.bc.c.863.14
          119.59odd241156.1.c.b.579.12
          119.66odd241156.1.d.a.1155.22
          119.73even481156.1.g.b.423.28
          119.80even481156.1.g.b.423.18
          119.81even123332.1.m.b.3039.12
          119.87odd241156.1.d.a.1155.12
          119.94odd241156.1.c.b.579.22
          119.101odd61156.1.f.b.327.12
          119.108even481156.1.g.b.399.28
          119.115odd1268.1.f.a.47.1&check;2
          140.3odd121700.1.n.a.599.12
          140.59even61700.1.p.a.1551.12
          140.87odd121700.1.n.b.599.12
          357.353even12612.1.l.a.523.12
          476.3odd481156.1.g.b.155.18
          476.31odd481156.1.g.b.155.28
          476.47even123332.1.bc.c.863.14
          476.59even241156.1.c.b.579.12
          476.87even241156.1.d.a.1155.12
          476.115even1268.1.f.a.47.1&check;2
          476.143odd481156.1.g.b.179.28
          476.199odd481156.1.g.b.423.18
          476.227odd481156.1.g.b.399.28
          476.251even43332.1.bc.c.2223.14
          476.283odd481156.1.g.b.399.18
          476.311odd481156.1.g.b.423.28
          476.319odd123332.1.m.b.3039.12
          476.339even61156.1.f.b.327.12
          476.367odd481156.1.g.b.179.18
          476.387odd12inner3332.1.bc.b.863.14
          476.395even121156.1.f.b.251.12
          476.423even241156.1.d.a.1155.22
          476.451even241156.1.c.b.579.22
          595.234odd121700.1.p.a.251.12
          595.353even121700.1.n.b.999.12
          595.472even121700.1.n.a.999.12
          952.115even121088.1.p.a.319.12
          952.829odd121088.1.p.a.319.12
          1428.1067odd12612.1.l.a.523.12
          2380.1067odd121700.1.n.a.999.12
          2380.1543odd121700.1.n.b.999.12
          2380.2019even121700.1.p.a.251.12
          \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisted newform
          TwistMinDimCharParityOrdType
          68.1.f.a.47.1&check;2119.115odd12
          68.1.f.a.47.1&check;2476.115even12
          68.1.f.a.55.1yes27.3odd6
          68.1.f.a.55.1yes228.3even6
          612.1.l.a.55.1221.17even6
          612.1.l.a.55.1284.59odd6
          612.1.l.a.523.12357.353even12
          612.1.l.a.523.121428.1067odd12
          1088.1.p.a.191.1256.3even6
          1088.1.p.a.191.1256.45odd6
          1088.1.p.a.319.12952.115even12
          1088.1.p.a.319.12952.829odd12
          1156.1.c.b.579.12119.59odd24
          1156.1.c.b.579.12476.59even24
          1156.1.c.b.579.22119.94odd24
          1156.1.c.b.579.22476.451even24
          1156.1.d.a.1155.12119.87odd24
          1156.1.d.a.1155.12476.87even24
          1156.1.d.a.1155.22119.66odd24
          1156.1.d.a.1155.22476.423even24
          1156.1.f.b.251.12119.38odd12
          1156.1.f.b.251.12476.395even12
          1156.1.f.b.327.12119.101odd6
          1156.1.f.b.327.12476.339even6
          1156.1.g.b.155.18119.3even48
          1156.1.g.b.155.18476.3odd48
          1156.1.g.b.155.28119.31even48
          1156.1.g.b.155.28476.31odd48
          1156.1.g.b.179.18119.10even48
          1156.1.g.b.179.18476.367odd48
          1156.1.g.b.179.28119.24even48
          1156.1.g.b.179.28476.143odd48
          1156.1.g.b.399.18119.45even48
          1156.1.g.b.399.18476.283odd48
          1156.1.g.b.399.28119.108even48
          1156.1.g.b.399.28476.227odd48
          1156.1.g.b.423.18119.80even48
          1156.1.g.b.423.18476.199odd48
          1156.1.g.b.423.28119.73even48
          1156.1.g.b.423.28476.311odd48
          1700.1.n.a.599.1235.3even12
          1700.1.n.a.599.12140.3odd12
          1700.1.n.a.999.12595.472even12
          1700.1.n.a.999.122380.1067odd12
          1700.1.n.b.599.1235.17even12
          1700.1.n.b.599.12140.87odd12
          1700.1.n.b.999.12595.353even12
          1700.1.n.b.999.122380.1543odd12
          1700.1.p.a.251.12595.234odd12
          1700.1.p.a.251.122380.2019even12
          1700.1.p.a.1551.1235.24odd6
          1700.1.p.a.1551.12140.59even6
          3332.1.m.b.2843.127.4even3
          3332.1.m.b.2843.1228.11odd6
          3332.1.m.b.3039.12119.81even12
          3332.1.m.b.3039.12476.319odd12
          3332.1.bc.b.667.147.2even3inner
          3332.1.bc.b.667.1428.23odd6inner
          3332.1.bc.b.863.14119.30even12inner
          3332.1.bc.b.863.14476.387odd12inner
          3332.1.bc.b.2027.141.1even1trivial
          3332.1.bc.b.2027.144.3odd2CM
          3332.1.bc.b.2223.1417.13even4inner
          3332.1.bc.b.2223.1468.47odd4inner
          3332.1.bc.c.667.147.5odd6
          3332.1.bc.c.667.1428.19even6
          3332.1.bc.c.863.14119.47odd12
          3332.1.bc.c.863.14476.47even12
          3332.1.bc.c.2027.147.6odd2
          3332.1.bc.c.2027.1428.27even2
          3332.1.bc.c.2223.14119.13odd4
          3332.1.bc.c.2223.14476.251even4
          ", "statics": {"paragraph": 22, "paragraph.text": 22, "table": 8, "title": 4}, "url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/3332/1/bc/b/2027/1/", "content": "Show commands: Magma/ PariGP/ SageMath\n\n[N,k,chi] = [3332,1,Mod(667,3332)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)\n\nfrom sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3332, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 4, 9])) N = Newforms(chi, 1, names=\"a\")\n\n//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter(\"3332.667\"); S:= CuspForms(chi, 1); N := Newforms(S);\n\n| Level
          : | \\( N \\) | \\(=\\) | \\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\) |\n|---|---|---|---|\n| Weight
          : | \\( k \\) | \\(=\\) | \\( 1 \\) |\n| Character orbit
          : | \\([\\chi]\\) | \\(=\\) | 3332.bc
          (of
          order
          \\(12\\),
          degree
          \\(4\\), not
          minimal
          ) |\n\n## Newform invariants\n\nsage:\u00a0f = N[0] \\# Warning: the index may be different\n\ngp:\u00a0f = lf[1] \\\\ Warning: the index may be different\n\n| Self dual
          : | no |\n|---|---|\n| Analytic conductor
          : | \\(1.66288462209\\) |\n| Analytic rank
          : | \\(0\\) |\n| Dimension
          : | \\(4\\) |\n| Coefficient field
          : | \\(\\Q(\\zeta_{12})\\) |\n| gp:\u00a0f.mod \\\\ as an extension of the character field | |\n| Defining polynomial
          : | \\( x^{4} - x^{2} + 1 \\) |\n| Coefficient ring
          : | \\(\\Z[a_1, a_2]\\) |\n| Coefficient ring index
          : | \\( 1 \\) |\n| Twist minimal
          : | no (minimal twist has level 68) |\n| Projective image
          : | \\(D_{4}\\) |\n| Projective field
          : | Galois closure of
          4.2.19652.1 |\n| Artin image
          : | $C_4\\wr C_2\\times C_6$ |\n| Artin field
          : | Galois closure of
          \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\) |\n\n## Embedding invariants\n\n| Embedding label | | | 2027.1 |\n|---|---|---|---|\n| Root | | | \\(0.866025 - 0.500000i\\) of defining polynomial |\n| Character | \\(\\chi\\) | \\(=\\) | 3332.2027 |\n| Dual form | | | 3332.1.bc.b.863.1 |\n\nsage:\u00a0f.q_expansion() \\# note that sage often uses an isomorphic number field\n\ngp:\u00a0mfcoefs(f, 20)\n\n| \\(f(q)\\) | \\(=\\) | \\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\) |\n|---|---|---|\n| \\(\\operatorname{Tr}(f)(q)\\) | \\(=\\) | \\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\) |\n\n## Character values\n\nWe give the values of \\(\\chi\\) on generators for \\(\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times\\).\n\n| \\(n\\) | \\(785\\) | \\(885\\) | \\(1667\\) |\n|---|---|---|---|\n| \\(\\chi(n)\\) | \\(e\\left(\\frac{3}{4}\\right)\\) | \\(e\\left(\\frac{2}{3}\\right)\\) | \\(-1\\) |\n\n## Coefficient data\n\nFor each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the Satake parameters\\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).\n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n
          \\(n\\)\\(a_n\\)\\(a_n / n^{(k-1)/2}\\)\\( \\alpha_n \\)\\( \\theta_n \\)
          \\(p\\)\\(a_p\\)\\(a_p / p^{(k-1)/2}\\)\\( \\alpha_p\\)\\( \\theta_p \\)
          \\(2\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(3\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(4\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(5\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(6\\)00
          \\(7\\)00
          \\(8\\)1.00000<br>i1.00000<br>i
          \\(9\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(10\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(11\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(12\\)00
          \\(13\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(14\\)00
          \\(15\\)00
          \\(16\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(17\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(18\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(19\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(20\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(21\\)00
          \\(22\\)00
          \\(23\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(24\\)00
          \\(25\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(26\\)00
          \\(27\\)00
          \\(28\\)00
          \\(29\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(30\\)00
          \\(31\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(32\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(33\\)00
          \\(34\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(35\\)00
          \\(36\\)1.00000<br>i1.00000<br>i
          \\(37\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(38\\)00
          \\(39\\)00
          \\(40\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(41\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(42\\)00
          \\(43\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(44\\)00
          \\(45\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(46\\)00
          \\(47\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(48\\)00
          \\(49\\)00
          \\(50\\)\u22121.00000\u22121.00000
          \\(51\\)00
          \\(52\\)00
          \\(53\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(54\\)00
          \\(55\\)00
          \\(56\\)00
          \\(57\\)00
          \\(58\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(59\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(60\\)00
          \\(61\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(62\\)00
          \\(63\\)00
          \\(64\\)\u22121.00000\u22121.00000
          \\(65\\)00
          \\(66\\)00
          \\(67\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(68\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(69\\)00
          \\(70\\)00
          \\(71\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(72\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(73\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(74\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(75\\)00
          \\(76\\)00
          \\(77\\)00
          \\(78\\)00
          \\(79\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(80\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(81\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(82\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(83\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(84\\)00
          \\(85\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(86\\)00
          \\(87\\)00
          \\(88\\)00
          \\(89\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(90\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(91\\)00
          \\(92\\)00
          \\(93\\)00
          \\(94\\)00
          \\(95\\)00
          \\(96\\)00
          \\(97\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(98\\)00
          \\(99\\)00
          \\(100\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(101\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(102\\)00
          \\(103\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(104\\)00
          \\(105\\)00
          \\(106\\)00
          \\(107\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(108\\)00
          \\(109\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(110\\)00
          \\(111\\)00
          \\(112\\)00
          \\(113\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(114\\)00
          \\(115\\)00
          \\(116\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(117\\)00
          \\(118\\)00
          \\(119\\)00
          \\(120\\)00
          \\(121\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(122\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(123\\)00
          \\(124\\)00
          \\(125\\)00
          \\(126\\)00
          \\(127\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(128\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(129\\)00
          \\(130\\)00
          \\(131\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(132\\)00
          \\(133\\)00
          \\(134\\)00
          \\(135\\)00
          \\(136\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(137\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(138\\)00
          \\(139\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(140\\)00
          \\(141\\)00
          \\(142\\)00
          \\(143\\)00
          \\(144\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(145\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(146\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(147\\)00
          \\(148\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(149\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \\(150\\)00
          \\(151\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(152\\)00
          \\(153\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(154\\)00
          \\(155\\)00
          \\(156\\)00
          \\(157\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(158\\)00
          \\(159\\)00
          \\(160\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(161\\)00
          \\(162\\)1.00000<br>i1.00000<br>i
          \\(163\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(164\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(165\\)00
          \\(166\\)00
          \\(167\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(168\\)00
          \\(169\\)\u22121.00000\u22121.00000
          \\(170\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(171\\)00
          \\(172\\)00
          \\(173\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(174\\)00
          \\(175\\)00
          \\(176\\)00
          \\(177\\)00
          \\(178\\)00
          \\(179\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(180\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(181\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(182\\)00
          \\(183\\)00
          \\(184\\)00
          \\(185\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(186\\)00
          \\(187\\)00
          \\(188\\)00
          \\(189\\)00
          \\(190\\)00
          \\(191\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(192\\)00
          \\(193\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(194\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(195\\)00
          \\(196\\)00
          \\(197\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(198\\)00
          \\(199\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(200\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(201\\)00
          \\(202\\)00
          \\(203\\)00
          \\(204\\)00
          \\(205\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
          \\(206\\)00
          \\(207\\)00
          \\(208\\)00
          \\(209\\)00
          \\(210\\)00
          \\(211\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(212\\)00
          \\(213\\)00
          \\(214\\)00
          \\(215\\)00
          \\(216\\)00
          \\(217\\)00
          \\(218\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(219\\)00
          \\(220\\)00
          \\(221\\)00
          \\(222\\)00
          \\(223\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(224\\)00
          \\(225\\)\u22121.00000\u22121.00000
          \\(226\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(227\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(228\\)00
          \\(229\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(230\\)00
          \\(231\\)00
          \\(232\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(233\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(234\\)00
          \\(235\\)00
          \\(236\\)00
          \\(237\\)00
          \\(238\\)00
          \\(239\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(240\\)00
          \\(241\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(242\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(243\\)00
          \\(244\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(245\\)00
          \\(246\\)00
          \\(247\\)00
          \\(248\\)00
          \\(249\\)00
          \\(250\\)00
          \\(251\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(252\\)00
          \\(253\\)00
          \\(254\\)00
          \\(255\\)00
          \\(256\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(257\\)1.73205\u22121.00000<br>i1.73205\u22121.00000<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(258\\)00
          \\(259\\)00
          \\(260\\)00
          \\(261\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(262\\)00
          \\(263\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(264\\)00
          \\(265\\)00
          \\(266\\)00
          \\(267\\)00
          \\(268\\)00
          \\(269\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(270\\)00
          \\(271\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(272\\)1.000001.00000
          \\(273\\)00
          \\(274\\)00
          \\(275\\)00
          \\(276\\)00
          \\(277\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(278\\)00
          \\(279\\)00
          \\(280\\)00
          \\(281\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(282\\)00
          \\(283\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(284\\)00
          \\(285\\)00
          \\(286\\)00
          \\(287\\)00
          \\(288\\)\u22121.00000\u22121.00000
          \\(289\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(290\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(291\\)00
          \\(292\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(293\\)2.000002.000001.00000\\(0\\)
          1.00000\\(0\\)
          \\(294\\)00
          \\(295\\)00
          \\(296\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(297\\)00
          \\(298\\)1.73205+1.00000<br>i1.73205+1.00000<br>i
          \\(299\\)00
          \\(300\\)00
          \\(301\\)00
          \\(302\\)00
          \\(303\\)00
          \\(304\\)00
          \\(305\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
          \\(306\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(307\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(308\\)00
          \\(309\\)00
          \\(310\\)00
          \\(311\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(312\\)00
          \\(313\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(314\\)2.00000<br>i2.00000<br>i
          \\(315\\)00
          \\(316\\)00
          \\(317\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(318\\)00
          \\(319\\)00
          \\(320\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(321\\)00
          \\(322\\)00
          \\(323\\)00
          \\(324\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(325\\)00
          \\(326\\)00
          \\(327\\)00
          \\(328\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(329\\)00
          \\(330\\)00
          \\(331\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(332\\)00
          \\(333\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(334\\)00
          \\(335\\)00
          \\(336\\)00
          \\(337\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(338\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(339\\)00
          \\(340\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(341\\)00
          \\(342\\)00
          \\(343\\)00
          \\(344\\)00
          \\(345\\)00
          \\(346\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(347\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(348\\)00
          \\(349\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(350\\)00
          \\(351\\)00
          \\(352\\)00
          \\(353\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(354\\)00
          \\(355\\)00
          \\(356\\)00
          \\(357\\)00
          \\(358\\)00
          \\(359\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(360\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(361\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(362\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(363\\)00
          \\(364\\)00
          \\(365\\)2.00000<br>i2.00000<br>i
          \\(366\\)00
          \\(367\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(368\\)00
          \\(369\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(370\\)2.000002.00000
          \\(371\\)00
          \\(372\\)00
          \\(373\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(374\\)00
          \\(375\\)00
          \\(376\\)00
          \\(377\\)00
          \\(378\\)00
          \\(379\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(380\\)00
          \\(381\\)00
          \\(382\\)00
          \\(383\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(384\\)00
          \\(385\\)00
          \\(386\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(387\\)00
          \\(388\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(389\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \\(390\\)00
          \\(391\\)00
          \\(392\\)00
          \\(393\\)00
          \\(394\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(395\\)00
          \\(396\\)00
          \\(397\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(398\\)00
          \\(399\\)00
          \\(400\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(401\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(402\\)00
          \\(403\\)00
          \\(404\\)00
          \\(405\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(406\\)00
          \\(407\\)00
          \\(408\\)00
          \\(409\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(410\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(411\\)00
          \\(412\\)00
          \\(413\\)00
          \\(414\\)00
          \\(415\\)00
          \\(416\\)00
          \\(417\\)00
          \\(418\\)00
          \\(419\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(420\\)00
          \\(421\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(422\\)00
          \\(423\\)00
          \\(424\\)00
          \\(425\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(426\\)00
          \\(427\\)00
          \\(428\\)00
          \\(429\\)00
          \\(430\\)00
          \\(431\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(432\\)00
          \\(433\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(434\\)00
          \\(435\\)00
          \\(436\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(437\\)00
          \\(438\\)00
          \\(439\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(440\\)00
          \\(441\\)00
          \\(442\\)00
          \\(443\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(444\\)00
          \\(445\\)00
          \\(446\\)00
          \\(447\\)00
          \\(448\\)00
          \\(449\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(450\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(451\\)00
          \\(452\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(453\\)00
          \\(454\\)00
          \\(455\\)00
          \\(456\\)00
          \\(457\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(458\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(459\\)00
          \\(460\\)00
          \\(461\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(462\\)00
          \\(463\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(464\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(465\\)00
          \\(466\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(467\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(468\\)00
          \\(469\\)00
          \\(470\\)00
          \\(471\\)00
          \\(472\\)00
          \\(473\\)00
          \\(474\\)00
          \\(475\\)00
          \\(476\\)00
          \\(477\\)00
          \\(478\\)00
          \\(479\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(480\\)00
          \\(481\\)00
          \\(482\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(483\\)00
          \\(484\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(485\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(486\\)00
          \\(487\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(488\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(489\\)00
          \\(490\\)00
          \\(491\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(492\\)00
          \\(493\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(494\\)00
          \\(495\\)00
          \\(496\\)00
          \\(497\\)00
          \\(498\\)00
          \\(499\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(500\\)00
          \\(501\\)00
          \\(502\\)00
          \\(503\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(504\\)00
          \\(505\\)00
          \\(506\\)00
          \\(507\\)00
          \\(508\\)00
          \\(509\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \\(510\\)00
          \\(511\\)00
          \\(512\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(513\\)00
          \\(514\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
          \\(515\\)00
          \\(516\\)00
          \\(517\\)00
          \\(518\\)00
          \\(519\\)00
          \\(520\\)00
          \\(521\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(522\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(523\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(524\\)00
          \\(525\\)00
          \\(526\\)00
          \\(527\\)00
          \\(528\\)00
          \\(529\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(530\\)00
          \\(531\\)00
          \\(532\\)00
          \\(533\\)00
          \\(534\\)00
          \\(535\\)00
          \\(536\\)00
          \\(537\\)00
          \\(538\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(539\\)00
          \\(540\\)00
          \\(541\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(542\\)00
          \\(543\\)00
          \\(544\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(545\\)\u22122.00000\u22122.00000
          \\(546\\)00
          \\(547\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(548\\)00
          \\(549\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(550\\)00
          \\(551\\)00
          \\(552\\)00
          \\(553\\)00
          \\(554\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(555\\)00
          \\(556\\)00
          \\(557\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(558\\)00
          \\(559\\)00
          \\(560\\)00
          \\(561\\)00
          \\(562\\)00
          \\(563\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(564\\)00
          \\(565\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
          \\(566\\)00
          \\(567\\)00
          \\(568\\)00
          \\(569\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(570\\)00
          \\(571\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(572\\)00
          \\(573\\)00
          \\(574\\)00
          \\(575\\)00
          \\(576\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(577\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(578\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(579\\)00
          \\(580\\)2.00000<br>i2.00000<br>i
          \\(581\\)00
          \\(582\\)00
          \\(583\\)00
          \\(584\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(585\\)00
          \\(586\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(587\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(588\\)00
          \\(589\\)00
          \\(590\\)00
          \\(591\\)00
          \\(592\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(593\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(594\\)00
          \\(595\\)00
          \\(596\\)\u22122.00000\u22122.00000
          \\(597\\)00
          \\(598\\)00
          \\(599\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(600\\)00
          \\(601\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(602\\)00
          \\(603\\)00
          \\(604\\)00
          \\(605\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(606\\)00
          \\(607\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(608\\)00
          \\(609\\)00
          \\(610\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(611\\)00
          \\(612\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(613\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(614\\)00
          \\(615\\)00
          \\(616\\)00
          \\(617\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(618\\)00
          \\(619\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(620\\)00
          \\(621\\)00
          \\(622\\)00
          \\(623\\)00
          \\(624\\)00
          \\(625\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(626\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(627\\)00
          \\(628\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(629\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(630\\)00
          \\(631\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(632\\)00
          \\(633\\)00
          \\(634\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(635\\)00
          \\(636\\)00
          \\(637\\)00
          \\(638\\)00
          \\(639\\)00
          \\(640\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(641\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(642\\)00
          \\(643\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(644\\)00
          \\(645\\)00
          \\(646\\)00
          \\(647\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(648\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(649\\)00
          \\(650\\)00
          \\(651\\)00
          \\(652\\)00
          \\(653\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(654\\)00
          \\(655\\)00
          \\(656\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(657\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(658\\)00
          \\(659\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(660\\)00
          \\(661\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(662\\)00
          \\(663\\)00
          \\(664\\)00
          \\(665\\)00
          \\(666\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(667\\)00
          \\(668\\)00
          \\(669\\)00
          \\(670\\)00
          \\(671\\)00
          \\(672\\)00
          \\(673\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(674\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(675\\)00
          \\(676\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(677\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(678\\)00
          \\(679\\)00
          \\(680\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(681\\)00
          \\(682\\)00
          \\(683\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(684\\)00
          \\(685\\)00
          \\(686\\)00
          \\(687\\)00
          \\(688\\)00
          \\(689\\)00
          \\(690\\)00
          \\(691\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(692\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(693\\)00
          \\(694\\)00
          \\(695\\)00
          \\(696\\)00
          \\(697\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(698\\)00
          \\(699\\)00
          \\(700\\)00
          \\(701\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(702\\)00
          \\(703\\)00
          \\(704\\)00
          \\(705\\)00
          \\(706\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(707\\)00
          \\(708\\)00
          \\(709\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(710\\)00
          \\(711\\)00
          \\(712\\)00
          \\(713\\)00
          \\(714\\)00
          \\(715\\)00
          \\(716\\)00
          \\(717\\)00
          \\(718\\)00
          \\(719\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(720\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(721\\)00
          \\(722\\)1.00000<br>i1.00000<br>i
          \\(723\\)00
          \\(724\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(725\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(726\\)00
          \\(727\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(728\\)00
          \\(729\\)1.00000<br>i1.00000<br>i
          \\(730\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(731\\)00
          \\(732\\)00
          \\(733\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(734\\)00
          \\(735\\)00
          \\(736\\)00
          \\(737\\)00
          \\(738\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(739\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(740\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(741\\)00
          \\(742\\)00
          \\(743\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(744\\)00
          \\(745\\)0.732051+2.73205<br>i0.732051+2.73205<br>i
          \\(746\\)00
          \\(747\\)00
          \\(748\\)00
          \\(749\\)00
          \\(750\\)00
          \\(751\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(752\\)00
          \\(753\\)00
          \\(754\\)00
          \\(755\\)00
          \\(756\\)00
          \\(757\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(758\\)00
          \\(759\\)00
          \\(760\\)00
          \\(761\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(762\\)00
          \\(763\\)00
          \\(764\\)00
          \\(765\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(766\\)00
          \\(767\\)00
          \\(768\\)00
          \\(769\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(770\\)00
          \\(771\\)00
          \\(772\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(773\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \\(774\\)00
          \\(775\\)00
          \\(776\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(777\\)00
          \\(778\\)2.000002.00000
          \\(779\\)00
          \\(780\\)00
          \\(781\\)00
          \\(782\\)00
          \\(783\\)00
          \\(784\\)00
          \\(785\\)\u22122.00000+2.00000<br>i\u22122.00000+2.00000<br>i
          \\(786\\)00
          \\(787\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(788\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(789\\)00
          \\(790\\)00
          \\(791\\)00
          \\(792\\)00
          \\(793\\)00
          \\(794\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(795\\)00
          \\(796\\)00
          \\(797\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(798\\)00
          \\(799\\)00
          \\(800\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(801\\)00
          \\(802\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(803\\)00
          \\(804\\)00
          \\(805\\)00
          \\(806\\)00
          \\(807\\)00
          \\(808\\)00
          \\(809\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(810\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(811\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(812\\)00
          \\(813\\)00
          \\(814\\)00
          \\(815\\)00
          \\(816\\)00
          \\(817\\)00
          \\(818\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(819\\)00
          \\(820\\)2.000002.00000
          \\(821\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(822\\)00
          \\(823\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(824\\)00
          \\(825\\)00
          \\(826\\)00
          \\(827\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(828\\)00
          \\(829\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(830\\)00
          \\(831\\)00
          \\(832\\)00
          \\(833\\)00
          \\(834\\)00
          \\(835\\)00
          \\(836\\)00
          \\(837\\)00
          \\(838\\)00
          \\(839\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(840\\)00
          \\(841\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(842\\)00
          \\(843\\)00
          \\(844\\)00
          \\(845\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(846\\)00
          \\(847\\)00
          \\(848\\)00
          \\(849\\)00
          \\(850\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(851\\)00
          \\(852\\)00
          \\(853\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(854\\)00
          \\(855\\)00
          \\(856\\)00
          \\(857\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(858\\)00
          \\(859\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(860\\)00
          \\(861\\)00
          \\(862\\)00
          \\(863\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(864\\)00
          \\(865\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(866\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(867\\)00
          \\(868\\)00
          \\(869\\)00
          \\(870\\)00
          \\(871\\)00
          \\(872\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(873\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(874\\)00
          \\(875\\)00
          \\(876\\)00
          \\(877\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(878\\)00
          \\(879\\)00
          \\(880\\)00
          \\(881\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(882\\)00
          \\(883\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(884\\)00
          \\(885\\)00
          \\(886\\)00
          \\(887\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(888\\)00
          \\(889\\)00
          \\(890\\)00
          \\(891\\)00
          \\(892\\)00
          \\(893\\)00
          \\(894\\)00
          \\(895\\)00
          \\(896\\)00
          \\(897\\)00
          \\(898\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(899\\)00
          \\(900\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(901\\)00
          \\(902\\)00
          \\(903\\)00
          \\(904\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(905\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(906\\)00
          \\(907\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(908\\)00
          \\(909\\)00
          \\(910\\)00
          \\(911\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(912\\)00
          \\(913\\)00
          \\(914\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(915\\)00
          \\(916\\)2.00000<br>i2.00000<br>i
          \\(917\\)00
          \\(918\\)00
          \\(919\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(920\\)00
          \\(921\\)00
          \\(922\\)00
          \\(923\\)00
          \\(924\\)00
          \\(925\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(926\\)00
          \\(927\\)00
          \\(928\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(929\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(930\\)00
          \\(931\\)00
          \\(932\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(933\\)00
          \\(934\\)00
          \\(935\\)00
          \\(936\\)00
          \\(937\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(938\\)00
          \\(939\\)00
          \\(940\\)00
          \\(941\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(942\\)00
          \\(943\\)00
          \\(944\\)00
          \\(945\\)00
          \\(946\\)00
          \\(947\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(948\\)00
          \\(949\\)00
          \\(950\\)00
          \\(951\\)00
          \\(952\\)00
          \\(953\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(954\\)00
          \\(955\\)00
          \\(956\\)00
          \\(957\\)00
          \\(958\\)00
          \\(959\\)00
          \\(960\\)00
          \\(961\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(962\\)00
          \\(963\\)00
          \\(964\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(965\\)2.000002.00000
          \\(966\\)00
          \\(967\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(968\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(969\\)00
          \\(970\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(971\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(972\\)00
          \\(973\\)00
          \\(974\\)00
          \\(975\\)00
          \\(976\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(977\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(978\\)00
          \\(979\\)00
          \\(980\\)00
          \\(981\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(982\\)00
          \\(983\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(984\\)00
          \\(985\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(986\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(987\\)00
          \\(988\\)00
          \\(989\\)00
          \\(990\\)00
          \\(991\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(992\\)00
          \\(993\\)00
          \\(994\\)00
          \\(995\\)00
          \\(996\\)00
          \\(997\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(998\\)00
          \\(999\\)00
          \n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n| By
          twisting character | | | | | | |\n|---|---|---|---|---|---|---|\n| Char | Parity | Ord | Type | Twist | Min | Dim |\n| 1.1 | even | 1 | trivial | 3332.1.bc.b.2027.1 | | 4 |\n| 4.3 | odd | 2 | CM | 3332.1.bc.b.2027.1 | | 4 |\n| 7.2 | even | 3 | inner | 3332.1.bc.b.667.1 | | 4 |\n| 7.3 | odd | 6 | | 68.1.f.a.55.1 | yes | 2 |\n| 7.4 | even | 3 | | 3332.1.m.b.2843.1 | | 2 |\n| 7.5 | odd | 6 | | 3332.1.bc.c.667.1 | | 4 |\n| 7.6 | odd | 2 | | 3332.1.bc.c.2027.1 | | 4 |\n| 17.13 | even | 4 | inner | 3332.1.bc.b.2223.1 | | 4 |\n| 21.17 | even | 6 | | 612.1.l.a.55.1 | | 2 |\n| 28.3 | even | 6 | | 68.1.f.a.55.1 | yes | 2 |\n| 28.11 | odd | 6 | | 3332.1.m.b.2843.1 | | 2 |\n| 28.19 | even | 6 | | 3332.1.bc.c.667.1 | | 4 |\n| 28.23 | odd | 6 | inner | 3332.1.bc.b.667.1 | | 4 |\n| 28.27 | even | 2 | | 3332.1.bc.c.2027.1 | | 4 |\n| 35.3 | even | 12 | | 1700.1.n.a.599.1 | | 2 |\n| 35.17 | even | 12 | | 1700.1.n.b.599.1 | | 2 |\n| 35.24 | odd | 6 | | 1700.1.p.a.1551.1 | | 2 |\n| 56.3 | even | 6 | | 1088.1.p.a.191.1 | | 2 |\n| 56.45 | odd | 6 | | 1088.1.p.a.191.1 | | 2 |\n| 68.47 | odd | 4 | inner | 3332.1.bc.b.2223.1 | | 4 |\n| 84.59 | odd | 6 | | 612.1.l.a.55.1 | | 2 |\n| 119.3 | even | 48 | | 1156.1.g.b.155.1 | | 8 |\n| 119.10 | even | 48 | | 1156.1.g.b.179.1 | | 8 |\n| 119.13 | odd | 4 | | 3332.1.bc.c.2223.1 | | 4 |\n| 119.24 | even | 48 | | 1156.1.g.b.179.2 | | 8 |\n| 119.30 | even | 12 | inner | 3332.1.bc.b.863.1 | | 4 |\n| 119.31 | even | 48 | | 1156.1.g.b.155.2 | | 8 |\n| 119.38 | odd | 12 | | 1156.1.f.b.251.1 | | 2 |\n| 119.45 | even | 48 | | 1156.1.g.b.399.1 | | 8 |\n| 119.47 | odd | 12 | | 3332.1.bc.c.863.1 | | 4 |\n| 119.59 | odd | 24 | | 1156.1.c.b.579.1 | | 2 |\n| 119.66 | odd | 24 | | 1156.1.d.a.1155.2 | | 2 |\n| 119.73 | even | 48 | | 1156.1.g.b.423.2 | | 8 |\n| 119.80 | even | 48 | | 1156.1.g.b.423.1 | | 8 |\n| 119.81 | even | 12 | | 3332.1.m.b.3039.1 | | 2 |\n| 119.87 | odd | 24 | | 1156.1.d.a.1155.1 | | 2 |\n| 119.94 | odd | 24 | | 1156.1.c.b.579.2 | | 2 |\n| 119.101 | odd | 6 | | 1156.1.f.b.327.1 | | 2 |\n| 119.108 | even | 48 | | 1156.1.g.b.399.2 | | 8 |\n| 119.115 | odd | 12 | | 68.1.f.a.47.1 | ✓ | 2 |\n| 140.3 | odd | 12 | | 1700.1.n.a.599.1 | | 2 |\n| 140.59 | even | 6 | | 1700.1.p.a.1551.1 | | 2 |\n| 140.87 | odd | 12 | | 1700.1.n.b.599.1 | | 2 |\n| 357.353 | even | 12 | | 612.1.l.a.523.1 | | 2 |\n| 476.3 | odd | 48 | | 1156.1.g.b.155.1 | | 8 |\n| 476.31 | odd | 48 | | 1156.1.g.b.155.2 | | 8 |\n| 476.47 | even | 12 | | 3332.1.bc.c.863.1 | | 4 |\n| 476.59 | even | 24 | | 1156.1.c.b.579.1 | | 2 |\n| 476.87 | even | 24 | | 1156.1.d.a.1155.1 | | 2 |\n| 476.115 | even | 12 | | 68.1.f.a.47.1 | ✓ | 2 |\n| 476.143 | odd | 48 | | 1156.1.g.b.179.2 | | 8 |\n| 476.199 | odd | 48 | | 1156.1.g.b.423.1 | | 8 |\n| 476.227 | odd | 48 | | 1156.1.g.b.399.2 | | 8 |\n| 476.251 | even | 4 | | 3332.1.bc.c.2223.1 | | 4 |\n| 476.283 | odd | 48 | | 1156.1.g.b.399.1 | | 8 |\n| 476.311 | odd | 48 | | 1156.1.g.b.423.2 | | 8 |\n| 476.319 | odd | 12 | | 3332.1.m.b.3039.1 | | 2 |\n| 476.339 | even | 6 | | 1156.1.f.b.327.1 | | 2 |\n| 476.367 | odd | 48 | | 1156.1.g.b.179.1 | | 8 |\n| 476.387 | odd | 12 | inner | 3332.1.bc.b.863.1 | | 4 |\n| 476.395 | even | 12 | | 1156.1.f.b.251.1 | | 2 |\n| 476.423 | even | 24 | | 1156.1.d.a.1155.2 | | 2 |\n| 476.451 | even | 24 | | 1156.1.c.b.579.2 | | 2 |\n| 595.234 | odd | 12 | | 1700.1.p.a.251.1 | | 2 |\n| 595.353 | even | 12 | | 1700.1.n.b.999.1 | | 2 |\n| 595.472 | even | 12 | | 1700.1.n.a.999.1 | | 2 |\n| 952.115 | even | 12 | | 1088.1.p.a.319.1 | | 2 |\n| 952.829 | odd | 12 | | 1088.1.p.a.319.1 | | 2 |\n| 1428.1067 | odd | 12 | | 612.1.l.a.523.1 | | 2 |\n| 2380.1067 | odd | 12 | | 1700.1.n.a.999.1 | | 2 |\n| 2380.1543 | odd | 12 | | 1700.1.n.b.999.1 | | 2 |\n| 2380.2019 | even | 12 | | 1700.1.p.a.251.1 | | 2 |\n\n| By
          twisted newform | | | | | | |\n|---|---|---|---|---|---|---|\n| Twist | Min | Dim | Char | Parity | Ord | Type |\n| 68.1.f.a.47.1 | ✓ | 2 | 119.115 | odd | 12 | |\n| 68.1.f.a.47.1 | ✓ | 2 | 476.115 | even | 12 | |\n| 68.1.f.a.55.1 | yes | 2 | 7.3 | odd | 6 | |\n| 68.1.f.a.55.1 | yes | 2 | 28.3 | even | 6 | |\n| 612.1.l.a.55.1 | | 2 | 21.17 | even | 6 | |\n| 612.1.l.a.55.1 | | 2 | 84.59 | odd | 6 | |\n| 612.1.l.a.523.1 | | 2 | 357.353 | even | 12 | |\n| 612.1.l.a.523.1 | | 2 | 1428.1067 | odd | 12 | |\n| 1088.1.p.a.191.1 | | 2 | 56.3 | even | 6 | |\n| 1088.1.p.a.191.1 | | 2 | 56.45 | odd | 6 | |\n| 1088.1.p.a.319.1 | | 2 | 952.115 | even | 12 | |\n| 1088.1.p.a.319.1 | | 2 | 952.829 | odd | 12 | |\n| 1156.1.c.b.579.1 | | 2 | 119.59 | odd | 24 | |\n| 1156.1.c.b.579.1 | | 2 | 476.59 | even | 24 | |\n| 1156.1.c.b.579.2 | | 2 | 119.94 | odd | 24 | |\n| 1156.1.c.b.579.2 | | 2 | 476.451 | even | 24 | |\n| 1156.1.d.a.1155.1 | | 2 | 119.87 | odd | 24 | |\n| 1156.1.d.a.1155.1 | | 2 | 476.87 | even | 24 | |\n| 1156.1.d.a.1155.2 | | 2 | 119.66 | odd | 24 | |\n| 1156.1.d.a.1155.2 | | 2 | 476.423 | even | 24 | |\n| 1156.1.f.b.251.1 | | 2 | 119.38 | odd | 12 | |\n| 1156.1.f.b.251.1 | | 2 | 476.395 | even | 12 | |\n| 1156.1.f.b.327.1 | | 2 | 119.101 | odd | 6 | |\n| 1156.1.f.b.327.1 | | 2 | 476.339 | even | 6 | |\n| 1156.1.g.b.155.1 | | 8 | 119.3 | even | 48 | |\n| 1156.1.g.b.155.1 | | 8 | 476.3 | odd | 48 | |\n| 1156.1.g.b.155.2 | | 8 | 119.31 | even | 48 | |\n| 1156.1.g.b.155.2 | | 8 | 476.31 | odd | 48 | |\n| 1156.1.g.b.179.1 | | 8 | 119.10 | even | 48 | |\n| 1156.1.g.b.179.1 | | 8 | 476.367 | odd | 48 | |\n| 1156.1.g.b.179.2 | | 8 | 119.24 | even | 48 | |\n| 1156.1.g.b.179.2 | | 8 | 476.143 | odd | 48 | |\n| 1156.1.g.b.399.1 | | 8 | 119.45 | even | 48 | |\n| 1156.1.g.b.399.1 | | 8 | 476.283 | odd | 48 | |\n| 1156.1.g.b.399.2 | | 8 | 119.108 | even | 48 | |\n| 1156.1.g.b.399.2 | | 8 | 476.227 | odd | 48 | |\n| 1156.1.g.b.423.1 | | 8 | 119.80 | even | 48 | |\n| 1156.1.g.b.423.1 | | 8 | 476.199 | odd | 48 | |\n| 1156.1.g.b.423.2 | | 8 | 119.73 | even | 48 | |\n| 1156.1.g.b.423.2 | | 8 | 476.311 | odd | 48 | |\n| 1700.1.n.a.599.1 | | 2 | 35.3 | even | 12 | |\n| 1700.1.n.a.599.1 | | 2 | 140.3 | odd | 12 | |\n| 1700.1.n.a.999.1 | | 2 | 595.472 | even | 12 | |\n| 1700.1.n.a.999.1 | | 2 | 2380.1067 | odd | 12 | |\n| 1700.1.n.b.599.1 | | 2 | 35.17 | even | 12 | |\n| 1700.1.n.b.599.1 | | 2 | 140.87 | odd | 12 | |\n| 1700.1.n.b.999.1 | | 2 | 595.353 | even | 12 | |\n| 1700.1.n.b.999.1 | | 2 | 2380.1543 | odd | 12 | |\n| 1700.1.p.a.251.1 | | 2 | 595.234 | odd | 12 | |\n| 1700.1.p.a.251.1 | | 2 | 2380.2019 | even | 12 | |\n| 1700.1.p.a.1551.1 | | 2 | 35.24 | odd | 6 | |\n| 1700.1.p.a.1551.1 | | 2 | 140.59 | even | 6 | |\n| 3332.1.m.b.2843.1 | | 2 | 7.4 | even | 3 | |\n| 3332.1.m.b.2843.1 | | 2 | 28.11 | odd | 6 | |\n| 3332.1.m.b.3039.1 | | 2 | 119.81 | even | 12 | |\n| 3332.1.m.b.3039.1 | | 2 | 476.319 | odd | 12 | |\n| 3332.1.bc.b.667.1 | | 4 | 7.2 | even | 3 | inner |\n| 3332.1.bc.b.667.1 | | 4 | 28.23 | odd | 6 | inner |\n| 3332.1.bc.b.863.1 | | 4 | 119.30 | even | 12 | inner |\n| 3332.1.bc.b.863.1 | | 4 | 476.387 | odd | 12 | inner |\n| 3332.1.bc.b.2027.1 | | 4 | 1.1 | even | 1 | trivial |\n| 3332.1.bc.b.2027.1 | | 4 | 4.3 | odd | 2 | CM |\n| 3332.1.bc.b.2223.1 | | 4 | 17.13 | even | 4 | inner |\n| 3332.1.bc.b.2223.1 | | 4 | 68.47 | odd | 4 | inner |\n| 3332.1.bc.c.667.1 | | 4 | 7.5 | odd | 6 | |\n| 3332.1.bc.c.667.1 | | 4 | 28.19 | even | 6 | |\n| 3332.1.bc.c.863.1 | | 4 | 119.47 | odd | 12 | |\n| 3332.1.bc.c.863.1 | | 4 | 476.47 | even | 12 | |\n| 3332.1.bc.c.2027.1 | | 4 | 7.6 | odd | 2 | |\n| 3332.1.bc.c.2027.1 | | 4 | 28.27 | even | 2 | |\n| 3332.1.bc.c.2223.1 | | 4 | 119.13 | odd | 4 | |\n| 3332.1.bc.c.2223.1 | | 4 | 476.251 | even | 4 | |\n", "html": "\n\n\n \n \n \n \n LMFDB - Embedded newform 3332.1.bc.b.2027.1 \n \n\n \n \n \n \n \n\n \n \n\n\n\n\n \n\n \n \n\n \n \n\n\n \n \n \n \n \n \n\n\n\n\n\n \n \n\n
          \n \n
          \n
          \n
          \n ⌂\n → Modular forms\n → Classical\n → Level 3332\n → Weight 1\n → Character orbit bc\n → Newform orbit b\n → Embedding 2027.1\n
          \n
          \n \n
          \n Citation\n ·\n Feedback\n ·\n Hide Menu\n \n
          \n
          \n\n
          \n
          Embedded newform 3332.1.bc.b.2027.1
          \n\n
          \n
          \n\n\n\n\n\n
          \n
          ↑
          \n

          Properties

          \n
          \n \n
          Label\n 3332.1.bc.b.2027.1
          \n
          Level\n $3332$
          Weight\n $1$
          Character\n 3332.2027
          Analytic conductor\n $1.663$
          Analytic rank\n $0$
          Dimension\n $4$
          Projective image\n $D_{4}$
          CM discriminant\n -4
          Inner twists\n $8$
          \n
          \n\n\n\n

          Related objects

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          \n \n
          \n\n\n\n

          Downloads

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          Learn more

          \n
          \n \n
          \n\n
          \n
          \n
          \n
          \n
          \n \n
          \n Show commands:\n Magma\n / PariGP\n / SageMath\n
          \n\n\n\n\n

          Newspace parameters

          \n\n
          comment: Compute space of new eigenforms
           
          \n
          [N,k,chi] = [3332,1,Mod(667,3332)]
           
          mf = mfinit([N,k,chi],0)
           
          lf = mfeigenbasis(mf)
           
          \n
          from sage.modular.dirichlet import DirichletCharacter
           
          H = DirichletGroup(3332, base_ring=CyclotomicField(12))
           
          chi = DirichletCharacter(H, H._module([6, 4, 9]))
           
          N = Newforms(chi, 1, names="a")
           
          \n
          //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
           
          chi := DirichletCharacter("3332.667");
           
          S:= CuspForms(chi, 1);
           
          N := Newforms(S);
           
          \n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
          Weight: \\( k \\) \\(=\\)\\( 1 \\)
          Character orbit: \\([\\chi]\\) \\(=\\) 3332.bc (of order \\(12\\), degree \\(4\\), not minimal)
          \n\n

          Newform invariants

          \n\n
          comment: select newform
           
          \n
          sage: f = N[0] # Warning: the index may be different
           
          \n
          gp: f = lf[1] \\\\ Warning: the index may be different
           
          \n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\t\n \n \t\n \t\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Self dual: no
          Analytic conductor: \\(1.66288462209\\)
          Analytic rank: \\(0\\)
          Dimension: \\(4\\)
          Coefficient field: \\(\\Q(\\zeta_{12})\\)
          \n
          comment: defining polynomial
           
          \n
          gp: f.mod \\\\ as an extension of the character field
           
          \n\n
          Defining polynomial: \n\n \\( x^{4} - x^{2} + 1 \\)\n \n\n \n \"Copy\n \n \n \"Toggle\n \n
          Coefficient ring: \\(\\Z[a_1, a_2]\\)
          Coefficient ring index: \\( 1 \\)
          Twist minimal: no (minimal twist has level 68)
          Projective image:\\(D_{4}\\)
          Projective field:Galois closure of 4.2.19652.1
          Artin image:$C_4\\wr C_2\\times C_6$
          Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
          \n\n\n

          Embedding invariants

          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Embedding label 2027.1
          Root\\(0.866025 - 0.500000i\\) of defining polynomial
          Character\\(\\chi\\)\\(=\\)3332.2027
          Dual form 3332.1.bc.b.863.1
          \n\n\n

          $q$-expansion

          \n
          \n
          comment: q-expansion
           
          \n
          sage: f.q_expansion() # note that sage often uses an isomorphic number field
           
          \n
          gp: mfcoefs(f, 20)
           
          \n\n
          \n \n \n \n \n \n \n \n \n \n \n \n
          \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
          \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\n\n \\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\n \n\n \n \"Copy\n \n \n \"Toggle\n \n
          \n
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          \n\n\n

          Character values

          \n

          We give the values of \\(\\chi\\) on generators for \\(\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times\\).

          \n\n \n \n \n \n \n \n \n \n \n \n \n \n
          \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
          \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
          \n\n\n

          Coefficient data

          \n\n

          For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the\nSatake parameters \\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).

          \n\n\n\n\n

          \n
          \n \n Display \\(a_p\\) with \\(p\\) up to:\n 50\n 250\n 1000\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n \n Display \\(a_n\\) with \\(n\\) up to:\n 50\n 250\n 1000\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n
          \n\n\n
          \n \n \n \n \n \n \n
          Significant digits:
          \n
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\n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n
          \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
          \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
          \n \\(2\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n\n \n\n \n\n \n\n
          \n\n \n\n \n\n \n\n
          \n \\(3\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(4\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
          \n \\(5\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(6\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(7\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n\n \n\n \n\n
          \n\n \n\n \n\n \n\n
          \n \\(8\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
          \n \\(9\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
          \n \\(10\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(11\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(12\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(13\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(14\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(15\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(16\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
          \n \\(17\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n \n\n \n\n \n\n \n\n
          \n\n \n\n \n\n \n\n
          \n \\(18\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
          \n \\(19\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(20\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(21\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(22\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(23\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
          \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
          \n \\(24\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
          \n \\(26\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(27\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(28\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(29\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n \\(30\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(31\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
          \n \\(33\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(34\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(35\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(36\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
          \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(38\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(39\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(40\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
          \n \\(41\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(42\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(43\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(44\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(45\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(46\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(47\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(48\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(49\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(50\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
          \n \\(51\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(52\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(53\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(54\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(55\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(56\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(57\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(58\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(59\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(60\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(61\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(62\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(63\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(64\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
          \n \\(65\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(66\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(67\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
          \n \\(69\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(70\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(71\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(72\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
          \n \\(73\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n \\(74\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(75\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(76\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(77\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(78\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(79\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
          \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
          \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(81\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
          \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(83\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(84\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(85\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(86\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(87\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(88\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(89\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(90\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(91\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(92\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(93\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(94\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(95\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(96\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(97\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n \\(98\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(99\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(100\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(101\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(102\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(103\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(104\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(105\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(106\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(107\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(108\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(109\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(110\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(111\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(112\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(113\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(114\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(115\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(116\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(117\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(118\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(119\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(120\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(121\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(123\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(124\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(125\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(126\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(127\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(128\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(129\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(130\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(131\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(132\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(133\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(134\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(135\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(136\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
          \n \\(137\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(138\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(139\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(140\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(141\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(142\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(143\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
          \n \\(145\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
          \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(147\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(148\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(149\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n \\(150\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(151\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(152\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(153\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(154\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(155\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(156\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(157\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n \\(158\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(159\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(160\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(161\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(162\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
          \n \\(163\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
          \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
          \n \\(164\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(165\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(166\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(167\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(168\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(169\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
          \n \\(170\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(171\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(172\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(174\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(175\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(176\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(177\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(178\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(179\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(180\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
          \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(182\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(183\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(184\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(185\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
          \n \\(186\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(187\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(188\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(189\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(190\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(191\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(192\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(193\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(194\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(195\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(196\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(198\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(199\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(200\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
          \n \\(201\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(202\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(203\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(204\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n \n 1.00000\n \n +\n \n 1.73205i\n
          \n \\(206\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(207\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(208\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(209\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(210\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(211\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(212\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(213\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(214\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(215\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(216\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(217\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(218\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(219\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(220\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(221\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(222\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(223\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(224\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(225\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
          \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(227\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(228\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(229\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(230\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(231\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(234\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(235\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(236\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(237\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(238\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(239\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(240\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(241\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n \\(242\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
          \n \\(243\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(244\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(245\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(246\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(247\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(248\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(249\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(250\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(251\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(252\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(253\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(254\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(255\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(256\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
          \n \\(257\\)\n \n 1.73205\n \n −\n \n 1.00000i\n \n 1.73205\n \n −\n \n 1.00000i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(258\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(259\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(260\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(261\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(262\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(263\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(264\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(265\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(266\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(267\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(268\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(270\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(271\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(272\\)\n \n 1.00000\n \n\n \n\n \n 1.00000\n \n\n \n\n
          \n \\(273\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(274\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(275\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(276\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(278\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(279\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(280\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(281\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(282\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(283\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(284\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(285\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(286\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(287\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(288\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
          \n \\(289\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
          \n \\(290\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
          \n \\(291\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(292\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
          \n \\(293\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n \\(294\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(295\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(296\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(297\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n \n 1.73205\n \n +\n \n 1.00000i\n
          \n \\(299\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(300\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(301\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(302\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(303\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(304\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(305\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n
          \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
          \n \\(307\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(308\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(309\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(310\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(311\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(312\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(313\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(314\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
          \n \\(315\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(316\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(317\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(318\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(319\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(321\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(322\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(323\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(324\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
          \n \\(325\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(326\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(327\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(328\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(329\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(330\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(331\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(332\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(333\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(334\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(335\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(336\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(337\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n \\(338\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(339\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(340\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(341\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(342\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(343\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(344\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(345\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(346\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(347\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(348\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(349\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(350\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(351\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(352\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(353\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(354\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(355\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(356\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(357\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(358\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(359\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(361\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
          \n \\(362\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
          \n \\(363\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(364\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(365\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
          \n \\(366\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(367\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(368\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(370\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
          \n \\(371\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(372\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(373\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(374\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(375\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(376\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(377\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(378\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(379\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(380\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(381\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(382\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(383\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(384\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(385\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(386\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(387\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(388\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(389\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n \\(390\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(391\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(392\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(393\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(394\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
          \n \\(395\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(396\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(397\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(398\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(399\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(400\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(401\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(402\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(403\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(404\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(405\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(406\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(407\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(408\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(409\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(410\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
          \n \\(411\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(412\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(413\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(414\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(415\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(416\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(417\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(418\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(419\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(420\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(421\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(422\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(423\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(424\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(425\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
          \n \\(426\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(427\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(428\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(429\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(430\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(431\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(432\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(433\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(434\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(435\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(436\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
          \n \\(437\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(438\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(439\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(440\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(441\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(442\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(443\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(444\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(445\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(446\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(447\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(448\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(449\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(450\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(451\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(452\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(453\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(454\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(455\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(456\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(457\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(458\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
          \n \\(459\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(460\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(461\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(462\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(463\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(464\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
          \n \\(465\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(466\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(467\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(468\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(469\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(470\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(471\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(472\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(473\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(474\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(475\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(476\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(477\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(478\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(479\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(480\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(481\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(483\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(484\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(485\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
          \n \\(486\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(487\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(489\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(490\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(491\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(492\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(494\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(495\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(496\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(497\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(498\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(499\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
          \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
          \n \\(500\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(501\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(502\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(503\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(504\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(505\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(506\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(507\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(508\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(509\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n \\(510\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(511\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(512\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(513\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(514\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n
          \n \\(515\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(516\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(517\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(518\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(519\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(520\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(521\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(522\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(523\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(524\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(525\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(526\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(527\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(528\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(529\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
          \n \\(530\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(531\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(532\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(533\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(534\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(535\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(536\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(537\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(538\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(539\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(540\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(542\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(543\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(544\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
          \n \\(545\\)\n \n −2.00000\n \n\n \n\n \n −2.00000\n \n\n \n\n
          \n \\(546\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(547\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(548\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(550\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(551\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(552\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(553\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(554\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(555\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(556\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(557\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(558\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(559\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(560\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(561\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(562\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(563\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(564\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n \n 1.00000\n \n +\n \n 1.73205i\n
          \n \\(566\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(567\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(568\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(569\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(570\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(571\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(572\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(573\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(574\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(575\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(576\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(577\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
          \n \\(579\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(580\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
          \n \\(581\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(582\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(583\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(584\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(585\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(586\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
          \n \\(587\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(588\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(589\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(590\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(591\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(592\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(593\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(594\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(595\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(596\\)\n \n −2.00000\n \n\n \n\n \n −2.00000\n \n\n \n\n
          \n \\(597\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(598\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(599\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(600\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(601\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(602\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(603\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(604\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(605\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(606\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(607\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(608\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(609\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(610\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
          \n \\(611\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(612\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
          \n \\(613\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(614\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(615\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(616\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(617\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(618\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(619\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(620\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(621\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(622\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(623\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(624\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(625\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
          \n \\(626\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(627\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(628\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
          \n \\(629\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(630\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(631\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(632\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(633\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(635\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(636\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(637\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(638\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(639\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(640\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(641\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n \\(642\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(643\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(644\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(645\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(646\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(647\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
          \n \\(649\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(650\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(651\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(652\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(653\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(654\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(655\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(656\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(658\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(659\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(660\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(661\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(662\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(663\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(664\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(665\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(666\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(667\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(668\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(669\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(670\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(671\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(672\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(673\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(674\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(675\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(676\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
          \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(678\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(679\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(681\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(682\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(683\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(684\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(685\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(686\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(687\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(688\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(689\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(690\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(691\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
          \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
          \n \\(692\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(693\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(694\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(695\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(696\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(697\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(698\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(699\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(700\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(701\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(702\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(703\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(704\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(705\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(706\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
          \n \\(707\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(708\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(709\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(710\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(711\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(712\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(713\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(714\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(715\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(716\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(717\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(718\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(719\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(720\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(721\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(722\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
          \n \\(723\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(724\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(725\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(726\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(727\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(728\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(729\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
          \n \\(730\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
          \n \\(731\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(732\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(733\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(734\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(735\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(736\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(737\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(738\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(739\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(740\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
          \n \\(741\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(742\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(743\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(744\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n \n 0.732051\n \n +\n \n 2.73205i\n
          \n \\(746\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(747\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(748\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(749\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(750\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(751\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(752\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(753\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(754\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(755\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(756\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(757\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(758\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(759\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(760\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(761\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(762\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(763\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(764\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(765\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(766\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(767\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(768\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(769\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(770\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(771\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(772\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(773\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n \\(774\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(775\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(777\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(778\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
          \n \\(779\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(780\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(781\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(782\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(783\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(784\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(785\\)\n \n −2.00000\n \n +\n \n 2.00000i\n \n −2.00000\n \n +\n \n 2.00000i\n
          \n \\(786\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(787\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(788\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(789\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(790\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(791\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(792\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(793\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(795\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(796\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(797\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(798\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(799\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
          \n \\(801\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(803\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(804\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(805\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(806\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(807\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(808\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(809\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(810\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
          \n \\(811\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(812\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(813\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(814\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(815\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(816\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(817\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(818\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
          \n \\(819\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(820\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
          \n \\(821\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(822\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(823\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(824\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(825\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(826\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(827\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(828\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(829\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n \\(830\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(831\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(832\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(833\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(834\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(835\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(836\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(837\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(838\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(839\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(840\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(841\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(842\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(843\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(844\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(846\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(847\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(848\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(849\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(850\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
          \n \\(851\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(852\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(853\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n \\(854\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(855\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(856\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(857\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(858\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(859\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(860\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(861\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(862\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(863\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(864\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(865\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
          \n \\(866\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
          \n \\(867\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(868\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(869\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(870\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(871\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(873\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(874\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(875\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(876\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(877\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(878\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(879\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(880\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(881\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(882\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(883\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(884\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(885\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(886\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(887\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(888\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(889\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(890\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(891\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(892\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(893\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(894\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(895\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(896\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(897\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(899\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(900\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
          \n \\(901\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(902\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(903\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(904\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(905\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
          \n \\(906\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(907\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(908\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(909\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(910\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(911\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(912\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(913\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(914\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
          \n \\(915\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(916\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
          \n \\(917\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(918\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(919\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(920\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(921\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(922\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(923\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(924\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(926\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(927\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(928\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(929\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(930\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(931\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(932\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(933\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(934\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(935\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(936\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(937\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(938\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(939\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(940\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(941\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(942\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(943\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(944\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(945\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(946\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(947\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(948\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(949\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(950\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(951\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(952\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(953\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(954\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(955\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(956\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(957\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(958\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(959\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(960\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(961\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
          \n \\(962\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(963\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(964\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
          \n \\(965\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
          \n \\(966\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(967\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
          \n \\(969\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(970\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
          \n \\(971\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(972\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(973\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(974\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(975\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(976\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(977\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(978\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(979\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(980\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(981\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(982\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(983\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(984\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(985\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
          \n \\(986\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(987\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(988\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(989\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(990\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(991\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(992\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(993\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(994\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(995\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(996\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(997\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(998\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(999\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n
          \n
          \n \n Display \\(a_p\\) with \\(p\\) up to:\n 50\n 250\n 1000\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n \n Display \\(a_n\\) with \\(n\\) up to:\n 50\n 250\n 1000\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n
          \n\n\n

          Twists

          \n\n\n\n\n \n \n \n \n \n \n \n\n\n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n
                 By twisting character
          CharParityOrdTypeTwistMinDim
          1.1even1trivial3332.1.bc.b.2027.14
          4.3odd2CM3332.1.bc.b.2027.14
          7.2even3inner3332.1.bc.b.667.14
          7.3odd668.1.f.a.55.1yes2
          7.4even33332.1.m.b.2843.12
          7.5odd63332.1.bc.c.667.14
          7.6odd23332.1.bc.c.2027.14
          17.13even4inner3332.1.bc.b.2223.14
          21.17even6612.1.l.a.55.12
          28.3even668.1.f.a.55.1yes2
          28.11odd63332.1.m.b.2843.12
          28.19even63332.1.bc.c.667.14
          28.23odd6inner3332.1.bc.b.667.14
          28.27even23332.1.bc.c.2027.14
          35.3even121700.1.n.a.599.12
          35.17even121700.1.n.b.599.12
          35.24odd61700.1.p.a.1551.12
          56.3even61088.1.p.a.191.12
          56.45odd61088.1.p.a.191.12
          68.47odd4inner3332.1.bc.b.2223.14
          84.59odd6612.1.l.a.55.12
          119.3even481156.1.g.b.155.18
          119.10even481156.1.g.b.179.18
          119.13odd43332.1.bc.c.2223.14
          119.24even481156.1.g.b.179.28
          119.30even12inner3332.1.bc.b.863.14
          119.31even481156.1.g.b.155.28
          119.38odd121156.1.f.b.251.12
          119.45even481156.1.g.b.399.18
          119.47odd123332.1.bc.c.863.14
          119.59odd241156.1.c.b.579.12
          119.66odd241156.1.d.a.1155.22
          119.73even481156.1.g.b.423.28
          119.80even481156.1.g.b.423.18
          119.81even123332.1.m.b.3039.12
          119.87odd241156.1.d.a.1155.12
          119.94odd241156.1.c.b.579.22
          119.101odd61156.1.f.b.327.12
          119.108even481156.1.g.b.399.28
          119.115odd1268.1.f.a.47.1✓2
          140.3odd121700.1.n.a.599.12
          140.59even61700.1.p.a.1551.12
          140.87odd121700.1.n.b.599.12
          357.353even12612.1.l.a.523.12
          476.3odd481156.1.g.b.155.18
          476.31odd481156.1.g.b.155.28
          476.47even123332.1.bc.c.863.14
          476.59even241156.1.c.b.579.12
          476.87even241156.1.d.a.1155.12
          476.115even1268.1.f.a.47.1✓2
          476.143odd481156.1.g.b.179.28
          476.199odd481156.1.g.b.423.18
          476.227odd481156.1.g.b.399.28
          476.251even43332.1.bc.c.2223.14
          476.283odd481156.1.g.b.399.18
          476.311odd481156.1.g.b.423.28
          476.319odd123332.1.m.b.3039.12
          476.339even61156.1.f.b.327.12
          476.367odd481156.1.g.b.179.18
          476.387odd12inner3332.1.bc.b.863.14
          476.395even121156.1.f.b.251.12
          476.423even241156.1.d.a.1155.22
          476.451even241156.1.c.b.579.22
          595.234odd121700.1.p.a.251.12
          595.353even121700.1.n.b.999.12
          595.472even121700.1.n.a.999.12
          952.115even121088.1.p.a.319.12
          952.829odd121088.1.p.a.319.12
          1428.1067odd12612.1.l.a.523.12
          2380.1067odd121700.1.n.a.999.12
          2380.1543odd121700.1.n.b.999.12
          2380.2019even121700.1.p.a.251.12
          \n
              
          \n\n\n\n\n \n \n \n \n \n \n \n\n\n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n
                  By twisted newform
          TwistMinDimCharParityOrdType
          68.1.f.a.47.1✓2119.115odd12
          68.1.f.a.47.1✓2476.115even12
          68.1.f.a.55.1yes27.3odd6
          68.1.f.a.55.1yes228.3even6
          612.1.l.a.55.1221.17even6
          612.1.l.a.55.1284.59odd6
          612.1.l.a.523.12357.353even12
          612.1.l.a.523.121428.1067odd12
          1088.1.p.a.191.1256.3even6
          1088.1.p.a.191.1256.45odd6
          1088.1.p.a.319.12952.115even12
          1088.1.p.a.319.12952.829odd12
          1156.1.c.b.579.12119.59odd24
          1156.1.c.b.579.12476.59even24
          1156.1.c.b.579.22119.94odd24
          1156.1.c.b.579.22476.451even24
          1156.1.d.a.1155.12119.87odd24
          1156.1.d.a.1155.12476.87even24
          1156.1.d.a.1155.22119.66odd24
          1156.1.d.a.1155.22476.423even24
          1156.1.f.b.251.12119.38odd12
          1156.1.f.b.251.12476.395even12
          1156.1.f.b.327.12119.101odd6
          1156.1.f.b.327.12476.339even6
          1156.1.g.b.155.18119.3even48
          1156.1.g.b.155.18476.3odd48
          1156.1.g.b.155.28119.31even48
          1156.1.g.b.155.28476.31odd48
          1156.1.g.b.179.18119.10even48
          1156.1.g.b.179.18476.367odd48
          1156.1.g.b.179.28119.24even48
          1156.1.g.b.179.28476.143odd48
          1156.1.g.b.399.18119.45even48
          1156.1.g.b.399.18476.283odd48
          1156.1.g.b.399.28119.108even48
          1156.1.g.b.399.28476.227odd48
          1156.1.g.b.423.18119.80even48
          1156.1.g.b.423.18476.199odd48
          1156.1.g.b.423.28119.73even48
          1156.1.g.b.423.28476.311odd48
          1700.1.n.a.599.1235.3even12
          1700.1.n.a.599.12140.3odd12
          1700.1.n.a.999.12595.472even12
          1700.1.n.a.999.122380.1067odd12
          1700.1.n.b.599.1235.17even12
          1700.1.n.b.599.12140.87odd12
          1700.1.n.b.999.12595.353even12
          1700.1.n.b.999.122380.1543odd12
          1700.1.p.a.251.12595.234odd12
          1700.1.p.a.251.122380.2019even12
          1700.1.p.a.1551.1235.24odd6
          1700.1.p.a.1551.12140.59even6
          3332.1.m.b.2843.127.4even3
          3332.1.m.b.2843.1228.11odd6
          3332.1.m.b.3039.12119.81even12
          3332.1.m.b.3039.12476.319odd12
          3332.1.bc.b.667.147.2even3inner
          3332.1.bc.b.667.1428.23odd6inner
          3332.1.bc.b.863.14119.30even12inner
          3332.1.bc.b.863.14476.387odd12inner
          3332.1.bc.b.2027.141.1even1trivial
          3332.1.bc.b.2027.144.3odd2CM
          3332.1.bc.b.2223.1417.13even4inner
          3332.1.bc.b.2223.1468.47odd4inner
          3332.1.bc.c.667.147.5odd6
          3332.1.bc.c.667.1428.19even6
          3332.1.bc.c.863.14119.47odd12
          3332.1.bc.c.863.14476.47even12
          3332.1.bc.c.2027.147.6odd2
          3332.1.bc.c.2027.1428.27even2
          3332.1.bc.c.2223.14119.13odd4
          3332.1.bc.c.2223.14476.251even4
          \n
          \n\n\n\n

          \n
          \n
          \n\n\n
          \n\n

          This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

          \n
          \n Contact\n ·\n Citation\n ·\n Acknowledgments\n ·\n Editorial Board\n ·\n Source\n ·\n SageMath version 10.1\n ·\n LMFDB Release 1.2.1\n
          \n
          \n\n"} +{"content_list": [[{"type": "paragraph", "raw_content": "
          \n Show commands:\n Magma\n / PariGP\n / SageMath
          ", "content": [{"c": "Show commands: Magma/ PariGP/ SageMath", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          [N,k,chi] = [3332,1,Mod(667,3332)]
          mf = mfinit([N,k,chi],0)
          lf = mfeigenbasis(mf)
          ", "content": [{"c": "[N,k,chi] = [3332,1,Mod(667,3332)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          from sage.modular.dirichlet import DirichletCharacter
          H = DirichletGroup(3332, base_ring=CyclotomicField(12))
          chi = DirichletCharacter(H, H._module([6, 4, 9]))
          N = Newforms(chi, 1, names=\"a\")
          ", "content": [{"c": "from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3332, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 4, 9])) N = Newforms(chi, 1, names=\"a\")", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
          chi := DirichletCharacter(\"3332.667\");
          S:= CuspForms(chi, 1);
          N := Newforms(S);
          ", "content": [{"c": "//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter(\"3332.667\"); S:= CuspForms(chi, 1); N := Newforms(S);", "t": "text"}]}, {"type": "table", "raw_content": "
          Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
          Weight: \\( k \\) \\(=\\)\\( 1 \\)
          Character orbit: \\([\\chi]\\) \\(=\\)3332.bc (of order \\(12\\), degree \\(4\\), not minimal)
          ", "content": {"html": "
          Level\\( N \\)\\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
          Weight\\( k \\)\\(=\\)\\( 1 \\)
          Character orbit\\([\\chi]\\)\\(=\\)3332.bc<br>order<br>degree<br>minimal
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

          Newform invariants

          ", "content": {"title_content": "Newform invariants", "level": "2"}}, {"type": "paragraph", "raw_content": "
          sage:\u00a0f = N[0] # Warning: the index may be different
          ", "content": [{"c": "sage:\u00a0f = N[0] # Warning: the index may be different", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          gp:\u00a0f = lf[1] \\\\ Warning: the index may be different
          ", "content": [{"c": "gp:\u00a0f = lf[1] \\\\ Warning: the index may be different", "t": "text"}]}, {"type": "table", "raw_content": "
          Self dual: no
          Analytic conductor: \\(1.66288462209\\)
          Analytic rank: \\(0\\)
          Dimension: \\(4\\)
          Coefficient field: \\(\\Q(\\zeta_{12})\\)
          gp:\u00a0f.mod \\\\ as an extension of the character field
          Defining polynomial: \\( x^{4} - x^{2} + 1 \\)\"Copy\"Toggle
          Coefficient ring: \\(\\Z[a_1, a_2]\\)
          Coefficient ring index: \\( 1 \\)
          Twist minimal: no (minimal twist has level 68)
          Projective image:\\(D_{4}\\)
          Projective field:Galois closure of 4.2.19652.1
          Artin image:$C_4\\wr C_2\\times C_6$
          Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
          ", "content": {"html": "
          Self dualno
          Analytic conductor\\(1.66288462209\\)
          Analytic rank\\(0\\)
          Dimension\\(4\\)
          Coefficient field\\(\\Q(\\zeta_{12})\\)
          gp:\u00a0f.mod \\\\ as an extension of the character field
          Defining polynomial\\( x^{4} - x^{2} + 1 \\)
          Coefficient ring\\(\\Z[a_1, a_2]\\)
          Coefficient ring index\\( 1 \\)
          Twist minimalno (minimal twist has level 68)
          Projective image\\(D_{4}\\)
          Projective fieldGalois closure of<br>4.2.19652.1
          Artin image$C_4\\wr C_2\\times C_6$
          Artin fieldGalois closure of<br>\\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
          ", "is_complex": true, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

          Embedding invariants

          ", "content": {"title_content": "Embedding invariants", "level": "2"}}, {"type": "table", "raw_content": "
          Embedding label 2027.1
          Root\\(0.866025 - 0.500000i\\) of defining polynomial
          Character\\(\\chi\\)\\(=\\)3332.2027
          Dual form3332.1.bc.b.863.1
          ", "content": {"html": "
          Embedding label2027.1
          Root\\(0.866025 - 0.500000i\\) of defining polynomial
          Character\\(\\chi\\)\\(=\\)3332.2027
          Dual form3332.1.bc.b.863.1
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "
          sage:\u00a0f.q_expansion() # note that sage often uses an isomorphic number field
          ", "content": [{"c": "sage:\u00a0f.q_expansion() # note that sage often uses an isomorphic number field", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          gp:\u00a0mfcoefs(f, 20)
          ", "content": [{"c": "gp:\u00a0mfcoefs(f, 20)", "t": "text"}]}, {"type": "table", "raw_content": "
          \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
          \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\"Copy\"Toggle
          ", "content": {"html": "
          \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
          \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

          Character values

          ", "content": {"title_content": "Character values", "level": "2"}}, {"type": "paragraph", "raw_content": "

          We give the values of \\chi on generators for \\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times.

          ", "content": [{"c": "We give the values of", "t": "text"}, {"c": "\\chi", "t": "equation-inline"}, {"c": "on generators for", "t": "text"}, {"c": "\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "table", "raw_content": "
          \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
          \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
          ", "content": {"html": "
          \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
          \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

          Coefficient data

          ", "content": {"title_content": "Coefficient data", "level": "2"}}, {"type": "paragraph", "raw_content": "

          For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the\nSatake parameters \\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).

          ", "content": [{"c": "For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the Satake parameters\\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "table", "raw_content": "
          \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
          \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
          \n \\(2\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(3\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(4\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(5\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(6\\)\n \n 0\n \n 0\n
          \n \\(7\\)\n \n 0\n \n 0\n
          \n \\(8\\)\n \n 1.00000i\n 1.00000i
          \n \\(9\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(10\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(11\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(12\\)\n \n 0\n \n 0\n
          \n \\(13\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(14\\)\n \n 0\n \n 0\n
          \n \\(15\\)\n \n 0\n \n 0\n
          \n \\(16\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(17\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(18\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(19\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(20\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(21\\)\n \n 0\n \n 0\n
          \n \\(22\\)\n \n 0\n \n 0\n
          \n \\(23\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(24\\)\n \n 0\n \n 0\n
          \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(26\\)\n \n 0\n \n 0\n
          \n \\(27\\)\n \n 0\n \n 0\n
          \n \\(28\\)\n \n 0\n \n 0\n
          \n \\(29\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(30\\)\n \n 0\n \n 0\n
          \n \\(31\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(33\\)\n \n 0\n \n 0\n
          \n \\(34\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(35\\)\n \n 0\n \n 0\n
          \n \\(36\\)\n \n 1.00000i\n 1.00000i
          \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(38\\)\n \n 0\n \n 0\n
          \n \\(39\\)\n \n 0\n \n 0\n
          \n \\(40\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(41\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(42\\)\n \n 0\n \n 0\n
          \n \\(43\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(44\\)\n \n 0\n \n 0\n
          \n \\(45\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(46\\)\n \n 0\n \n 0\n
          \n \\(47\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(48\\)\n \n 0\n \n 0\n
          \n \\(49\\)\n \n 0\n \n 0\n
          \n \\(50\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(51\\)\n \n 0\n \n 0\n
          \n \\(52\\)\n \n 0\n \n 0\n
          \n \\(53\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(54\\)\n \n 0\n \n 0\n
          \n \\(55\\)\n \n 0\n \n 0\n
          \n \\(56\\)\n \n 0\n \n 0\n
          \n \\(57\\)\n \n 0\n \n 0\n
          \n \\(58\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(59\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(60\\)\n \n 0\n \n 0\n
          \n \\(61\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(62\\)\n \n 0\n \n 0\n
          \n \\(63\\)\n \n 0\n \n 0\n
          \n \\(64\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(65\\)\n \n 0\n \n 0\n
          \n \\(66\\)\n \n 0\n \n 0\n
          \n \\(67\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(69\\)\n \n 0\n \n 0\n
          \n \\(70\\)\n \n 0\n \n 0\n
          \n \\(71\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(72\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(73\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(74\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(75\\)\n \n 0\n \n 0\n
          \n \\(76\\)\n \n 0\n \n 0\n
          \n \\(77\\)\n \n 0\n \n 0\n
          \n \\(78\\)\n \n 0\n \n 0\n
          \n \\(79\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(81\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(83\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(84\\)\n \n 0\n \n 0\n
          \n \\(85\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(86\\)\n \n 0\n \n 0\n
          \n \\(87\\)\n \n 0\n \n 0\n
          \n \\(88\\)\n \n 0\n \n 0\n
          \n \\(89\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(90\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(91\\)\n \n 0\n \n 0\n
          \n \\(92\\)\n \n 0\n \n 0\n
          \n \\(93\\)\n \n 0\n \n 0\n
          \n \\(94\\)\n \n 0\n \n 0\n
          \n \\(95\\)\n \n 0\n \n 0\n
          \n \\(96\\)\n \n 0\n \n 0\n
          \n \\(97\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(98\\)\n \n 0\n \n 0\n
          \n \\(99\\)\n \n 0\n \n 0\n
          \n \\(100\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(101\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(102\\)\n \n 0\n \n 0\n
          \n \\(103\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(104\\)\n \n 0\n \n 0\n
          \n \\(105\\)\n \n 0\n \n 0\n
          \n \\(106\\)\n \n 0\n \n 0\n
          \n \\(107\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(108\\)\n \n 0\n \n 0\n
          \n \\(109\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(110\\)\n \n 0\n \n 0\n
          \n \\(111\\)\n \n 0\n \n 0\n
          \n \\(112\\)\n \n 0\n \n 0\n
          \n \\(113\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(114\\)\n \n 0\n \n 0\n
          \n \\(115\\)\n \n 0\n \n 0\n
          \n \\(116\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(117\\)\n \n 0\n \n 0\n
          \n \\(118\\)\n \n 0\n \n 0\n
          \n \\(119\\)\n \n 0\n \n 0\n
          \n \\(120\\)\n \n 0\n \n 0\n
          \n \\(121\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(123\\)\n \n 0\n \n 0\n
          \n \\(124\\)\n \n 0\n \n 0\n
          \n \\(125\\)\n \n 0\n \n 0\n
          \n \\(126\\)\n \n 0\n \n 0\n
          \n \\(127\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(128\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(129\\)\n \n 0\n \n 0\n
          \n \\(130\\)\n \n 0\n \n 0\n
          \n \\(131\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(132\\)\n \n 0\n \n 0\n
          \n \\(133\\)\n \n 0\n \n 0\n
          \n \\(134\\)\n \n 0\n \n 0\n
          \n \\(135\\)\n \n 0\n \n 0\n
          \n \\(136\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(137\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(138\\)\n \n 0\n \n 0\n
          \n \\(139\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(140\\)\n \n 0\n \n 0\n
          \n \\(141\\)\n \n 0\n \n 0\n
          \n \\(142\\)\n \n 0\n \n 0\n
          \n \\(143\\)\n \n 0\n \n 0\n
          \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(145\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(147\\)\n \n 0\n \n 0\n
          \n \\(148\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(149\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \\(150\\)\n \n 0\n \n 0\n
          \n \\(151\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(152\\)\n \n 0\n \n 0\n
          \n \\(153\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(154\\)\n \n 0\n \n 0\n
          \n \\(155\\)\n \n 0\n \n 0\n
          \n \\(156\\)\n \n 0\n \n 0\n
          \n \\(157\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(158\\)\n \n 0\n \n 0\n
          \n \\(159\\)\n \n 0\n \n 0\n
          \n \\(160\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(161\\)\n \n 0\n \n 0\n
          \n \\(162\\)\n \n 1.00000i\n 1.00000i
          \n \\(163\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(164\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(165\\)\n \n 0\n \n 0\n
          \n \\(166\\)\n \n 0\n \n 0\n
          \n \\(167\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(168\\)\n \n 0\n \n 0\n
          \n \\(169\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(170\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(171\\)\n \n 0\n \n 0\n
          \n \\(172\\)\n \n 0\n \n 0\n
          \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(174\\)\n \n 0\n \n 0\n
          \n \\(175\\)\n \n 0\n \n 0\n
          \n \\(176\\)\n \n 0\n \n 0\n
          \n \\(177\\)\n \n 0\n \n 0\n
          \n \\(178\\)\n \n 0\n \n 0\n
          \n \\(179\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(180\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(182\\)\n \n 0\n \n 0\n
          \n \\(183\\)\n \n 0\n \n 0\n
          \n \\(184\\)\n \n 0\n \n 0\n
          \n \\(185\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(186\\)\n \n 0\n \n 0\n
          \n \\(187\\)\n \n 0\n \n 0\n
          \n \\(188\\)\n \n 0\n \n 0\n
          \n \\(189\\)\n \n 0\n \n 0\n
          \n \\(190\\)\n \n 0\n \n 0\n
          \n \\(191\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(192\\)\n \n 0\n \n 0\n
          \n \\(193\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(194\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(195\\)\n \n 0\n \n 0\n
          \n \\(196\\)\n \n 0\n \n 0\n
          \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(198\\)\n \n 0\n \n 0\n
          \n \\(199\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(200\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(201\\)\n \n 0\n \n 0\n
          \n \\(202\\)\n \n 0\n \n 0\n
          \n \\(203\\)\n \n 0\n \n 0\n
          \n \\(204\\)\n \n 0\n \n 0\n
          \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
          \n \\(206\\)\n \n 0\n \n 0\n
          \n \\(207\\)\n \n 0\n \n 0\n
          \n \\(208\\)\n \n 0\n \n 0\n
          \n \\(209\\)\n \n 0\n \n 0\n
          \n \\(210\\)\n \n 0\n \n 0\n
          \n \\(211\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(212\\)\n \n 0\n \n 0\n
          \n \\(213\\)\n \n 0\n \n 0\n
          \n \\(214\\)\n \n 0\n \n 0\n
          \n \\(215\\)\n \n 0\n \n 0\n
          \n \\(216\\)\n \n 0\n \n 0\n
          \n \\(217\\)\n \n 0\n \n 0\n
          \n \\(218\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(219\\)\n \n 0\n \n 0\n
          \n \\(220\\)\n \n 0\n \n 0\n
          \n \\(221\\)\n \n 0\n \n 0\n
          \n \\(222\\)\n \n 0\n \n 0\n
          \n \\(223\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(224\\)\n \n 0\n \n 0\n
          \n \\(225\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(227\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(228\\)\n \n 0\n \n 0\n
          \n \\(229\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(230\\)\n \n 0\n \n 0\n
          \n \\(231\\)\n \n 0\n \n 0\n
          \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(234\\)\n \n 0\n \n 0\n
          \n \\(235\\)\n \n 0\n \n 0\n
          \n \\(236\\)\n \n 0\n \n 0\n
          \n \\(237\\)\n \n 0\n \n 0\n
          \n \\(238\\)\n \n 0\n \n 0\n
          \n \\(239\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(240\\)\n \n 0\n \n 0\n
          \n \\(241\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(242\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(243\\)\n \n 0\n \n 0\n
          \n \\(244\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(245\\)\n \n 0\n \n 0\n
          \n \\(246\\)\n \n 0\n \n 0\n
          \n \\(247\\)\n \n 0\n \n 0\n
          \n \\(248\\)\n \n 0\n \n 0\n
          \n \\(249\\)\n \n 0\n \n 0\n
          \n \\(250\\)\n \n 0\n \n 0\n
          \n \\(251\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(252\\)\n \n 0\n \n 0\n
          \n \\(253\\)\n \n 0\n \n 0\n
          \n \\(254\\)\n \n 0\n \n 0\n
          \n \\(255\\)\n \n 0\n \n 0\n
          \n \\(256\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(257\\)\n \n 1.73205\n \n \u2212\n \n 1.00000i\n 1.73205\n \n \u2212\n \n 1.00000i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(258\\)\n \n 0\n \n 0\n
          \n \\(259\\)\n \n 0\n \n 0\n
          \n \\(260\\)\n \n 0\n \n 0\n
          \n \\(261\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(262\\)\n \n 0\n \n 0\n
          \n \\(263\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(264\\)\n \n 0\n \n 0\n
          \n \\(265\\)\n \n 0\n \n 0\n
          \n \\(266\\)\n \n 0\n \n 0\n
          \n \\(267\\)\n \n 0\n \n 0\n
          \n \\(268\\)\n \n 0\n \n 0\n
          \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(270\\)\n \n 0\n \n 0\n
          \n \\(271\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(272\\)\n \n 1.00000\n \n 1.00000\n
          \n \\(273\\)\n \n 0\n \n 0\n
          \n \\(274\\)\n \n 0\n \n 0\n
          \n \\(275\\)\n \n 0\n \n 0\n
          \n \\(276\\)\n \n 0\n \n 0\n
          \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(278\\)\n \n 0\n \n 0\n
          \n \\(279\\)\n \n 0\n \n 0\n
          \n \\(280\\)\n \n 0\n \n 0\n
          \n \\(281\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(282\\)\n \n 0\n \n 0\n
          \n \\(283\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(284\\)\n \n 0\n \n 0\n
          \n \\(285\\)\n \n 0\n \n 0\n
          \n \\(286\\)\n \n 0\n \n 0\n
          \n \\(287\\)\n \n 0\n \n 0\n
          \n \\(288\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(289\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(290\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(291\\)\n \n 0\n \n 0\n
          \n \\(292\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(293\\)\n \n 2.00000\n \n 2.00000\n \n 1.00000\n \n \\(0\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(294\\)\n \n 0\n \n 0\n
          \n \\(295\\)\n \n 0\n \n 0\n
          \n \\(296\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(297\\)\n \n 0\n \n 0\n
          \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n 1.73205\n \n +\n \n 1.00000i
          \n \\(299\\)\n \n 0\n \n 0\n
          \n \\(300\\)\n \n 0\n \n 0\n
          \n \\(301\\)\n \n 0\n \n 0\n
          \n \\(302\\)\n \n 0\n \n 0\n
          \n \\(303\\)\n \n 0\n \n 0\n
          \n \\(304\\)\n \n 0\n \n 0\n
          \n \\(305\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
          \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(307\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(308\\)\n \n 0\n \n 0\n
          \n \\(309\\)\n \n 0\n \n 0\n
          \n \\(310\\)\n \n 0\n \n 0\n
          \n \\(311\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(312\\)\n \n 0\n \n 0\n
          \n \\(313\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(314\\)\n \n 2.00000i\n 2.00000i
          \n \\(315\\)\n \n 0\n \n 0\n
          \n \\(316\\)\n \n 0\n \n 0\n
          \n \\(317\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(318\\)\n \n 0\n \n 0\n
          \n \\(319\\)\n \n 0\n \n 0\n
          \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(321\\)\n \n 0\n \n 0\n
          \n \\(322\\)\n \n 0\n \n 0\n
          \n \\(323\\)\n \n 0\n \n 0\n
          \n \\(324\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(325\\)\n \n 0\n \n 0\n
          \n \\(326\\)\n \n 0\n \n 0\n
          \n \\(327\\)\n \n 0\n \n 0\n
          \n \\(328\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(329\\)\n \n 0\n \n 0\n
          \n \\(330\\)\n \n 0\n \n 0\n
          \n \\(331\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(332\\)\n \n 0\n \n 0\n
          \n \\(333\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(334\\)\n \n 0\n \n 0\n
          \n \\(335\\)\n \n 0\n \n 0\n
          \n \\(336\\)\n \n 0\n \n 0\n
          \n \\(337\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(338\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(339\\)\n \n 0\n \n 0\n
          \n \\(340\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(341\\)\n \n 0\n \n 0\n
          \n \\(342\\)\n \n 0\n \n 0\n
          \n \\(343\\)\n \n 0\n \n 0\n
          \n \\(344\\)\n \n 0\n \n 0\n
          \n \\(345\\)\n \n 0\n \n 0\n
          \n \\(346\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(347\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(348\\)\n \n 0\n \n 0\n
          \n \\(349\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(350\\)\n \n 0\n \n 0\n
          \n \\(351\\)\n \n 0\n \n 0\n
          \n \\(352\\)\n \n 0\n \n 0\n
          \n \\(353\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(354\\)\n \n 0\n \n 0\n
          \n \\(355\\)\n \n 0\n \n 0\n
          \n \\(356\\)\n \n 0\n \n 0\n
          \n \\(357\\)\n \n 0\n \n 0\n
          \n \\(358\\)\n \n 0\n \n 0\n
          \n \\(359\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(361\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(362\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(363\\)\n \n 0\n \n 0\n
          \n \\(364\\)\n \n 0\n \n 0\n
          \n \\(365\\)\n \n 2.00000i\n 2.00000i
          \n \\(366\\)\n \n 0\n \n 0\n
          \n \\(367\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(368\\)\n \n 0\n \n 0\n
          \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(370\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(371\\)\n \n 0\n \n 0\n
          \n \\(372\\)\n \n 0\n \n 0\n
          \n \\(373\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(374\\)\n \n 0\n \n 0\n
          \n \\(375\\)\n \n 0\n \n 0\n
          \n \\(376\\)\n \n 0\n \n 0\n
          \n \\(377\\)\n \n 0\n \n 0\n
          \n \\(378\\)\n \n 0\n \n 0\n
          \n \\(379\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(380\\)\n \n 0\n \n 0\n
          \n \\(381\\)\n \n 0\n \n 0\n
          \n \\(382\\)\n \n 0\n \n 0\n
          \n \\(383\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(384\\)\n \n 0\n \n 0\n
          \n \\(385\\)\n \n 0\n \n 0\n
          \n \\(386\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(387\\)\n \n 0\n \n 0\n
          \n \\(388\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(389\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \\(390\\)\n \n 0\n \n 0\n
          \n \\(391\\)\n \n 0\n \n 0\n
          \n \\(392\\)\n \n 0\n \n 0\n
          \n \\(393\\)\n \n 0\n \n 0\n
          \n \\(394\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(395\\)\n \n 0\n \n 0\n
          \n \\(396\\)\n \n 0\n \n 0\n
          \n \\(397\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(398\\)\n \n 0\n \n 0\n
          \n \\(399\\)\n \n 0\n \n 0\n
          \n \\(400\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(401\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(402\\)\n \n 0\n \n 0\n
          \n \\(403\\)\n \n 0\n \n 0\n
          \n \\(404\\)\n \n 0\n \n 0\n
          \n \\(405\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(406\\)\n \n 0\n \n 0\n
          \n \\(407\\)\n \n 0\n \n 0\n
          \n \\(408\\)\n \n 0\n \n 0\n
          \n \\(409\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(410\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(411\\)\n \n 0\n \n 0\n
          \n \\(412\\)\n \n 0\n \n 0\n
          \n \\(413\\)\n \n 0\n \n 0\n
          \n \\(414\\)\n \n 0\n \n 0\n
          \n \\(415\\)\n \n 0\n \n 0\n
          \n \\(416\\)\n \n 0\n \n 0\n
          \n \\(417\\)\n \n 0\n \n 0\n
          \n \\(418\\)\n \n 0\n \n 0\n
          \n \\(419\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(420\\)\n \n 0\n \n 0\n
          \n \\(421\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(422\\)\n \n 0\n \n 0\n
          \n \\(423\\)\n \n 0\n \n 0\n
          \n \\(424\\)\n \n 0\n \n 0\n
          \n \\(425\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(426\\)\n \n 0\n \n 0\n
          \n \\(427\\)\n \n 0\n \n 0\n
          \n \\(428\\)\n \n 0\n \n 0\n
          \n \\(429\\)\n \n 0\n \n 0\n
          \n \\(430\\)\n \n 0\n \n 0\n
          \n \\(431\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(432\\)\n \n 0\n \n 0\n
          \n \\(433\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(434\\)\n \n 0\n \n 0\n
          \n \\(435\\)\n \n 0\n \n 0\n
          \n \\(436\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(437\\)\n \n 0\n \n 0\n
          \n \\(438\\)\n \n 0\n \n 0\n
          \n \\(439\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(440\\)\n \n 0\n \n 0\n
          \n \\(441\\)\n \n 0\n \n 0\n
          \n \\(442\\)\n \n 0\n \n 0\n
          \n \\(443\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(444\\)\n \n 0\n \n 0\n
          \n \\(445\\)\n \n 0\n \n 0\n
          \n \\(446\\)\n \n 0\n \n 0\n
          \n \\(447\\)\n \n 0\n \n 0\n
          \n \\(448\\)\n \n 0\n \n 0\n
          \n \\(449\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(450\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(451\\)\n \n 0\n \n 0\n
          \n \\(452\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(453\\)\n \n 0\n \n 0\n
          \n \\(454\\)\n \n 0\n \n 0\n
          \n \\(455\\)\n \n 0\n \n 0\n
          \n \\(456\\)\n \n 0\n \n 0\n
          \n \\(457\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(458\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(459\\)\n \n 0\n \n 0\n
          \n \\(460\\)\n \n 0\n \n 0\n
          \n \\(461\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(462\\)\n \n 0\n \n 0\n
          \n \\(463\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(464\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(465\\)\n \n 0\n \n 0\n
          \n \\(466\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(467\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(468\\)\n \n 0\n \n 0\n
          \n \\(469\\)\n \n 0\n \n 0\n
          \n \\(470\\)\n \n 0\n \n 0\n
          \n \\(471\\)\n \n 0\n \n 0\n
          \n \\(472\\)\n \n 0\n \n 0\n
          \n \\(473\\)\n \n 0\n \n 0\n
          \n \\(474\\)\n \n 0\n \n 0\n
          \n \\(475\\)\n \n 0\n \n 0\n
          \n \\(476\\)\n \n 0\n \n 0\n
          \n \\(477\\)\n \n 0\n \n 0\n
          \n \\(478\\)\n \n 0\n \n 0\n
          \n \\(479\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(480\\)\n \n 0\n \n 0\n
          \n \\(481\\)\n \n 0\n \n 0\n
          \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(483\\)\n \n 0\n \n 0\n
          \n \\(484\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(485\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(486\\)\n \n 0\n \n 0\n
          \n \\(487\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(489\\)\n \n 0\n \n 0\n
          \n \\(490\\)\n \n 0\n \n 0\n
          \n \\(491\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(492\\)\n \n 0\n \n 0\n
          \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(494\\)\n \n 0\n \n 0\n
          \n \\(495\\)\n \n 0\n \n 0\n
          \n \\(496\\)\n \n 0\n \n 0\n
          \n \\(497\\)\n \n 0\n \n 0\n
          \n \\(498\\)\n \n 0\n \n 0\n
          \n \\(499\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(500\\)\n \n 0\n \n 0\n
          \n \\(501\\)\n \n 0\n \n 0\n
          \n \\(502\\)\n \n 0\n \n 0\n
          \n \\(503\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(504\\)\n \n 0\n \n 0\n
          \n \\(505\\)\n \n 0\n \n 0\n
          \n \\(506\\)\n \n 0\n \n 0\n
          \n \\(507\\)\n \n 0\n \n 0\n
          \n \\(508\\)\n \n 0\n \n 0\n
          \n \\(509\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \\(510\\)\n \n 0\n \n 0\n
          \n \\(511\\)\n \n 0\n \n 0\n
          \n \\(512\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(513\\)\n \n 0\n \n 0\n
          \n \\(514\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
          \n \\(515\\)\n \n 0\n \n 0\n
          \n \\(516\\)\n \n 0\n \n 0\n
          \n \\(517\\)\n \n 0\n \n 0\n
          \n \\(518\\)\n \n 0\n \n 0\n
          \n \\(519\\)\n \n 0\n \n 0\n
          \n \\(520\\)\n \n 0\n \n 0\n
          \n \\(521\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(522\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(523\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(524\\)\n \n 0\n \n 0\n
          \n \\(525\\)\n \n 0\n \n 0\n
          \n \\(526\\)\n \n 0\n \n 0\n
          \n \\(527\\)\n \n 0\n \n 0\n
          \n \\(528\\)\n \n 0\n \n 0\n
          \n \\(529\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(530\\)\n \n 0\n \n 0\n
          \n \\(531\\)\n \n 0\n \n 0\n
          \n \\(532\\)\n \n 0\n \n 0\n
          \n \\(533\\)\n \n 0\n \n 0\n
          \n \\(534\\)\n \n 0\n \n 0\n
          \n \\(535\\)\n \n 0\n \n 0\n
          \n \\(536\\)\n \n 0\n \n 0\n
          \n \\(537\\)\n \n 0\n \n 0\n
          \n \\(538\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(539\\)\n \n 0\n \n 0\n
          \n \\(540\\)\n \n 0\n \n 0\n
          \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(542\\)\n \n 0\n \n 0\n
          \n \\(543\\)\n \n 0\n \n 0\n
          \n \\(544\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(545\\)\n \n \u22122.00000\n \n \u22122.00000\n
          \n \\(546\\)\n \n 0\n \n 0\n
          \n \\(547\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(548\\)\n \n 0\n \n 0\n
          \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(550\\)\n \n 0\n \n 0\n
          \n \\(551\\)\n \n 0\n \n 0\n
          \n \\(552\\)\n \n 0\n \n 0\n
          \n \\(553\\)\n \n 0\n \n 0\n
          \n \\(554\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(555\\)\n \n 0\n \n 0\n
          \n \\(556\\)\n \n 0\n \n 0\n
          \n \\(557\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(558\\)\n \n 0\n \n 0\n
          \n \\(559\\)\n \n 0\n \n 0\n
          \n \\(560\\)\n \n 0\n \n 0\n
          \n \\(561\\)\n \n 0\n \n 0\n
          \n \\(562\\)\n \n 0\n \n 0\n
          \n \\(563\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(564\\)\n \n 0\n \n 0\n
          \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
          \n \\(566\\)\n \n 0\n \n 0\n
          \n \\(567\\)\n \n 0\n \n 0\n
          \n \\(568\\)\n \n 0\n \n 0\n
          \n \\(569\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(570\\)\n \n 0\n \n 0\n
          \n \\(571\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(572\\)\n \n 0\n \n 0\n
          \n \\(573\\)\n \n 0\n \n 0\n
          \n \\(574\\)\n \n 0\n \n 0\n
          \n \\(575\\)\n \n 0\n \n 0\n
          \n \\(576\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(577\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(579\\)\n \n 0\n \n 0\n
          \n \\(580\\)\n \n 2.00000i\n 2.00000i
          \n \\(581\\)\n \n 0\n \n 0\n
          \n \\(582\\)\n \n 0\n \n 0\n
          \n \\(583\\)\n \n 0\n \n 0\n
          \n \\(584\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(585\\)\n \n 0\n \n 0\n
          \n \\(586\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(587\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(588\\)\n \n 0\n \n 0\n
          \n \\(589\\)\n \n 0\n \n 0\n
          \n \\(590\\)\n \n 0\n \n 0\n
          \n \\(591\\)\n \n 0\n \n 0\n
          \n \\(592\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(593\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(594\\)\n \n 0\n \n 0\n
          \n \\(595\\)\n \n 0\n \n 0\n
          \n \\(596\\)\n \n \u22122.00000\n \n \u22122.00000\n
          \n \\(597\\)\n \n 0\n \n 0\n
          \n \\(598\\)\n \n 0\n \n 0\n
          \n \\(599\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(600\\)\n \n 0\n \n 0\n
          \n \\(601\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(602\\)\n \n 0\n \n 0\n
          \n \\(603\\)\n \n 0\n \n 0\n
          \n \\(604\\)\n \n 0\n \n 0\n
          \n \\(605\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(606\\)\n \n 0\n \n 0\n
          \n \\(607\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(608\\)\n \n 0\n \n 0\n
          \n \\(609\\)\n \n 0\n \n 0\n
          \n \\(610\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(611\\)\n \n 0\n \n 0\n
          \n \\(612\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(613\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(614\\)\n \n 0\n \n 0\n
          \n \\(615\\)\n \n 0\n \n 0\n
          \n \\(616\\)\n \n 0\n \n 0\n
          \n \\(617\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(618\\)\n \n 0\n \n 0\n
          \n \\(619\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(620\\)\n \n 0\n \n 0\n
          \n \\(621\\)\n \n 0\n \n 0\n
          \n \\(622\\)\n \n 0\n \n 0\n
          \n \\(623\\)\n \n 0\n \n 0\n
          \n \\(624\\)\n \n 0\n \n 0\n
          \n \\(625\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(626\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(627\\)\n \n 0\n \n 0\n
          \n \\(628\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(629\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(630\\)\n \n 0\n \n 0\n
          \n \\(631\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(632\\)\n \n 0\n \n 0\n
          \n \\(633\\)\n \n 0\n \n 0\n
          \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(635\\)\n \n 0\n \n 0\n
          \n \\(636\\)\n \n 0\n \n 0\n
          \n \\(637\\)\n \n 0\n \n 0\n
          \n \\(638\\)\n \n 0\n \n 0\n
          \n \\(639\\)\n \n 0\n \n 0\n
          \n \\(640\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(641\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(642\\)\n \n 0\n \n 0\n
          \n \\(643\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(644\\)\n \n 0\n \n 0\n
          \n \\(645\\)\n \n 0\n \n 0\n
          \n \\(646\\)\n \n 0\n \n 0\n
          \n \\(647\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(649\\)\n \n 0\n \n 0\n
          \n \\(650\\)\n \n 0\n \n 0\n
          \n \\(651\\)\n \n 0\n \n 0\n
          \n \\(652\\)\n \n 0\n \n 0\n
          \n \\(653\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(654\\)\n \n 0\n \n 0\n
          \n \\(655\\)\n \n 0\n \n 0\n
          \n \\(656\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(658\\)\n \n 0\n \n 0\n
          \n \\(659\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(660\\)\n \n 0\n \n 0\n
          \n \\(661\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(662\\)\n \n 0\n \n 0\n
          \n \\(663\\)\n \n 0\n \n 0\n
          \n \\(664\\)\n \n 0\n \n 0\n
          \n \\(665\\)\n \n 0\n \n 0\n
          \n \\(666\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(667\\)\n \n 0\n \n 0\n
          \n \\(668\\)\n \n 0\n \n 0\n
          \n \\(669\\)\n \n 0\n \n 0\n
          \n \\(670\\)\n \n 0\n \n 0\n
          \n \\(671\\)\n \n 0\n \n 0\n
          \n \\(672\\)\n \n 0\n \n 0\n
          \n \\(673\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(674\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(675\\)\n \n 0\n \n 0\n
          \n \\(676\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(678\\)\n \n 0\n \n 0\n
          \n \\(679\\)\n \n 0\n \n 0\n
          \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(681\\)\n \n 0\n \n 0\n
          \n \\(682\\)\n \n 0\n \n 0\n
          \n \\(683\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(684\\)\n \n 0\n \n 0\n
          \n \\(685\\)\n \n 0\n \n 0\n
          \n \\(686\\)\n \n 0\n \n 0\n
          \n \\(687\\)\n \n 0\n \n 0\n
          \n \\(688\\)\n \n 0\n \n 0\n
          \n \\(689\\)\n \n 0\n \n 0\n
          \n \\(690\\)\n \n 0\n \n 0\n
          \n \\(691\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(692\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(693\\)\n \n 0\n \n 0\n
          \n \\(694\\)\n \n 0\n \n 0\n
          \n \\(695\\)\n \n 0\n \n 0\n
          \n \\(696\\)\n \n 0\n \n 0\n
          \n \\(697\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(698\\)\n \n 0\n \n 0\n
          \n \\(699\\)\n \n 0\n \n 0\n
          \n \\(700\\)\n \n 0\n \n 0\n
          \n \\(701\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(702\\)\n \n 0\n \n 0\n
          \n \\(703\\)\n \n 0\n \n 0\n
          \n \\(704\\)\n \n 0\n \n 0\n
          \n \\(705\\)\n \n 0\n \n 0\n
          \n \\(706\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(707\\)\n \n 0\n \n 0\n
          \n \\(708\\)\n \n 0\n \n 0\n
          \n \\(709\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(710\\)\n \n 0\n \n 0\n
          \n \\(711\\)\n \n 0\n \n 0\n
          \n \\(712\\)\n \n 0\n \n 0\n
          \n \\(713\\)\n \n 0\n \n 0\n
          \n \\(714\\)\n \n 0\n \n 0\n
          \n \\(715\\)\n \n 0\n \n 0\n
          \n \\(716\\)\n \n 0\n \n 0\n
          \n \\(717\\)\n \n 0\n \n 0\n
          \n \\(718\\)\n \n 0\n \n 0\n
          \n \\(719\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(720\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(721\\)\n \n 0\n \n 0\n
          \n \\(722\\)\n \n 1.00000i\n 1.00000i
          \n \\(723\\)\n \n 0\n \n 0\n
          \n \\(724\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(725\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(726\\)\n \n 0\n \n 0\n
          \n \\(727\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(728\\)\n \n 0\n \n 0\n
          \n \\(729\\)\n \n 1.00000i\n 1.00000i
          \n \\(730\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(731\\)\n \n 0\n \n 0\n
          \n \\(732\\)\n \n 0\n \n 0\n
          \n \\(733\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(734\\)\n \n 0\n \n 0\n
          \n \\(735\\)\n \n 0\n \n 0\n
          \n \\(736\\)\n \n 0\n \n 0\n
          \n \\(737\\)\n \n 0\n \n 0\n
          \n \\(738\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(739\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(740\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(741\\)\n \n 0\n \n 0\n
          \n \\(742\\)\n \n 0\n \n 0\n
          \n \\(743\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(744\\)\n \n 0\n \n 0\n
          \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n 0.732051\n \n +\n \n 2.73205i
          \n \\(746\\)\n \n 0\n \n 0\n
          \n \\(747\\)\n \n 0\n \n 0\n
          \n \\(748\\)\n \n 0\n \n 0\n
          \n \\(749\\)\n \n 0\n \n 0\n
          \n \\(750\\)\n \n 0\n \n 0\n
          \n \\(751\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(752\\)\n \n 0\n \n 0\n
          \n \\(753\\)\n \n 0\n \n 0\n
          \n \\(754\\)\n \n 0\n \n 0\n
          \n \\(755\\)\n \n 0\n \n 0\n
          \n \\(756\\)\n \n 0\n \n 0\n
          \n \\(757\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(758\\)\n \n 0\n \n 0\n
          \n \\(759\\)\n \n 0\n \n 0\n
          \n \\(760\\)\n \n 0\n \n 0\n
          \n \\(761\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(762\\)\n \n 0\n \n 0\n
          \n \\(763\\)\n \n 0\n \n 0\n
          \n \\(764\\)\n \n 0\n \n 0\n
          \n \\(765\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(766\\)\n \n 0\n \n 0\n
          \n \\(767\\)\n \n 0\n \n 0\n
          \n \\(768\\)\n \n 0\n \n 0\n
          \n \\(769\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(770\\)\n \n 0\n \n 0\n
          \n \\(771\\)\n \n 0\n \n 0\n
          \n \\(772\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(773\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \\(774\\)\n \n 0\n \n 0\n
          \n \\(775\\)\n \n 0\n \n 0\n
          \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(777\\)\n \n 0\n \n 0\n
          \n \\(778\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(779\\)\n \n 0\n \n 0\n
          \n \\(780\\)\n \n 0\n \n 0\n
          \n \\(781\\)\n \n 0\n \n 0\n
          \n \\(782\\)\n \n 0\n \n 0\n
          \n \\(783\\)\n \n 0\n \n 0\n
          \n \\(784\\)\n \n 0\n \n 0\n
          \n \\(785\\)\n \n \u22122.00000\n \n +\n \n 2.00000i\n \u22122.00000\n \n +\n \n 2.00000i
          \n \\(786\\)\n \n 0\n \n 0\n
          \n \\(787\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(788\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(789\\)\n \n 0\n \n 0\n
          \n \\(790\\)\n \n 0\n \n 0\n
          \n \\(791\\)\n \n 0\n \n 0\n
          \n \\(792\\)\n \n 0\n \n 0\n
          \n \\(793\\)\n \n 0\n \n 0\n
          \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(795\\)\n \n 0\n \n 0\n
          \n \\(796\\)\n \n 0\n \n 0\n
          \n \\(797\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(798\\)\n \n 0\n \n 0\n
          \n \\(799\\)\n \n 0\n \n 0\n
          \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(801\\)\n \n 0\n \n 0\n
          \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(803\\)\n \n 0\n \n 0\n
          \n \\(804\\)\n \n 0\n \n 0\n
          \n \\(805\\)\n \n 0\n \n 0\n
          \n \\(806\\)\n \n 0\n \n 0\n
          \n \\(807\\)\n \n 0\n \n 0\n
          \n \\(808\\)\n \n 0\n \n 0\n
          \n \\(809\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(810\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(811\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(812\\)\n \n 0\n \n 0\n
          \n \\(813\\)\n \n 0\n \n 0\n
          \n \\(814\\)\n \n 0\n \n 0\n
          \n \\(815\\)\n \n 0\n \n 0\n
          \n \\(816\\)\n \n 0\n \n 0\n
          \n \\(817\\)\n \n 0\n \n 0\n
          \n \\(818\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(819\\)\n \n 0\n \n 0\n
          \n \\(820\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(821\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(822\\)\n \n 0\n \n 0\n
          \n \\(823\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(824\\)\n \n 0\n \n 0\n
          \n \\(825\\)\n \n 0\n \n 0\n
          \n \\(826\\)\n \n 0\n \n 0\n
          \n \\(827\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(828\\)\n \n 0\n \n 0\n
          \n \\(829\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(830\\)\n \n 0\n \n 0\n
          \n \\(831\\)\n \n 0\n \n 0\n
          \n \\(832\\)\n \n 0\n \n 0\n
          \n \\(833\\)\n \n 0\n \n 0\n
          \n \\(834\\)\n \n 0\n \n 0\n
          \n \\(835\\)\n \n 0\n \n 0\n
          \n \\(836\\)\n \n 0\n \n 0\n
          \n \\(837\\)\n \n 0\n \n 0\n
          \n \\(838\\)\n \n 0\n \n 0\n
          \n \\(839\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(840\\)\n \n 0\n \n 0\n
          \n \\(841\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(842\\)\n \n 0\n \n 0\n
          \n \\(843\\)\n \n 0\n \n 0\n
          \n \\(844\\)\n \n 0\n \n 0\n
          \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(846\\)\n \n 0\n \n 0\n
          \n \\(847\\)\n \n 0\n \n 0\n
          \n \\(848\\)\n \n 0\n \n 0\n
          \n \\(849\\)\n \n 0\n \n 0\n
          \n \\(850\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(851\\)\n \n 0\n \n 0\n
          \n \\(852\\)\n \n 0\n \n 0\n
          \n \\(853\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(854\\)\n \n 0\n \n 0\n
          \n \\(855\\)\n \n 0\n \n 0\n
          \n \\(856\\)\n \n 0\n \n 0\n
          \n \\(857\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(858\\)\n \n 0\n \n 0\n
          \n \\(859\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(860\\)\n \n 0\n \n 0\n
          \n \\(861\\)\n \n 0\n \n 0\n
          \n \\(862\\)\n \n 0\n \n 0\n
          \n \\(863\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(864\\)\n \n 0\n \n 0\n
          \n \\(865\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(866\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(867\\)\n \n 0\n \n 0\n
          \n \\(868\\)\n \n 0\n \n 0\n
          \n \\(869\\)\n \n 0\n \n 0\n
          \n \\(870\\)\n \n 0\n \n 0\n
          \n \\(871\\)\n \n 0\n \n 0\n
          \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(873\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(874\\)\n \n 0\n \n 0\n
          \n \\(875\\)\n \n 0\n \n 0\n
          \n \\(876\\)\n \n 0\n \n 0\n
          \n \\(877\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(878\\)\n \n 0\n \n 0\n
          \n \\(879\\)\n \n 0\n \n 0\n
          \n \\(880\\)\n \n 0\n \n 0\n
          \n \\(881\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(882\\)\n \n 0\n \n 0\n
          \n \\(883\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(884\\)\n \n 0\n \n 0\n
          \n \\(885\\)\n \n 0\n \n 0\n
          \n \\(886\\)\n \n 0\n \n 0\n
          \n \\(887\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(888\\)\n \n 0\n \n 0\n
          \n \\(889\\)\n \n 0\n \n 0\n
          \n \\(890\\)\n \n 0\n \n 0\n
          \n \\(891\\)\n \n 0\n \n 0\n
          \n \\(892\\)\n \n 0\n \n 0\n
          \n \\(893\\)\n \n 0\n \n 0\n
          \n \\(894\\)\n \n 0\n \n 0\n
          \n \\(895\\)\n \n 0\n \n 0\n
          \n \\(896\\)\n \n 0\n \n 0\n
          \n \\(897\\)\n \n 0\n \n 0\n
          \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(899\\)\n \n 0\n \n 0\n
          \n \\(900\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(901\\)\n \n 0\n \n 0\n
          \n \\(902\\)\n \n 0\n \n 0\n
          \n \\(903\\)\n \n 0\n \n 0\n
          \n \\(904\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(905\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(906\\)\n \n 0\n \n 0\n
          \n \\(907\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(908\\)\n \n 0\n \n 0\n
          \n \\(909\\)\n \n 0\n \n 0\n
          \n \\(910\\)\n \n 0\n \n 0\n
          \n \\(911\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(912\\)\n \n 0\n \n 0\n
          \n \\(913\\)\n \n 0\n \n 0\n
          \n \\(914\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(915\\)\n \n 0\n \n 0\n
          \n \\(916\\)\n \n 2.00000i\n 2.00000i
          \n \\(917\\)\n \n 0\n \n 0\n
          \n \\(918\\)\n \n 0\n \n 0\n
          \n \\(919\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(920\\)\n \n 0\n \n 0\n
          \n \\(921\\)\n \n 0\n \n 0\n
          \n \\(922\\)\n \n 0\n \n 0\n
          \n \\(923\\)\n \n 0\n \n 0\n
          \n \\(924\\)\n \n 0\n \n 0\n
          \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(926\\)\n \n 0\n \n 0\n
          \n \\(927\\)\n \n 0\n \n 0\n
          \n \\(928\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(929\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(930\\)\n \n 0\n \n 0\n
          \n \\(931\\)\n \n 0\n \n 0\n
          \n \\(932\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(933\\)\n \n 0\n \n 0\n
          \n \\(934\\)\n \n 0\n \n 0\n
          \n \\(935\\)\n \n 0\n \n 0\n
          \n \\(936\\)\n \n 0\n \n 0\n
          \n \\(937\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(938\\)\n \n 0\n \n 0\n
          \n \\(939\\)\n \n 0\n \n 0\n
          \n \\(940\\)\n \n 0\n \n 0\n
          \n \\(941\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(942\\)\n \n 0\n \n 0\n
          \n \\(943\\)\n \n 0\n \n 0\n
          \n \\(944\\)\n \n 0\n \n 0\n
          \n \\(945\\)\n \n 0\n \n 0\n
          \n \\(946\\)\n \n 0\n \n 0\n
          \n \\(947\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(948\\)\n \n 0\n \n 0\n
          \n \\(949\\)\n \n 0\n \n 0\n
          \n \\(950\\)\n \n 0\n \n 0\n
          \n \\(951\\)\n \n 0\n \n 0\n
          \n \\(952\\)\n \n 0\n \n 0\n
          \n \\(953\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(954\\)\n \n 0\n \n 0\n
          \n \\(955\\)\n \n 0\n \n 0\n
          \n \\(956\\)\n \n 0\n \n 0\n
          \n \\(957\\)\n \n 0\n \n 0\n
          \n \\(958\\)\n \n 0\n \n 0\n
          \n \\(959\\)\n \n 0\n \n 0\n
          \n \\(960\\)\n \n 0\n \n 0\n
          \n \\(961\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(962\\)\n \n 0\n \n 0\n
          \n \\(963\\)\n \n 0\n \n 0\n
          \n \\(964\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(965\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(966\\)\n \n 0\n \n 0\n
          \n \\(967\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(969\\)\n \n 0\n \n 0\n
          \n \\(970\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(971\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(972\\)\n \n 0\n \n 0\n
          \n \\(973\\)\n \n 0\n \n 0\n
          \n \\(974\\)\n \n 0\n \n 0\n
          \n \\(975\\)\n \n 0\n \n 0\n
          \n \\(976\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(977\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(978\\)\n \n 0\n \n 0\n
          \n \\(979\\)\n \n 0\n \n 0\n
          \n \\(980\\)\n \n 0\n \n 0\n
          \n \\(981\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(982\\)\n \n 0\n \n 0\n
          \n \\(983\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(984\\)\n \n 0\n \n 0\n
          \n \\(985\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(986\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(987\\)\n \n 0\n \n 0\n
          \n \\(988\\)\n \n 0\n \n 0\n
          \n \\(989\\)\n \n 0\n \n 0\n
          \n \\(990\\)\n \n 0\n \n 0\n
          \n \\(991\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(992\\)\n \n 0\n \n 0\n
          \n \\(993\\)\n \n 0\n \n 0\n
          \n \\(994\\)\n \n 0\n \n 0\n
          \n \\(995\\)\n \n 0\n \n 0\n
          \n \\(996\\)\n \n 0\n \n 0\n
          \n \\(997\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(998\\)\n \n 0\n \n 0\n
          \n \\(999\\)\n \n 0\n \n 0\n
          ", "content": {"html": "
          \\(n\\)\\(a_n\\)\\(a_n / n^{(k-1)/2}\\)\\( \\alpha_n \\)\\( \\theta_n \\)
          \\(p\\)\\(a_p\\)\\(a_p / p^{(k-1)/2}\\)\\( \\alpha_p\\)\\( \\theta_p \\)
          \\(2\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(3\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(4\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(5\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(6\\)00
          \\(7\\)00
          \\(8\\)1.00000<br>i1.00000<br>i
          \\(9\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(10\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(11\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(12\\)00
          \\(13\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(14\\)00
          \\(15\\)00
          \\(16\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(17\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(18\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(19\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(20\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(21\\)00
          \\(22\\)00
          \\(23\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(24\\)00
          \\(25\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(26\\)00
          \\(27\\)00
          \\(28\\)00
          \\(29\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(30\\)00
          \\(31\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(32\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(33\\)00
          \\(34\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(35\\)00
          \\(36\\)1.00000<br>i1.00000<br>i
          \\(37\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(38\\)00
          \\(39\\)00
          \\(40\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(41\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(42\\)00
          \\(43\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(44\\)00
          \\(45\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(46\\)00
          \\(47\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(48\\)00
          \\(49\\)00
          \\(50\\)\u22121.00000\u22121.00000
          \\(51\\)00
          \\(52\\)00
          \\(53\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(54\\)00
          \\(55\\)00
          \\(56\\)00
          \\(57\\)00
          \\(58\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(59\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(60\\)00
          \\(61\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(62\\)00
          \\(63\\)00
          \\(64\\)\u22121.00000\u22121.00000
          \\(65\\)00
          \\(66\\)00
          \\(67\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(68\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(69\\)00
          \\(70\\)00
          \\(71\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(72\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(73\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(74\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(75\\)00
          \\(76\\)00
          \\(77\\)00
          \\(78\\)00
          \\(79\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(80\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(81\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(82\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(83\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(84\\)00
          \\(85\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(86\\)00
          \\(87\\)00
          \\(88\\)00
          \\(89\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(90\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(91\\)00
          \\(92\\)00
          \\(93\\)00
          \\(94\\)00
          \\(95\\)00
          \\(96\\)00
          \\(97\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(98\\)00
          \\(99\\)00
          \\(100\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(101\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(102\\)00
          \\(103\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(104\\)00
          \\(105\\)00
          \\(106\\)00
          \\(107\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(108\\)00
          \\(109\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(110\\)00
          \\(111\\)00
          \\(112\\)00
          \\(113\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(114\\)00
          \\(115\\)00
          \\(116\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(117\\)00
          \\(118\\)00
          \\(119\\)00
          \\(120\\)00
          \\(121\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(122\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(123\\)00
          \\(124\\)00
          \\(125\\)00
          \\(126\\)00
          \\(127\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(128\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(129\\)00
          \\(130\\)00
          \\(131\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(132\\)00
          \\(133\\)00
          \\(134\\)00
          \\(135\\)00
          \\(136\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(137\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(138\\)00
          \\(139\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(140\\)00
          \\(141\\)00
          \\(142\\)00
          \\(143\\)00
          \\(144\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(145\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(146\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(147\\)00
          \\(148\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(149\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \\(150\\)00
          \\(151\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(152\\)00
          \\(153\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(154\\)00
          \\(155\\)00
          \\(156\\)00
          \\(157\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(158\\)00
          \\(159\\)00
          \\(160\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(161\\)00
          \\(162\\)1.00000<br>i1.00000<br>i
          \\(163\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(164\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(165\\)00
          \\(166\\)00
          \\(167\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(168\\)00
          \\(169\\)\u22121.00000\u22121.00000
          \\(170\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(171\\)00
          \\(172\\)00
          \\(173\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(174\\)00
          \\(175\\)00
          \\(176\\)00
          \\(177\\)00
          \\(178\\)00
          \\(179\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(180\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(181\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(182\\)00
          \\(183\\)00
          \\(184\\)00
          \\(185\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(186\\)00
          \\(187\\)00
          \\(188\\)00
          \\(189\\)00
          \\(190\\)00
          \\(191\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(192\\)00
          \\(193\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(194\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(195\\)00
          \\(196\\)00
          \\(197\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(198\\)00
          \\(199\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(200\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(201\\)00
          \\(202\\)00
          \\(203\\)00
          \\(204\\)00
          \\(205\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
          \\(206\\)00
          \\(207\\)00
          \\(208\\)00
          \\(209\\)00
          \\(210\\)00
          \\(211\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(212\\)00
          \\(213\\)00
          \\(214\\)00
          \\(215\\)00
          \\(216\\)00
          \\(217\\)00
          \\(218\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(219\\)00
          \\(220\\)00
          \\(221\\)00
          \\(222\\)00
          \\(223\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(224\\)00
          \\(225\\)\u22121.00000\u22121.00000
          \\(226\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(227\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(228\\)00
          \\(229\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(230\\)00
          \\(231\\)00
          \\(232\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(233\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(234\\)00
          \\(235\\)00
          \\(236\\)00
          \\(237\\)00
          \\(238\\)00
          \\(239\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(240\\)00
          \\(241\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(242\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(243\\)00
          \\(244\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(245\\)00
          \\(246\\)00
          \\(247\\)00
          \\(248\\)00
          \\(249\\)00
          \\(250\\)00
          \\(251\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(252\\)00
          \\(253\\)00
          \\(254\\)00
          \\(255\\)00
          \\(256\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(257\\)1.73205\u22121.00000<br>i1.73205\u22121.00000<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(258\\)00
          \\(259\\)00
          \\(260\\)00
          \\(261\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(262\\)00
          \\(263\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(264\\)00
          \\(265\\)00
          \\(266\\)00
          \\(267\\)00
          \\(268\\)00
          \\(269\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(270\\)00
          \\(271\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(272\\)1.000001.00000
          \\(273\\)00
          \\(274\\)00
          \\(275\\)00
          \\(276\\)00
          \\(277\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(278\\)00
          \\(279\\)00
          \\(280\\)00
          \\(281\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(282\\)00
          \\(283\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(284\\)00
          \\(285\\)00
          \\(286\\)00
          \\(287\\)00
          \\(288\\)\u22121.00000\u22121.00000
          \\(289\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(290\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(291\\)00
          \\(292\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(293\\)2.000002.000001.00000\\(0\\)
          1.00000\\(0\\)
          \\(294\\)00
          \\(295\\)00
          \\(296\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(297\\)00
          \\(298\\)1.73205+1.00000<br>i1.73205+1.00000<br>i
          \\(299\\)00
          \\(300\\)00
          \\(301\\)00
          \\(302\\)00
          \\(303\\)00
          \\(304\\)00
          \\(305\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
          \\(306\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(307\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(308\\)00
          \\(309\\)00
          \\(310\\)00
          \\(311\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(312\\)00
          \\(313\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(314\\)2.00000<br>i2.00000<br>i
          \\(315\\)00
          \\(316\\)00
          \\(317\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(318\\)00
          \\(319\\)00
          \\(320\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(321\\)00
          \\(322\\)00
          \\(323\\)00
          \\(324\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(325\\)00
          \\(326\\)00
          \\(327\\)00
          \\(328\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(329\\)00
          \\(330\\)00
          \\(331\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(332\\)00
          \\(333\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(334\\)00
          \\(335\\)00
          \\(336\\)00
          \\(337\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(338\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(339\\)00
          \\(340\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(341\\)00
          \\(342\\)00
          \\(343\\)00
          \\(344\\)00
          \\(345\\)00
          \\(346\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(347\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(348\\)00
          \\(349\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(350\\)00
          \\(351\\)00
          \\(352\\)00
          \\(353\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(354\\)00
          \\(355\\)00
          \\(356\\)00
          \\(357\\)00
          \\(358\\)00
          \\(359\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(360\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(361\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(362\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(363\\)00
          \\(364\\)00
          \\(365\\)2.00000<br>i2.00000<br>i
          \\(366\\)00
          \\(367\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(368\\)00
          \\(369\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(370\\)2.000002.00000
          \\(371\\)00
          \\(372\\)00
          \\(373\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(374\\)00
          \\(375\\)00
          \\(376\\)00
          \\(377\\)00
          \\(378\\)00
          \\(379\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(380\\)00
          \\(381\\)00
          \\(382\\)00
          \\(383\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(384\\)00
          \\(385\\)00
          \\(386\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(387\\)00
          \\(388\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(389\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \\(390\\)00
          \\(391\\)00
          \\(392\\)00
          \\(393\\)00
          \\(394\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(395\\)00
          \\(396\\)00
          \\(397\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(398\\)00
          \\(399\\)00
          \\(400\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(401\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(402\\)00
          \\(403\\)00
          \\(404\\)00
          \\(405\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(406\\)00
          \\(407\\)00
          \\(408\\)00
          \\(409\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(410\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(411\\)00
          \\(412\\)00
          \\(413\\)00
          \\(414\\)00
          \\(415\\)00
          \\(416\\)00
          \\(417\\)00
          \\(418\\)00
          \\(419\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(420\\)00
          \\(421\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(422\\)00
          \\(423\\)00
          \\(424\\)00
          \\(425\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(426\\)00
          \\(427\\)00
          \\(428\\)00
          \\(429\\)00
          \\(430\\)00
          \\(431\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(432\\)00
          \\(433\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(434\\)00
          \\(435\\)00
          \\(436\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(437\\)00
          \\(438\\)00
          \\(439\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(440\\)00
          \\(441\\)00
          \\(442\\)00
          \\(443\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(444\\)00
          \\(445\\)00
          \\(446\\)00
          \\(447\\)00
          \\(448\\)00
          \\(449\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(450\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(451\\)00
          \\(452\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(453\\)00
          \\(454\\)00
          \\(455\\)00
          \\(456\\)00
          \\(457\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(458\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(459\\)00
          \\(460\\)00
          \\(461\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(462\\)00
          \\(463\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(464\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(465\\)00
          \\(466\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(467\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(468\\)00
          \\(469\\)00
          \\(470\\)00
          \\(471\\)00
          \\(472\\)00
          \\(473\\)00
          \\(474\\)00
          \\(475\\)00
          \\(476\\)00
          \\(477\\)00
          \\(478\\)00
          \\(479\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(480\\)00
          \\(481\\)00
          \\(482\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(483\\)00
          \\(484\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(485\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(486\\)00
          \\(487\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(488\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(489\\)00
          \\(490\\)00
          \\(491\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(492\\)00
          \\(493\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(494\\)00
          \\(495\\)00
          \\(496\\)00
          \\(497\\)00
          \\(498\\)00
          \\(499\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(500\\)00
          \\(501\\)00
          \\(502\\)00
          \\(503\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(504\\)00
          \\(505\\)00
          \\(506\\)00
          \\(507\\)00
          \\(508\\)00
          \\(509\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \\(510\\)00
          \\(511\\)00
          \\(512\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(513\\)00
          \\(514\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
          \\(515\\)00
          \\(516\\)00
          \\(517\\)00
          \\(518\\)00
          \\(519\\)00
          \\(520\\)00
          \\(521\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(522\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(523\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(524\\)00
          \\(525\\)00
          \\(526\\)00
          \\(527\\)00
          \\(528\\)00
          \\(529\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(530\\)00
          \\(531\\)00
          \\(532\\)00
          \\(533\\)00
          \\(534\\)00
          \\(535\\)00
          \\(536\\)00
          \\(537\\)00
          \\(538\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(539\\)00
          \\(540\\)00
          \\(541\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(542\\)00
          \\(543\\)00
          \\(544\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(545\\)\u22122.00000\u22122.00000
          \\(546\\)00
          \\(547\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(548\\)00
          \\(549\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(550\\)00
          \\(551\\)00
          \\(552\\)00
          \\(553\\)00
          \\(554\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(555\\)00
          \\(556\\)00
          \\(557\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(558\\)00
          \\(559\\)00
          \\(560\\)00
          \\(561\\)00
          \\(562\\)00
          \\(563\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(564\\)00
          \\(565\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
          \\(566\\)00
          \\(567\\)00
          \\(568\\)00
          \\(569\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(570\\)00
          \\(571\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(572\\)00
          \\(573\\)00
          \\(574\\)00
          \\(575\\)00
          \\(576\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(577\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(578\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(579\\)00
          \\(580\\)2.00000<br>i2.00000<br>i
          \\(581\\)00
          \\(582\\)00
          \\(583\\)00
          \\(584\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(585\\)00
          \\(586\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(587\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(588\\)00
          \\(589\\)00
          \\(590\\)00
          \\(591\\)00
          \\(592\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(593\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(594\\)00
          \\(595\\)00
          \\(596\\)\u22122.00000\u22122.00000
          \\(597\\)00
          \\(598\\)00
          \\(599\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(600\\)00
          \\(601\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(602\\)00
          \\(603\\)00
          \\(604\\)00
          \\(605\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(606\\)00
          \\(607\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(608\\)00
          \\(609\\)00
          \\(610\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(611\\)00
          \\(612\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(613\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(614\\)00
          \\(615\\)00
          \\(616\\)00
          \\(617\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(618\\)00
          \\(619\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(620\\)00
          \\(621\\)00
          \\(622\\)00
          \\(623\\)00
          \\(624\\)00
          \\(625\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(626\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(627\\)00
          \\(628\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(629\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(630\\)00
          \\(631\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(632\\)00
          \\(633\\)00
          \\(634\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(635\\)00
          \\(636\\)00
          \\(637\\)00
          \\(638\\)00
          \\(639\\)00
          \\(640\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(641\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(642\\)00
          \\(643\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(644\\)00
          \\(645\\)00
          \\(646\\)00
          \\(647\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(648\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(649\\)00
          \\(650\\)00
          \\(651\\)00
          \\(652\\)00
          \\(653\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(654\\)00
          \\(655\\)00
          \\(656\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(657\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(658\\)00
          \\(659\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(660\\)00
          \\(661\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(662\\)00
          \\(663\\)00
          \\(664\\)00
          \\(665\\)00
          \\(666\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(667\\)00
          \\(668\\)00
          \\(669\\)00
          \\(670\\)00
          \\(671\\)00
          \\(672\\)00
          \\(673\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(674\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(675\\)00
          \\(676\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(677\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(678\\)00
          \\(679\\)00
          \\(680\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(681\\)00
          \\(682\\)00
          \\(683\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(684\\)00
          \\(685\\)00
          \\(686\\)00
          \\(687\\)00
          \\(688\\)00
          \\(689\\)00
          \\(690\\)00
          \\(691\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(692\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(693\\)00
          \\(694\\)00
          \\(695\\)00
          \\(696\\)00
          \\(697\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(698\\)00
          \\(699\\)00
          \\(700\\)00
          \\(701\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(702\\)00
          \\(703\\)00
          \\(704\\)00
          \\(705\\)00
          \\(706\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(707\\)00
          \\(708\\)00
          \\(709\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(710\\)00
          \\(711\\)00
          \\(712\\)00
          \\(713\\)00
          \\(714\\)00
          \\(715\\)00
          \\(716\\)00
          \\(717\\)00
          \\(718\\)00
          \\(719\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(720\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(721\\)00
          \\(722\\)1.00000<br>i1.00000<br>i
          \\(723\\)00
          \\(724\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(725\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(726\\)00
          \\(727\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(728\\)00
          \\(729\\)1.00000<br>i1.00000<br>i
          \\(730\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(731\\)00
          \\(732\\)00
          \\(733\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(734\\)00
          \\(735\\)00
          \\(736\\)00
          \\(737\\)00
          \\(738\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(739\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(740\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(741\\)00
          \\(742\\)00
          \\(743\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(744\\)00
          \\(745\\)0.732051+2.73205<br>i0.732051+2.73205<br>i
          \\(746\\)00
          \\(747\\)00
          \\(748\\)00
          \\(749\\)00
          \\(750\\)00
          \\(751\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(752\\)00
          \\(753\\)00
          \\(754\\)00
          \\(755\\)00
          \\(756\\)00
          \\(757\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(758\\)00
          \\(759\\)00
          \\(760\\)00
          \\(761\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(762\\)00
          \\(763\\)00
          \\(764\\)00
          \\(765\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(766\\)00
          \\(767\\)00
          \\(768\\)00
          \\(769\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(770\\)00
          \\(771\\)00
          \\(772\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(773\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \\(774\\)00
          \\(775\\)00
          \\(776\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(777\\)00
          \\(778\\)2.000002.00000
          \\(779\\)00
          \\(780\\)00
          \\(781\\)00
          \\(782\\)00
          \\(783\\)00
          \\(784\\)00
          \\(785\\)\u22122.00000+2.00000<br>i\u22122.00000+2.00000<br>i
          \\(786\\)00
          \\(787\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(788\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(789\\)00
          \\(790\\)00
          \\(791\\)00
          \\(792\\)00
          \\(793\\)00
          \\(794\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(795\\)00
          \\(796\\)00
          \\(797\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(798\\)00
          \\(799\\)00
          \\(800\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(801\\)00
          \\(802\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(803\\)00
          \\(804\\)00
          \\(805\\)00
          \\(806\\)00
          \\(807\\)00
          \\(808\\)00
          \\(809\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(810\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(811\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(812\\)00
          \\(813\\)00
          \\(814\\)00
          \\(815\\)00
          \\(816\\)00
          \\(817\\)00
          \\(818\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(819\\)00
          \\(820\\)2.000002.00000
          \\(821\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(822\\)00
          \\(823\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(824\\)00
          \\(825\\)00
          \\(826\\)00
          \\(827\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(828\\)00
          \\(829\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(830\\)00
          \\(831\\)00
          \\(832\\)00
          \\(833\\)00
          \\(834\\)00
          \\(835\\)00
          \\(836\\)00
          \\(837\\)00
          \\(838\\)00
          \\(839\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(840\\)00
          \\(841\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(842\\)00
          \\(843\\)00
          \\(844\\)00
          \\(845\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(846\\)00
          \\(847\\)00
          \\(848\\)00
          \\(849\\)00
          \\(850\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(851\\)00
          \\(852\\)00
          \\(853\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(854\\)00
          \\(855\\)00
          \\(856\\)00
          \\(857\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(858\\)00
          \\(859\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(860\\)00
          \\(861\\)00
          \\(862\\)00
          \\(863\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(864\\)00
          \\(865\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(866\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(867\\)00
          \\(868\\)00
          \\(869\\)00
          \\(870\\)00
          \\(871\\)00
          \\(872\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(873\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(874\\)00
          \\(875\\)00
          \\(876\\)00
          \\(877\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(878\\)00
          \\(879\\)00
          \\(880\\)00
          \\(881\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(882\\)00
          \\(883\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(884\\)00
          \\(885\\)00
          \\(886\\)00
          \\(887\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(888\\)00
          \\(889\\)00
          \\(890\\)00
          \\(891\\)00
          \\(892\\)00
          \\(893\\)00
          \\(894\\)00
          \\(895\\)00
          \\(896\\)00
          \\(897\\)00
          \\(898\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(899\\)00
          \\(900\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(901\\)00
          \\(902\\)00
          \\(903\\)00
          \\(904\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(905\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(906\\)00
          \\(907\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(908\\)00
          \\(909\\)00
          \\(910\\)00
          \\(911\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(912\\)00
          \\(913\\)00
          \\(914\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(915\\)00
          \\(916\\)2.00000<br>i2.00000<br>i
          \\(917\\)00
          \\(918\\)00
          \\(919\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(920\\)00
          \\(921\\)00
          \\(922\\)00
          \\(923\\)00
          \\(924\\)00
          \\(925\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(926\\)00
          \\(927\\)00
          \\(928\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(929\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(930\\)00
          \\(931\\)00
          \\(932\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(933\\)00
          \\(934\\)00
          \\(935\\)00
          \\(936\\)00
          \\(937\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(938\\)00
          \\(939\\)00
          \\(940\\)00
          \\(941\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(942\\)00
          \\(943\\)00
          \\(944\\)00
          \\(945\\)00
          \\(946\\)00
          \\(947\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(948\\)00
          \\(949\\)00
          \\(950\\)00
          \\(951\\)00
          \\(952\\)00
          \\(953\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(954\\)00
          \\(955\\)00
          \\(956\\)00
          \\(957\\)00
          \\(958\\)00
          \\(959\\)00
          \\(960\\)00
          \\(961\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(962\\)00
          \\(963\\)00
          \\(964\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(965\\)2.000002.00000
          \\(966\\)00
          \\(967\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(968\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(969\\)00
          \\(970\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(971\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(972\\)00
          \\(973\\)00
          \\(974\\)00
          \\(975\\)00
          \\(976\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(977\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(978\\)00
          \\(979\\)00
          \\(980\\)00
          \\(981\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(982\\)00
          \\(983\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(984\\)00
          \\(985\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(986\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(987\\)00
          \\(988\\)00
          \\(989\\)00
          \\(990\\)00
          \\(991\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(992\\)00
          \\(993\\)00
          \\(994\\)00
          \\(995\\)00
          \\(996\\)00
          \\(997\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(998\\)00
          \\(999\\)00
          ", "is_complex": true, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "table", "raw_content": "
          \u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisting character
          CharParityOrdTypeTwistMinDim
          1.1even1trivial3332.1.bc.b.2027.14
          4.3odd2CM3332.1.bc.b.2027.14
          7.2even3inner3332.1.bc.b.667.14
          7.3odd668.1.f.a.55.1yes2
          7.4even33332.1.m.b.2843.12
          7.5odd63332.1.bc.c.667.14
          7.6odd23332.1.bc.c.2027.14
          17.13even4inner3332.1.bc.b.2223.14
          21.17even6612.1.l.a.55.12
          28.3even668.1.f.a.55.1yes2
          28.11odd63332.1.m.b.2843.12
          28.19even63332.1.bc.c.667.14
          28.23odd6inner3332.1.bc.b.667.14
          28.27even23332.1.bc.c.2027.14
          35.3even121700.1.n.a.599.12
          35.17even121700.1.n.b.599.12
          35.24odd61700.1.p.a.1551.12
          56.3even61088.1.p.a.191.12
          56.45odd61088.1.p.a.191.12
          68.47odd4inner3332.1.bc.b.2223.14
          84.59odd6612.1.l.a.55.12
          119.3even481156.1.g.b.155.18
          119.10even481156.1.g.b.179.18
          119.13odd43332.1.bc.c.2223.14
          119.24even481156.1.g.b.179.28
          119.30even12inner3332.1.bc.b.863.14
          119.31even481156.1.g.b.155.28
          119.38odd121156.1.f.b.251.12
          119.45even481156.1.g.b.399.18
          119.47odd123332.1.bc.c.863.14
          119.59odd241156.1.c.b.579.12
          119.66odd241156.1.d.a.1155.22
          119.73even481156.1.g.b.423.28
          119.80even481156.1.g.b.423.18
          119.81even123332.1.m.b.3039.12
          119.87odd241156.1.d.a.1155.12
          119.94odd241156.1.c.b.579.22
          119.101odd61156.1.f.b.327.12
          119.108even481156.1.g.b.399.28
          119.115odd1268.1.f.a.47.1&check;2
          140.3odd121700.1.n.a.599.12
          140.59even61700.1.p.a.1551.12
          140.87odd121700.1.n.b.599.12
          357.353even12612.1.l.a.523.12
          476.3odd481156.1.g.b.155.18
          476.31odd481156.1.g.b.155.28
          476.47even123332.1.bc.c.863.14
          476.59even241156.1.c.b.579.12
          476.87even241156.1.d.a.1155.12
          476.115even1268.1.f.a.47.1&check;2
          476.143odd481156.1.g.b.179.28
          476.199odd481156.1.g.b.423.18
          476.227odd481156.1.g.b.399.28
          476.251even43332.1.bc.c.2223.14
          476.283odd481156.1.g.b.399.18
          476.311odd481156.1.g.b.423.28
          476.319odd123332.1.m.b.3039.12
          476.339even61156.1.f.b.327.12
          476.367odd481156.1.g.b.179.18
          476.387odd12inner3332.1.bc.b.863.14
          476.395even121156.1.f.b.251.12
          476.423even241156.1.d.a.1155.22
          476.451even241156.1.c.b.579.22
          595.234odd121700.1.p.a.251.12
          595.353even121700.1.n.b.999.12
          595.472even121700.1.n.a.999.12
          952.115even121088.1.p.a.319.12
          952.829odd121088.1.p.a.319.12
          1428.1067odd12612.1.l.a.523.12
          2380.1067odd121700.1.n.a.999.12
          2380.1543odd121700.1.n.b.999.12
          2380.2019even121700.1.p.a.251.12
          ", "content": {"html": "
          By<br>twisting character
          CharParityOrdTypeTwistMinDim
          1.1even1trivial3332.1.bc.b.2027.14
          4.3odd2CM3332.1.bc.b.2027.14
          7.2even3inner3332.1.bc.b.667.14
          7.3odd668.1.f.a.55.1yes2
          7.4even33332.1.m.b.2843.12
          7.5odd63332.1.bc.c.667.14
          7.6odd23332.1.bc.c.2027.14
          17.13even4inner3332.1.bc.b.2223.14
          21.17even6612.1.l.a.55.12
          28.3even668.1.f.a.55.1yes2
          28.11odd63332.1.m.b.2843.12
          28.19even63332.1.bc.c.667.14
          28.23odd6inner3332.1.bc.b.667.14
          28.27even23332.1.bc.c.2027.14
          35.3even121700.1.n.a.599.12
          35.17even121700.1.n.b.599.12
          35.24odd61700.1.p.a.1551.12
          56.3even61088.1.p.a.191.12
          56.45odd61088.1.p.a.191.12
          68.47odd4inner3332.1.bc.b.2223.14
          84.59odd6612.1.l.a.55.12
          119.3even481156.1.g.b.155.18
          119.10even481156.1.g.b.179.18
          119.13odd43332.1.bc.c.2223.14
          119.24even481156.1.g.b.179.28
          119.30even12inner3332.1.bc.b.863.14
          119.31even481156.1.g.b.155.28
          119.38odd121156.1.f.b.251.12
          119.45even481156.1.g.b.399.18
          119.47odd123332.1.bc.c.863.14
          119.59odd241156.1.c.b.579.12
          119.66odd241156.1.d.a.1155.22
          119.73even481156.1.g.b.423.28
          119.80even481156.1.g.b.423.18
          119.81even123332.1.m.b.3039.12
          119.87odd241156.1.d.a.1155.12
          119.94odd241156.1.c.b.579.22
          119.101odd61156.1.f.b.327.12
          119.108even481156.1.g.b.399.28
          119.115odd1268.1.f.a.47.1&check;2
          140.3odd121700.1.n.a.599.12
          140.59even61700.1.p.a.1551.12
          140.87odd121700.1.n.b.599.12
          357.353even12612.1.l.a.523.12
          476.3odd481156.1.g.b.155.18
          476.31odd481156.1.g.b.155.28
          476.47even123332.1.bc.c.863.14
          476.59even241156.1.c.b.579.12
          476.87even241156.1.d.a.1155.12
          476.115even1268.1.f.a.47.1&check;2
          476.143odd481156.1.g.b.179.28
          476.199odd481156.1.g.b.423.18
          476.227odd481156.1.g.b.399.28
          476.251even43332.1.bc.c.2223.14
          476.283odd481156.1.g.b.399.18
          476.311odd481156.1.g.b.423.28
          476.319odd123332.1.m.b.3039.12
          476.339even61156.1.f.b.327.12
          476.367odd481156.1.g.b.179.18
          476.387odd12inner3332.1.bc.b.863.14
          476.395even121156.1.f.b.251.12
          476.423even241156.1.d.a.1155.22
          476.451even241156.1.c.b.579.22
          595.234odd121700.1.p.a.251.12
          595.353even121700.1.n.b.999.12
          595.472even121700.1.n.a.999.12
          952.115even121088.1.p.a.319.12
          952.829odd121088.1.p.a.319.12
          1428.1067odd12612.1.l.a.523.12
          2380.1067odd121700.1.n.a.999.12
          2380.1543odd121700.1.n.b.999.12
          2380.2019even121700.1.p.a.251.12
          ", "is_complex": true, "table_nest_level": "1"}}, {"type": "table", "raw_content": "
          \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisted newform
          TwistMinDimCharParityOrdType
          68.1.f.a.47.1&check;2119.115odd12
          68.1.f.a.47.1&check;2476.115even12
          68.1.f.a.55.1yes27.3odd6
          68.1.f.a.55.1yes228.3even6
          612.1.l.a.55.1221.17even6
          612.1.l.a.55.1284.59odd6
          612.1.l.a.523.12357.353even12
          612.1.l.a.523.121428.1067odd12
          1088.1.p.a.191.1256.3even6
          1088.1.p.a.191.1256.45odd6
          1088.1.p.a.319.12952.115even12
          1088.1.p.a.319.12952.829odd12
          1156.1.c.b.579.12119.59odd24
          1156.1.c.b.579.12476.59even24
          1156.1.c.b.579.22119.94odd24
          1156.1.c.b.579.22476.451even24
          1156.1.d.a.1155.12119.87odd24
          1156.1.d.a.1155.12476.87even24
          1156.1.d.a.1155.22119.66odd24
          1156.1.d.a.1155.22476.423even24
          1156.1.f.b.251.12119.38odd12
          1156.1.f.b.251.12476.395even12
          1156.1.f.b.327.12119.101odd6
          1156.1.f.b.327.12476.339even6
          1156.1.g.b.155.18119.3even48
          1156.1.g.b.155.18476.3odd48
          1156.1.g.b.155.28119.31even48
          1156.1.g.b.155.28476.31odd48
          1156.1.g.b.179.18119.10even48
          1156.1.g.b.179.18476.367odd48
          1156.1.g.b.179.28119.24even48
          1156.1.g.b.179.28476.143odd48
          1156.1.g.b.399.18119.45even48
          1156.1.g.b.399.18476.283odd48
          1156.1.g.b.399.28119.108even48
          1156.1.g.b.399.28476.227odd48
          1156.1.g.b.423.18119.80even48
          1156.1.g.b.423.18476.199odd48
          1156.1.g.b.423.28119.73even48
          1156.1.g.b.423.28476.311odd48
          1700.1.n.a.599.1235.3even12
          1700.1.n.a.599.12140.3odd12
          1700.1.n.a.999.12595.472even12
          1700.1.n.a.999.122380.1067odd12
          1700.1.n.b.599.1235.17even12
          1700.1.n.b.599.12140.87odd12
          1700.1.n.b.999.12595.353even12
          1700.1.n.b.999.122380.1543odd12
          1700.1.p.a.251.12595.234odd12
          1700.1.p.a.251.122380.2019even12
          1700.1.p.a.1551.1235.24odd6
          1700.1.p.a.1551.12140.59even6
          3332.1.m.b.2843.127.4even3
          3332.1.m.b.2843.1228.11odd6
          3332.1.m.b.3039.12119.81even12
          3332.1.m.b.3039.12476.319odd12
          3332.1.bc.b.667.147.2even3inner
          3332.1.bc.b.667.1428.23odd6inner
          3332.1.bc.b.863.14119.30even12inner
          3332.1.bc.b.863.14476.387odd12inner
          3332.1.bc.b.2027.141.1even1trivial
          3332.1.bc.b.2027.144.3odd2CM
          3332.1.bc.b.2223.1417.13even4inner
          3332.1.bc.b.2223.1468.47odd4inner
          3332.1.bc.c.667.147.5odd6
          3332.1.bc.c.667.1428.19even6
          3332.1.bc.c.863.14119.47odd12
          3332.1.bc.c.863.14476.47even12
          3332.1.bc.c.2027.147.6odd2
          3332.1.bc.c.2027.1428.27even2
          3332.1.bc.c.2223.14119.13odd4
          3332.1.bc.c.2223.14476.251even4
          ", "content": {"html": "
          By<br>twisted newform
          TwistMinDimCharParityOrdType
          68.1.f.a.47.1&check;2119.115odd12
          68.1.f.a.47.1&check;2476.115even12
          68.1.f.a.55.1yes27.3odd6
          68.1.f.a.55.1yes228.3even6
          612.1.l.a.55.1221.17even6
          612.1.l.a.55.1284.59odd6
          612.1.l.a.523.12357.353even12
          612.1.l.a.523.121428.1067odd12
          1088.1.p.a.191.1256.3even6
          1088.1.p.a.191.1256.45odd6
          1088.1.p.a.319.12952.115even12
          1088.1.p.a.319.12952.829odd12
          1156.1.c.b.579.12119.59odd24
          1156.1.c.b.579.12476.59even24
          1156.1.c.b.579.22119.94odd24
          1156.1.c.b.579.22476.451even24
          1156.1.d.a.1155.12119.87odd24
          1156.1.d.a.1155.12476.87even24
          1156.1.d.a.1155.22119.66odd24
          1156.1.d.a.1155.22476.423even24
          1156.1.f.b.251.12119.38odd12
          1156.1.f.b.251.12476.395even12
          1156.1.f.b.327.12119.101odd6
          1156.1.f.b.327.12476.339even6
          1156.1.g.b.155.18119.3even48
          1156.1.g.b.155.18476.3odd48
          1156.1.g.b.155.28119.31even48
          1156.1.g.b.155.28476.31odd48
          1156.1.g.b.179.18119.10even48
          1156.1.g.b.179.18476.367odd48
          1156.1.g.b.179.28119.24even48
          1156.1.g.b.179.28476.143odd48
          1156.1.g.b.399.18119.45even48
          1156.1.g.b.399.18476.283odd48
          1156.1.g.b.399.28119.108even48
          1156.1.g.b.399.28476.227odd48
          1156.1.g.b.423.18119.80even48
          1156.1.g.b.423.18476.199odd48
          1156.1.g.b.423.28119.73even48
          1156.1.g.b.423.28476.311odd48
          1700.1.n.a.599.1235.3even12
          1700.1.n.a.599.12140.3odd12
          1700.1.n.a.999.12595.472even12
          1700.1.n.a.999.122380.1067odd12
          1700.1.n.b.599.1235.17even12
          1700.1.n.b.599.12140.87odd12
          1700.1.n.b.999.12595.353even12
          1700.1.n.b.999.122380.1543odd12
          1700.1.p.a.251.12595.234odd12
          1700.1.p.a.251.122380.2019even12
          1700.1.p.a.1551.1235.24odd6
          1700.1.p.a.1551.12140.59even6
          3332.1.m.b.2843.127.4even3
          3332.1.m.b.2843.1228.11odd6
          3332.1.m.b.3039.12119.81even12
          3332.1.m.b.3039.12476.319odd12
          3332.1.bc.b.667.147.2even3inner
          3332.1.bc.b.667.1428.23odd6inner
          3332.1.bc.b.863.14119.30even12inner
          3332.1.bc.b.863.14476.387odd12inner
          3332.1.bc.b.2027.141.1even1trivial
          3332.1.bc.b.2027.144.3odd2CM
          3332.1.bc.b.2223.1417.13even4inner
          3332.1.bc.b.2223.1468.47odd4inner
          3332.1.bc.c.667.147.5odd6
          3332.1.bc.c.667.1428.19even6
          3332.1.bc.c.863.14119.47odd12
          3332.1.bc.c.863.14476.47even12
          3332.1.bc.c.2027.147.6odd2
          3332.1.bc.c.2027.1428.27even2
          3332.1.bc.c.2223.14119.13odd4
          3332.1.bc.c.2223.14476.251even4
          ", "is_complex": true, "table_nest_level": "1"}}]], "main_html": "
          \n Show commands:\n Magma\n / PariGP\n / SageMath
          [N,k,chi] = [3332,1,Mod(667,3332)]
          mf = mfinit([N,k,chi],0)
          lf = mfeigenbasis(mf)
          from sage.modular.dirichlet import DirichletCharacter
          H = DirichletGroup(3332, base_ring=CyclotomicField(12))
          chi = DirichletCharacter(H, H._module([6, 4, 9]))
          N = Newforms(chi, 1, names=\"a\")
          //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
          chi := DirichletCharacter(\"3332.667\");
          S:= CuspForms(chi, 1);
          N := Newforms(S);
          Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
          Weight: \\( k \\) \\(=\\)\\( 1 \\)
          Character orbit: \\([\\chi]\\) \\(=\\)3332.bc (of order \\(12\\), degree \\(4\\), not minimal)

          Newform invariants

          sage:\u00a0f = N[0] # Warning: the index may be different
          gp:\u00a0f = lf[1] \\\\ Warning: the index may be different
          Self dual: no
          Analytic conductor: \\(1.66288462209\\)
          Analytic rank: \\(0\\)
          Dimension: \\(4\\)
          Coefficient field: \\(\\Q(\\zeta_{12})\\)
          gp:\u00a0f.mod \\\\ as an extension of the character field
          Defining polynomial: \\( x^{4} - x^{2} + 1 \\)\"Copy\"Toggle
          Coefficient ring: \\(\\Z[a_1, a_2]\\)
          Coefficient ring index: \\( 1 \\)
          Twist minimal: no (minimal twist has level 68)
          Projective image:\\(D_{4}\\)
          Projective field:Galois closure of 4.2.19652.1
          Artin image:$C_4\\wr C_2\\times C_6$
          Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)

          Embedding invariants

          Embedding label 2027.1
          Root\\(0.866025 - 0.500000i\\) of defining polynomial
          Character\\(\\chi\\)\\(=\\)3332.2027
          Dual form3332.1.bc.b.863.1
          sage:\u00a0f.q_expansion() # note that sage often uses an isomorphic number field
          gp:\u00a0mfcoefs(f, 20)
          \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
          \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\"Copy\"Toggle

          Character values

          We give the values of \\chi on generators for \\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times.

          \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
          \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)

          Coefficient data

          For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the\nSatake parameters \\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).

          \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
          \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
          \n \\(2\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(3\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(4\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(5\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(6\\)\n \n 0\n \n 0\n
          \n \\(7\\)\n \n 0\n \n 0\n
          \n \\(8\\)\n \n 1.00000i\n 1.00000i
          \n \\(9\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(10\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(11\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(12\\)\n \n 0\n \n 0\n
          \n \\(13\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(14\\)\n \n 0\n \n 0\n
          \n \\(15\\)\n \n 0\n \n 0\n
          \n \\(16\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(17\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(18\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(19\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(20\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(21\\)\n \n 0\n \n 0\n
          \n \\(22\\)\n \n 0\n \n 0\n
          \n \\(23\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(24\\)\n \n 0\n \n 0\n
          \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(26\\)\n \n 0\n \n 0\n
          \n \\(27\\)\n \n 0\n \n 0\n
          \n \\(28\\)\n \n 0\n \n 0\n
          \n \\(29\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(30\\)\n \n 0\n \n 0\n
          \n \\(31\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(33\\)\n \n 0\n \n 0\n
          \n \\(34\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(35\\)\n \n 0\n \n 0\n
          \n \\(36\\)\n \n 1.00000i\n 1.00000i
          \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(38\\)\n \n 0\n \n 0\n
          \n \\(39\\)\n \n 0\n \n 0\n
          \n \\(40\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(41\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(42\\)\n \n 0\n \n 0\n
          \n \\(43\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(44\\)\n \n 0\n \n 0\n
          \n \\(45\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(46\\)\n \n 0\n \n 0\n
          \n \\(47\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(48\\)\n \n 0\n \n 0\n
          \n \\(49\\)\n \n 0\n \n 0\n
          \n \\(50\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(51\\)\n \n 0\n \n 0\n
          \n \\(52\\)\n \n 0\n \n 0\n
          \n \\(53\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(54\\)\n \n 0\n \n 0\n
          \n \\(55\\)\n \n 0\n \n 0\n
          \n \\(56\\)\n \n 0\n \n 0\n
          \n \\(57\\)\n \n 0\n \n 0\n
          \n \\(58\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(59\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(60\\)\n \n 0\n \n 0\n
          \n \\(61\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(62\\)\n \n 0\n \n 0\n
          \n \\(63\\)\n \n 0\n \n 0\n
          \n \\(64\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(65\\)\n \n 0\n \n 0\n
          \n \\(66\\)\n \n 0\n \n 0\n
          \n \\(67\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(69\\)\n \n 0\n \n 0\n
          \n \\(70\\)\n \n 0\n \n 0\n
          \n \\(71\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(72\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(73\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(74\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(75\\)\n \n 0\n \n 0\n
          \n \\(76\\)\n \n 0\n \n 0\n
          \n \\(77\\)\n \n 0\n \n 0\n
          \n \\(78\\)\n \n 0\n \n 0\n
          \n \\(79\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(81\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(83\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(84\\)\n \n 0\n \n 0\n
          \n \\(85\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(86\\)\n \n 0\n \n 0\n
          \n \\(87\\)\n \n 0\n \n 0\n
          \n \\(88\\)\n \n 0\n \n 0\n
          \n \\(89\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(90\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(91\\)\n \n 0\n \n 0\n
          \n \\(92\\)\n \n 0\n \n 0\n
          \n \\(93\\)\n \n 0\n \n 0\n
          \n \\(94\\)\n \n 0\n \n 0\n
          \n \\(95\\)\n \n 0\n \n 0\n
          \n \\(96\\)\n \n 0\n \n 0\n
          \n \\(97\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(98\\)\n \n 0\n \n 0\n
          \n \\(99\\)\n \n 0\n \n 0\n
          \n \\(100\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(101\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(102\\)\n \n 0\n \n 0\n
          \n \\(103\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(104\\)\n \n 0\n \n 0\n
          \n \\(105\\)\n \n 0\n \n 0\n
          \n \\(106\\)\n \n 0\n \n 0\n
          \n \\(107\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(108\\)\n \n 0\n \n 0\n
          \n \\(109\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(110\\)\n \n 0\n \n 0\n
          \n \\(111\\)\n \n 0\n \n 0\n
          \n \\(112\\)\n \n 0\n \n 0\n
          \n \\(113\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(114\\)\n \n 0\n \n 0\n
          \n \\(115\\)\n \n 0\n \n 0\n
          \n \\(116\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(117\\)\n \n 0\n \n 0\n
          \n \\(118\\)\n \n 0\n \n 0\n
          \n \\(119\\)\n \n 0\n \n 0\n
          \n \\(120\\)\n \n 0\n \n 0\n
          \n \\(121\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(123\\)\n \n 0\n \n 0\n
          \n \\(124\\)\n \n 0\n \n 0\n
          \n \\(125\\)\n \n 0\n \n 0\n
          \n \\(126\\)\n \n 0\n \n 0\n
          \n \\(127\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(128\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(129\\)\n \n 0\n \n 0\n
          \n \\(130\\)\n \n 0\n \n 0\n
          \n \\(131\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(132\\)\n \n 0\n \n 0\n
          \n \\(133\\)\n \n 0\n \n 0\n
          \n \\(134\\)\n \n 0\n \n 0\n
          \n \\(135\\)\n \n 0\n \n 0\n
          \n \\(136\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(137\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(138\\)\n \n 0\n \n 0\n
          \n \\(139\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(140\\)\n \n 0\n \n 0\n
          \n \\(141\\)\n \n 0\n \n 0\n
          \n \\(142\\)\n \n 0\n \n 0\n
          \n \\(143\\)\n \n 0\n \n 0\n
          \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(145\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(147\\)\n \n 0\n \n 0\n
          \n \\(148\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(149\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \\(150\\)\n \n 0\n \n 0\n
          \n \\(151\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(152\\)\n \n 0\n \n 0\n
          \n \\(153\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(154\\)\n \n 0\n \n 0\n
          \n \\(155\\)\n \n 0\n \n 0\n
          \n \\(156\\)\n \n 0\n \n 0\n
          \n \\(157\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(158\\)\n \n 0\n \n 0\n
          \n \\(159\\)\n \n 0\n \n 0\n
          \n \\(160\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(161\\)\n \n 0\n \n 0\n
          \n \\(162\\)\n \n 1.00000i\n 1.00000i
          \n \\(163\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(164\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(165\\)\n \n 0\n \n 0\n
          \n \\(166\\)\n \n 0\n \n 0\n
          \n \\(167\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(168\\)\n \n 0\n \n 0\n
          \n \\(169\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(170\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(171\\)\n \n 0\n \n 0\n
          \n \\(172\\)\n \n 0\n \n 0\n
          \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(174\\)\n \n 0\n \n 0\n
          \n \\(175\\)\n \n 0\n \n 0\n
          \n \\(176\\)\n \n 0\n \n 0\n
          \n \\(177\\)\n \n 0\n \n 0\n
          \n \\(178\\)\n \n 0\n \n 0\n
          \n \\(179\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(180\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(182\\)\n \n 0\n \n 0\n
          \n \\(183\\)\n \n 0\n \n 0\n
          \n \\(184\\)\n \n 0\n \n 0\n
          \n \\(185\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(186\\)\n \n 0\n \n 0\n
          \n \\(187\\)\n \n 0\n \n 0\n
          \n \\(188\\)\n \n 0\n \n 0\n
          \n \\(189\\)\n \n 0\n \n 0\n
          \n \\(190\\)\n \n 0\n \n 0\n
          \n \\(191\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(192\\)\n \n 0\n \n 0\n
          \n \\(193\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(194\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(195\\)\n \n 0\n \n 0\n
          \n \\(196\\)\n \n 0\n \n 0\n
          \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(198\\)\n \n 0\n \n 0\n
          \n \\(199\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(200\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(201\\)\n \n 0\n \n 0\n
          \n \\(202\\)\n \n 0\n \n 0\n
          \n \\(203\\)\n \n 0\n \n 0\n
          \n \\(204\\)\n \n 0\n \n 0\n
          \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
          \n \\(206\\)\n \n 0\n \n 0\n
          \n \\(207\\)\n \n 0\n \n 0\n
          \n \\(208\\)\n \n 0\n \n 0\n
          \n \\(209\\)\n \n 0\n \n 0\n
          \n \\(210\\)\n \n 0\n \n 0\n
          \n \\(211\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(212\\)\n \n 0\n \n 0\n
          \n \\(213\\)\n \n 0\n \n 0\n
          \n \\(214\\)\n \n 0\n \n 0\n
          \n \\(215\\)\n \n 0\n \n 0\n
          \n \\(216\\)\n \n 0\n \n 0\n
          \n \\(217\\)\n \n 0\n \n 0\n
          \n \\(218\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(219\\)\n \n 0\n \n 0\n
          \n \\(220\\)\n \n 0\n \n 0\n
          \n \\(221\\)\n \n 0\n \n 0\n
          \n \\(222\\)\n \n 0\n \n 0\n
          \n \\(223\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(224\\)\n \n 0\n \n 0\n
          \n \\(225\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(227\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(228\\)\n \n 0\n \n 0\n
          \n \\(229\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(230\\)\n \n 0\n \n 0\n
          \n \\(231\\)\n \n 0\n \n 0\n
          \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(234\\)\n \n 0\n \n 0\n
          \n \\(235\\)\n \n 0\n \n 0\n
          \n \\(236\\)\n \n 0\n \n 0\n
          \n \\(237\\)\n \n 0\n \n 0\n
          \n \\(238\\)\n \n 0\n \n 0\n
          \n \\(239\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(240\\)\n \n 0\n \n 0\n
          \n \\(241\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(242\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(243\\)\n \n 0\n \n 0\n
          \n \\(244\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(245\\)\n \n 0\n \n 0\n
          \n \\(246\\)\n \n 0\n \n 0\n
          \n \\(247\\)\n \n 0\n \n 0\n
          \n \\(248\\)\n \n 0\n \n 0\n
          \n \\(249\\)\n \n 0\n \n 0\n
          \n \\(250\\)\n \n 0\n \n 0\n
          \n \\(251\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(252\\)\n \n 0\n \n 0\n
          \n \\(253\\)\n \n 0\n \n 0\n
          \n \\(254\\)\n \n 0\n \n 0\n
          \n \\(255\\)\n \n 0\n \n 0\n
          \n \\(256\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(257\\)\n \n 1.73205\n \n \u2212\n \n 1.00000i\n 1.73205\n \n \u2212\n \n 1.00000i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(258\\)\n \n 0\n \n 0\n
          \n \\(259\\)\n \n 0\n \n 0\n
          \n \\(260\\)\n \n 0\n \n 0\n
          \n \\(261\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(262\\)\n \n 0\n \n 0\n
          \n \\(263\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(264\\)\n \n 0\n \n 0\n
          \n \\(265\\)\n \n 0\n \n 0\n
          \n \\(266\\)\n \n 0\n \n 0\n
          \n \\(267\\)\n \n 0\n \n 0\n
          \n \\(268\\)\n \n 0\n \n 0\n
          \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(270\\)\n \n 0\n \n 0\n
          \n \\(271\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(272\\)\n \n 1.00000\n \n 1.00000\n
          \n \\(273\\)\n \n 0\n \n 0\n
          \n \\(274\\)\n \n 0\n \n 0\n
          \n \\(275\\)\n \n 0\n \n 0\n
          \n \\(276\\)\n \n 0\n \n 0\n
          \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(278\\)\n \n 0\n \n 0\n
          \n \\(279\\)\n \n 0\n \n 0\n
          \n \\(280\\)\n \n 0\n \n 0\n
          \n \\(281\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(282\\)\n \n 0\n \n 0\n
          \n \\(283\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(284\\)\n \n 0\n \n 0\n
          \n \\(285\\)\n \n 0\n \n 0\n
          \n \\(286\\)\n \n 0\n \n 0\n
          \n \\(287\\)\n \n 0\n \n 0\n
          \n \\(288\\)\n \n \u22121.00000\n \n \u22121.00000\n
          \n \\(289\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(290\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(291\\)\n \n 0\n \n 0\n
          \n \\(292\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(293\\)\n \n 2.00000\n \n 2.00000\n \n 1.00000\n \n \\(0\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(294\\)\n \n 0\n \n 0\n
          \n \\(295\\)\n \n 0\n \n 0\n
          \n \\(296\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(297\\)\n \n 0\n \n 0\n
          \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n 1.73205\n \n +\n \n 1.00000i
          \n \\(299\\)\n \n 0\n \n 0\n
          \n \\(300\\)\n \n 0\n \n 0\n
          \n \\(301\\)\n \n 0\n \n 0\n
          \n \\(302\\)\n \n 0\n \n 0\n
          \n \\(303\\)\n \n 0\n \n 0\n
          \n \\(304\\)\n \n 0\n \n 0\n
          \n \\(305\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
          \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(307\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(308\\)\n \n 0\n \n 0\n
          \n \\(309\\)\n \n 0\n \n 0\n
          \n \\(310\\)\n \n 0\n \n 0\n
          \n \\(311\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(312\\)\n \n 0\n \n 0\n
          \n \\(313\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(314\\)\n \n 2.00000i\n 2.00000i
          \n \\(315\\)\n \n 0\n \n 0\n
          \n \\(316\\)\n \n 0\n \n 0\n
          \n \\(317\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(318\\)\n \n 0\n \n 0\n
          \n \\(319\\)\n \n 0\n \n 0\n
          \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(321\\)\n \n 0\n \n 0\n
          \n \\(322\\)\n \n 0\n \n 0\n
          \n \\(323\\)\n \n 0\n \n 0\n
          \n \\(324\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(325\\)\n \n 0\n \n 0\n
          \n \\(326\\)\n \n 0\n \n 0\n
          \n \\(327\\)\n \n 0\n \n 0\n
          \n \\(328\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(329\\)\n \n 0\n \n 0\n
          \n \\(330\\)\n \n 0\n \n 0\n
          \n \\(331\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(332\\)\n \n 0\n \n 0\n
          \n \\(333\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(334\\)\n \n 0\n \n 0\n
          \n \\(335\\)\n \n 0\n \n 0\n
          \n \\(336\\)\n \n 0\n \n 0\n
          \n \\(337\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(338\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(339\\)\n \n 0\n \n 0\n
          \n \\(340\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(341\\)\n \n 0\n \n 0\n
          \n \\(342\\)\n \n 0\n \n 0\n
          \n \\(343\\)\n \n 0\n \n 0\n
          \n \\(344\\)\n \n 0\n \n 0\n
          \n \\(345\\)\n \n 0\n \n 0\n
          \n \\(346\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(347\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(348\\)\n \n 0\n \n 0\n
          \n \\(349\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(350\\)\n \n 0\n \n 0\n
          \n \\(351\\)\n \n 0\n \n 0\n
          \n \\(352\\)\n \n 0\n \n 0\n
          \n \\(353\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(354\\)\n \n 0\n \n 0\n
          \n \\(355\\)\n \n 0\n \n 0\n
          \n \\(356\\)\n \n 0\n \n 0\n
          \n \\(357\\)\n \n 0\n \n 0\n
          \n \\(358\\)\n \n 0\n \n 0\n
          \n \\(359\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(361\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(362\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(363\\)\n \n 0\n \n 0\n
          \n \\(364\\)\n \n 0\n \n 0\n
          \n \\(365\\)\n \n 2.00000i\n 2.00000i
          \n \\(366\\)\n \n 0\n \n 0\n
          \n \\(367\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(368\\)\n \n 0\n \n 0\n
          \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(370\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(371\\)\n \n 0\n \n 0\n
          \n \\(372\\)\n \n 0\n \n 0\n
          \n \\(373\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(374\\)\n \n 0\n \n 0\n
          \n \\(375\\)\n \n 0\n \n 0\n
          \n \\(376\\)\n \n 0\n \n 0\n
          \n \\(377\\)\n \n 0\n \n 0\n
          \n \\(378\\)\n \n 0\n \n 0\n
          \n \\(379\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(380\\)\n \n 0\n \n 0\n
          \n \\(381\\)\n \n 0\n \n 0\n
          \n \\(382\\)\n \n 0\n \n 0\n
          \n \\(383\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(384\\)\n \n 0\n \n 0\n
          \n \\(385\\)\n \n 0\n \n 0\n
          \n \\(386\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(387\\)\n \n 0\n \n 0\n
          \n \\(388\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(389\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \\(390\\)\n \n 0\n \n 0\n
          \n \\(391\\)\n \n 0\n \n 0\n
          \n \\(392\\)\n \n 0\n \n 0\n
          \n \\(393\\)\n \n 0\n \n 0\n
          \n \\(394\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(395\\)\n \n 0\n \n 0\n
          \n \\(396\\)\n \n 0\n \n 0\n
          \n \\(397\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(398\\)\n \n 0\n \n 0\n
          \n \\(399\\)\n \n 0\n \n 0\n
          \n \\(400\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(401\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(402\\)\n \n 0\n \n 0\n
          \n \\(403\\)\n \n 0\n \n 0\n
          \n \\(404\\)\n \n 0\n \n 0\n
          \n \\(405\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(406\\)\n \n 0\n \n 0\n
          \n \\(407\\)\n \n 0\n \n 0\n
          \n \\(408\\)\n \n 0\n \n 0\n
          \n \\(409\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(410\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(411\\)\n \n 0\n \n 0\n
          \n \\(412\\)\n \n 0\n \n 0\n
          \n \\(413\\)\n \n 0\n \n 0\n
          \n \\(414\\)\n \n 0\n \n 0\n
          \n \\(415\\)\n \n 0\n \n 0\n
          \n \\(416\\)\n \n 0\n \n 0\n
          \n \\(417\\)\n \n 0\n \n 0\n
          \n \\(418\\)\n \n 0\n \n 0\n
          \n \\(419\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(420\\)\n \n 0\n \n 0\n
          \n \\(421\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(422\\)\n \n 0\n \n 0\n
          \n \\(423\\)\n \n 0\n \n 0\n
          \n \\(424\\)\n \n 0\n \n 0\n
          \n \\(425\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(426\\)\n \n 0\n \n 0\n
          \n \\(427\\)\n \n 0\n \n 0\n
          \n \\(428\\)\n \n 0\n \n 0\n
          \n \\(429\\)\n \n 0\n \n 0\n
          \n \\(430\\)\n \n 0\n \n 0\n
          \n \\(431\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(432\\)\n \n 0\n \n 0\n
          \n \\(433\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(434\\)\n \n 0\n \n 0\n
          \n \\(435\\)\n \n 0\n \n 0\n
          \n \\(436\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(437\\)\n \n 0\n \n 0\n
          \n \\(438\\)\n \n 0\n \n 0\n
          \n \\(439\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(440\\)\n \n 0\n \n 0\n
          \n \\(441\\)\n \n 0\n \n 0\n
          \n \\(442\\)\n \n 0\n \n 0\n
          \n \\(443\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(444\\)\n \n 0\n \n 0\n
          \n \\(445\\)\n \n 0\n \n 0\n
          \n \\(446\\)\n \n 0\n \n 0\n
          \n \\(447\\)\n \n 0\n \n 0\n
          \n \\(448\\)\n \n 0\n \n 0\n
          \n \\(449\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(450\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(451\\)\n \n 0\n \n 0\n
          \n \\(452\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(453\\)\n \n 0\n \n 0\n
          \n \\(454\\)\n \n 0\n \n 0\n
          \n \\(455\\)\n \n 0\n \n 0\n
          \n \\(456\\)\n \n 0\n \n 0\n
          \n \\(457\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(458\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(459\\)\n \n 0\n \n 0\n
          \n \\(460\\)\n \n 0\n \n 0\n
          \n \\(461\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(462\\)\n \n 0\n \n 0\n
          \n \\(463\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(464\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(465\\)\n \n 0\n \n 0\n
          \n \\(466\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(467\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(468\\)\n \n 0\n \n 0\n
          \n \\(469\\)\n \n 0\n \n 0\n
          \n \\(470\\)\n \n 0\n \n 0\n
          \n \\(471\\)\n \n 0\n \n 0\n
          \n \\(472\\)\n \n 0\n \n 0\n
          \n \\(473\\)\n \n 0\n \n 0\n
          \n \\(474\\)\n \n 0\n \n 0\n
          \n \\(475\\)\n \n 0\n \n 0\n
          \n \\(476\\)\n \n 0\n \n 0\n
          \n \\(477\\)\n \n 0\n \n 0\n
          \n \\(478\\)\n \n 0\n \n 0\n
          \n \\(479\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(480\\)\n \n 0\n \n 0\n
          \n \\(481\\)\n \n 0\n \n 0\n
          \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(483\\)\n \n 0\n \n 0\n
          \n \\(484\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(485\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(486\\)\n \n 0\n \n 0\n
          \n \\(487\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(489\\)\n \n 0\n \n 0\n
          \n \\(490\\)\n \n 0\n \n 0\n
          \n \\(491\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(492\\)\n \n 0\n \n 0\n
          \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(494\\)\n \n 0\n \n 0\n
          \n \\(495\\)\n \n 0\n \n 0\n
          \n \\(496\\)\n \n 0\n \n 0\n
          \n \\(497\\)\n \n 0\n \n 0\n
          \n \\(498\\)\n \n 0\n \n 0\n
          \n \\(499\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(500\\)\n \n 0\n \n 0\n
          \n \\(501\\)\n \n 0\n \n 0\n
          \n \\(502\\)\n \n 0\n \n 0\n
          \n \\(503\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(504\\)\n \n 0\n \n 0\n
          \n \\(505\\)\n \n 0\n \n 0\n
          \n \\(506\\)\n \n 0\n \n 0\n
          \n \\(507\\)\n \n 0\n \n 0\n
          \n \\(508\\)\n \n 0\n \n 0\n
          \n \\(509\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \\(510\\)\n \n 0\n \n 0\n
          \n \\(511\\)\n \n 0\n \n 0\n
          \n \\(512\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(513\\)\n \n 0\n \n 0\n
          \n \\(514\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
          \n \\(515\\)\n \n 0\n \n 0\n
          \n \\(516\\)\n \n 0\n \n 0\n
          \n \\(517\\)\n \n 0\n \n 0\n
          \n \\(518\\)\n \n 0\n \n 0\n
          \n \\(519\\)\n \n 0\n \n 0\n
          \n \\(520\\)\n \n 0\n \n 0\n
          \n \\(521\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(522\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(523\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(524\\)\n \n 0\n \n 0\n
          \n \\(525\\)\n \n 0\n \n 0\n
          \n \\(526\\)\n \n 0\n \n 0\n
          \n \\(527\\)\n \n 0\n \n 0\n
          \n \\(528\\)\n \n 0\n \n 0\n
          \n \\(529\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(530\\)\n \n 0\n \n 0\n
          \n \\(531\\)\n \n 0\n \n 0\n
          \n \\(532\\)\n \n 0\n \n 0\n
          \n \\(533\\)\n \n 0\n \n 0\n
          \n \\(534\\)\n \n 0\n \n 0\n
          \n \\(535\\)\n \n 0\n \n 0\n
          \n \\(536\\)\n \n 0\n \n 0\n
          \n \\(537\\)\n \n 0\n \n 0\n
          \n \\(538\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(539\\)\n \n 0\n \n 0\n
          \n \\(540\\)\n \n 0\n \n 0\n
          \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(542\\)\n \n 0\n \n 0\n
          \n \\(543\\)\n \n 0\n \n 0\n
          \n \\(544\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(545\\)\n \n \u22122.00000\n \n \u22122.00000\n
          \n \\(546\\)\n \n 0\n \n 0\n
          \n \\(547\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(548\\)\n \n 0\n \n 0\n
          \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(550\\)\n \n 0\n \n 0\n
          \n \\(551\\)\n \n 0\n \n 0\n
          \n \\(552\\)\n \n 0\n \n 0\n
          \n \\(553\\)\n \n 0\n \n 0\n
          \n \\(554\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(555\\)\n \n 0\n \n 0\n
          \n \\(556\\)\n \n 0\n \n 0\n
          \n \\(557\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(558\\)\n \n 0\n \n 0\n
          \n \\(559\\)\n \n 0\n \n 0\n
          \n \\(560\\)\n \n 0\n \n 0\n
          \n \\(561\\)\n \n 0\n \n 0\n
          \n \\(562\\)\n \n 0\n \n 0\n
          \n \\(563\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(564\\)\n \n 0\n \n 0\n
          \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
          \n \\(566\\)\n \n 0\n \n 0\n
          \n \\(567\\)\n \n 0\n \n 0\n
          \n \\(568\\)\n \n 0\n \n 0\n
          \n \\(569\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(570\\)\n \n 0\n \n 0\n
          \n \\(571\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(572\\)\n \n 0\n \n 0\n
          \n \\(573\\)\n \n 0\n \n 0\n
          \n \\(574\\)\n \n 0\n \n 0\n
          \n \\(575\\)\n \n 0\n \n 0\n
          \n \\(576\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
          \n \\(577\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(579\\)\n \n 0\n \n 0\n
          \n \\(580\\)\n \n 2.00000i\n 2.00000i
          \n \\(581\\)\n \n 0\n \n 0\n
          \n \\(582\\)\n \n 0\n \n 0\n
          \n \\(583\\)\n \n 0\n \n 0\n
          \n \\(584\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(585\\)\n \n 0\n \n 0\n
          \n \\(586\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(587\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(588\\)\n \n 0\n \n 0\n
          \n \\(589\\)\n \n 0\n \n 0\n
          \n \\(590\\)\n \n 0\n \n 0\n
          \n \\(591\\)\n \n 0\n \n 0\n
          \n \\(592\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(593\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(594\\)\n \n 0\n \n 0\n
          \n \\(595\\)\n \n 0\n \n 0\n
          \n \\(596\\)\n \n \u22122.00000\n \n \u22122.00000\n
          \n \\(597\\)\n \n 0\n \n 0\n
          \n \\(598\\)\n \n 0\n \n 0\n
          \n \\(599\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(600\\)\n \n 0\n \n 0\n
          \n \\(601\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(602\\)\n \n 0\n \n 0\n
          \n \\(603\\)\n \n 0\n \n 0\n
          \n \\(604\\)\n \n 0\n \n 0\n
          \n \\(605\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(606\\)\n \n 0\n \n 0\n
          \n \\(607\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(608\\)\n \n 0\n \n 0\n
          \n \\(609\\)\n \n 0\n \n 0\n
          \n \\(610\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(611\\)\n \n 0\n \n 0\n
          \n \\(612\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
          \n \\(613\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(614\\)\n \n 0\n \n 0\n
          \n \\(615\\)\n \n 0\n \n 0\n
          \n \\(616\\)\n \n 0\n \n 0\n
          \n \\(617\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(618\\)\n \n 0\n \n 0\n
          \n \\(619\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(620\\)\n \n 0\n \n 0\n
          \n \\(621\\)\n \n 0\n \n 0\n
          \n \\(622\\)\n \n 0\n \n 0\n
          \n \\(623\\)\n \n 0\n \n 0\n
          \n \\(624\\)\n \n 0\n \n 0\n
          \n \\(625\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
          \n \\(626\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
          \n \\(627\\)\n \n 0\n \n 0\n
          \n \\(628\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(629\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(630\\)\n \n 0\n \n 0\n
          \n \\(631\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(632\\)\n \n 0\n \n 0\n
          \n \\(633\\)\n \n 0\n \n 0\n
          \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(635\\)\n \n 0\n \n 0\n
          \n \\(636\\)\n \n 0\n \n 0\n
          \n \\(637\\)\n \n 0\n \n 0\n
          \n \\(638\\)\n \n 0\n \n 0\n
          \n \\(639\\)\n \n 0\n \n 0\n
          \n \\(640\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(641\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(642\\)\n \n 0\n \n 0\n
          \n \\(643\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(644\\)\n \n 0\n \n 0\n
          \n \\(645\\)\n \n 0\n \n 0\n
          \n \\(646\\)\n \n 0\n \n 0\n
          \n \\(647\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
          \n \\(649\\)\n \n 0\n \n 0\n
          \n \\(650\\)\n \n 0\n \n 0\n
          \n \\(651\\)\n \n 0\n \n 0\n
          \n \\(652\\)\n \n 0\n \n 0\n
          \n \\(653\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \\(654\\)\n \n 0\n \n 0\n
          \n \\(655\\)\n \n 0\n \n 0\n
          \n \\(656\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(658\\)\n \n 0\n \n 0\n
          \n \\(659\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(660\\)\n \n 0\n \n 0\n
          \n \\(661\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(662\\)\n \n 0\n \n 0\n
          \n \\(663\\)\n \n 0\n \n 0\n
          \n \\(664\\)\n \n 0\n \n 0\n
          \n \\(665\\)\n \n 0\n \n 0\n
          \n \\(666\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(667\\)\n \n 0\n \n 0\n
          \n \\(668\\)\n \n 0\n \n 0\n
          \n \\(669\\)\n \n 0\n \n 0\n
          \n \\(670\\)\n \n 0\n \n 0\n
          \n \\(671\\)\n \n 0\n \n 0\n
          \n \\(672\\)\n \n 0\n \n 0\n
          \n \\(673\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(674\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(675\\)\n \n 0\n \n 0\n
          \n \\(676\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(678\\)\n \n 0\n \n 0\n
          \n \\(679\\)\n \n 0\n \n 0\n
          \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(681\\)\n \n 0\n \n 0\n
          \n \\(682\\)\n \n 0\n \n 0\n
          \n \\(683\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(684\\)\n \n 0\n \n 0\n
          \n \\(685\\)\n \n 0\n \n 0\n
          \n \\(686\\)\n \n 0\n \n 0\n
          \n \\(687\\)\n \n 0\n \n 0\n
          \n \\(688\\)\n \n 0\n \n 0\n
          \n \\(689\\)\n \n 0\n \n 0\n
          \n \\(690\\)\n \n 0\n \n 0\n
          \n \\(691\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
          \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
          \n \\(692\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(693\\)\n \n 0\n \n 0\n
          \n \\(694\\)\n \n 0\n \n 0\n
          \n \\(695\\)\n \n 0\n \n 0\n
          \n \\(696\\)\n \n 0\n \n 0\n
          \n \\(697\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(698\\)\n \n 0\n \n 0\n
          \n \\(699\\)\n \n 0\n \n 0\n
          \n \\(700\\)\n \n 0\n \n 0\n
          \n \\(701\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(702\\)\n \n 0\n \n 0\n
          \n \\(703\\)\n \n 0\n \n 0\n
          \n \\(704\\)\n \n 0\n \n 0\n
          \n \\(705\\)\n \n 0\n \n 0\n
          \n \\(706\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(707\\)\n \n 0\n \n 0\n
          \n \\(708\\)\n \n 0\n \n 0\n
          \n \\(709\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(710\\)\n \n 0\n \n 0\n
          \n \\(711\\)\n \n 0\n \n 0\n
          \n \\(712\\)\n \n 0\n \n 0\n
          \n \\(713\\)\n \n 0\n \n 0\n
          \n \\(714\\)\n \n 0\n \n 0\n
          \n \\(715\\)\n \n 0\n \n 0\n
          \n \\(716\\)\n \n 0\n \n 0\n
          \n \\(717\\)\n \n 0\n \n 0\n
          \n \\(718\\)\n \n 0\n \n 0\n
          \n \\(719\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(720\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(721\\)\n \n 0\n \n 0\n
          \n \\(722\\)\n \n 1.00000i\n 1.00000i
          \n \\(723\\)\n \n 0\n \n 0\n
          \n \\(724\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(725\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(726\\)\n \n 0\n \n 0\n
          \n \\(727\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(728\\)\n \n 0\n \n 0\n
          \n \\(729\\)\n \n 1.00000i\n 1.00000i
          \n \\(730\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(731\\)\n \n 0\n \n 0\n
          \n \\(732\\)\n \n 0\n \n 0\n
          \n \\(733\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(734\\)\n \n 0\n \n 0\n
          \n \\(735\\)\n \n 0\n \n 0\n
          \n \\(736\\)\n \n 0\n \n 0\n
          \n \\(737\\)\n \n 0\n \n 0\n
          \n \\(738\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(739\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(740\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
          \n \\(741\\)\n \n 0\n \n 0\n
          \n \\(742\\)\n \n 0\n \n 0\n
          \n \\(743\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(744\\)\n \n 0\n \n 0\n
          \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n 0.732051\n \n +\n \n 2.73205i
          \n \\(746\\)\n \n 0\n \n 0\n
          \n \\(747\\)\n \n 0\n \n 0\n
          \n \\(748\\)\n \n 0\n \n 0\n
          \n \\(749\\)\n \n 0\n \n 0\n
          \n \\(750\\)\n \n 0\n \n 0\n
          \n \\(751\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(752\\)\n \n 0\n \n 0\n
          \n \\(753\\)\n \n 0\n \n 0\n
          \n \\(754\\)\n \n 0\n \n 0\n
          \n \\(755\\)\n \n 0\n \n 0\n
          \n \\(756\\)\n \n 0\n \n 0\n
          \n \\(757\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(758\\)\n \n 0\n \n 0\n
          \n \\(759\\)\n \n 0\n \n 0\n
          \n \\(760\\)\n \n 0\n \n 0\n
          \n \\(761\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(762\\)\n \n 0\n \n 0\n
          \n \\(763\\)\n \n 0\n \n 0\n
          \n \\(764\\)\n \n 0\n \n 0\n
          \n \\(765\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(766\\)\n \n 0\n \n 0\n
          \n \\(767\\)\n \n 0\n \n 0\n
          \n \\(768\\)\n \n 0\n \n 0\n
          \n \\(769\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(770\\)\n \n 0\n \n 0\n
          \n \\(771\\)\n \n 0\n \n 0\n
          \n \\(772\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(773\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
          \n \\(774\\)\n \n 0\n \n 0\n
          \n \\(775\\)\n \n 0\n \n 0\n
          \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(777\\)\n \n 0\n \n 0\n
          \n \\(778\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(779\\)\n \n 0\n \n 0\n
          \n \\(780\\)\n \n 0\n \n 0\n
          \n \\(781\\)\n \n 0\n \n 0\n
          \n \\(782\\)\n \n 0\n \n 0\n
          \n \\(783\\)\n \n 0\n \n 0\n
          \n \\(784\\)\n \n 0\n \n 0\n
          \n \\(785\\)\n \n \u22122.00000\n \n +\n \n 2.00000i\n \u22122.00000\n \n +\n \n 2.00000i
          \n \\(786\\)\n \n 0\n \n 0\n
          \n \\(787\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(788\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(789\\)\n \n 0\n \n 0\n
          \n \\(790\\)\n \n 0\n \n 0\n
          \n \\(791\\)\n \n 0\n \n 0\n
          \n \\(792\\)\n \n 0\n \n 0\n
          \n \\(793\\)\n \n 0\n \n 0\n
          \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(795\\)\n \n 0\n \n 0\n
          \n \\(796\\)\n \n 0\n \n 0\n
          \n \\(797\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(798\\)\n \n 0\n \n 0\n
          \n \\(799\\)\n \n 0\n \n 0\n
          \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(801\\)\n \n 0\n \n 0\n
          \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(803\\)\n \n 0\n \n 0\n
          \n \\(804\\)\n \n 0\n \n 0\n
          \n \\(805\\)\n \n 0\n \n 0\n
          \n \\(806\\)\n \n 0\n \n 0\n
          \n \\(807\\)\n \n 0\n \n 0\n
          \n \\(808\\)\n \n 0\n \n 0\n
          \n \\(809\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(810\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
          \n \\(811\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
          \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
          \n \\(812\\)\n \n 0\n \n 0\n
          \n \\(813\\)\n \n 0\n \n 0\n
          \n \\(814\\)\n \n 0\n \n 0\n
          \n \\(815\\)\n \n 0\n \n 0\n
          \n \\(816\\)\n \n 0\n \n 0\n
          \n \\(817\\)\n \n 0\n \n 0\n
          \n \\(818\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
          \n \\(819\\)\n \n 0\n \n 0\n
          \n \\(820\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(821\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(822\\)\n \n 0\n \n 0\n
          \n \\(823\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(824\\)\n \n 0\n \n 0\n
          \n \\(825\\)\n \n 0\n \n 0\n
          \n \\(826\\)\n \n 0\n \n 0\n
          \n \\(827\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(828\\)\n \n 0\n \n 0\n
          \n \\(829\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \\(830\\)\n \n 0\n \n 0\n
          \n \\(831\\)\n \n 0\n \n 0\n
          \n \\(832\\)\n \n 0\n \n 0\n
          \n \\(833\\)\n \n 0\n \n 0\n
          \n \\(834\\)\n \n 0\n \n 0\n
          \n \\(835\\)\n \n 0\n \n 0\n
          \n \\(836\\)\n \n 0\n \n 0\n
          \n \\(837\\)\n \n 0\n \n 0\n
          \n \\(838\\)\n \n 0\n \n 0\n
          \n \\(839\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(840\\)\n \n 0\n \n 0\n
          \n \\(841\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
          \n \\(842\\)\n \n 0\n \n 0\n
          \n \\(843\\)\n \n 0\n \n 0\n
          \n \\(844\\)\n \n 0\n \n 0\n
          \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(846\\)\n \n 0\n \n 0\n
          \n \\(847\\)\n \n 0\n \n 0\n
          \n \\(848\\)\n \n 0\n \n 0\n
          \n \\(849\\)\n \n 0\n \n 0\n
          \n \\(850\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
          \n \\(851\\)\n \n 0\n \n 0\n
          \n \\(852\\)\n \n 0\n \n 0\n
          \n \\(853\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n \\(0\\)\n
          \n \\(854\\)\n \n 0\n \n 0\n
          \n \\(855\\)\n \n 0\n \n 0\n
          \n \\(856\\)\n \n 0\n \n 0\n
          \n \\(857\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(858\\)\n \n 0\n \n 0\n
          \n \\(859\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(860\\)\n \n 0\n \n 0\n
          \n \\(861\\)\n \n 0\n \n 0\n
          \n \\(862\\)\n \n 0\n \n 0\n
          \n \\(863\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(864\\)\n \n 0\n \n 0\n
          \n \\(865\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
          \n \\(866\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(867\\)\n \n 0\n \n 0\n
          \n \\(868\\)\n \n 0\n \n 0\n
          \n \\(869\\)\n \n 0\n \n 0\n
          \n \\(870\\)\n \n 0\n \n 0\n
          \n \\(871\\)\n \n 0\n \n 0\n
          \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
          \n \\(873\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
          \n \\(874\\)\n \n 0\n \n 0\n
          \n \\(875\\)\n \n 0\n \n 0\n
          \n \\(876\\)\n \n 0\n \n 0\n
          \n \\(877\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(878\\)\n \n 0\n \n 0\n
          \n \\(879\\)\n \n 0\n \n 0\n
          \n \\(880\\)\n \n 0\n \n 0\n
          \n \\(881\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(882\\)\n \n 0\n \n 0\n
          \n \\(883\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(884\\)\n \n 0\n \n 0\n
          \n \\(885\\)\n \n 0\n \n 0\n
          \n \\(886\\)\n \n 0\n \n 0\n
          \n \\(887\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(888\\)\n \n 0\n \n 0\n
          \n \\(889\\)\n \n 0\n \n 0\n
          \n \\(890\\)\n \n 0\n \n 0\n
          \n \\(891\\)\n \n 0\n \n 0\n
          \n \\(892\\)\n \n 0\n \n 0\n
          \n \\(893\\)\n \n 0\n \n 0\n
          \n \\(894\\)\n \n 0\n \n 0\n
          \n \\(895\\)\n \n 0\n \n 0\n
          \n \\(896\\)\n \n 0\n \n 0\n
          \n \\(897\\)\n \n 0\n \n 0\n
          \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
          \n \\(899\\)\n \n 0\n \n 0\n
          \n \\(900\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
          \n \\(901\\)\n \n 0\n \n 0\n
          \n \\(902\\)\n \n 0\n \n 0\n
          \n \\(903\\)\n \n 0\n \n 0\n
          \n \\(904\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(905\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(906\\)\n \n 0\n \n 0\n
          \n \\(907\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
          \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
          \n \\(908\\)\n \n 0\n \n 0\n
          \n \\(909\\)\n \n 0\n \n 0\n
          \n \\(910\\)\n \n 0\n \n 0\n
          \n \\(911\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
          \n \\(912\\)\n \n 0\n \n 0\n
          \n \\(913\\)\n \n 0\n \n 0\n
          \n \\(914\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(915\\)\n \n 0\n \n 0\n
          \n \\(916\\)\n \n 2.00000i\n 2.00000i
          \n \\(917\\)\n \n 0\n \n 0\n
          \n \\(918\\)\n \n 0\n \n 0\n
          \n \\(919\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \\(920\\)\n \n 0\n \n 0\n
          \n \\(921\\)\n \n 0\n \n 0\n
          \n \\(922\\)\n \n 0\n \n 0\n
          \n \\(923\\)\n \n 0\n \n 0\n
          \n \\(924\\)\n \n 0\n \n 0\n
          \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
          \n \\(926\\)\n \n 0\n \n 0\n
          \n \\(927\\)\n \n 0\n \n 0\n
          \n \\(928\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
          \n \\(929\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(930\\)\n \n 0\n \n 0\n
          \n \\(931\\)\n \n 0\n \n 0\n
          \n \\(932\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
          \n \\(933\\)\n \n 0\n \n 0\n
          \n \\(934\\)\n \n 0\n \n 0\n
          \n \\(935\\)\n \n 0\n \n 0\n
          \n \\(936\\)\n \n 0\n \n 0\n
          \n \\(937\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
          \n \u22121.00000\n \n \\(\\pi\\)\n
          \n \\(938\\)\n \n 0\n \n 0\n
          \n \\(939\\)\n \n 0\n \n 0\n
          \n \\(940\\)\n \n 0\n \n 0\n
          \n \\(941\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(942\\)\n \n 0\n \n 0\n
          \n \\(943\\)\n \n 0\n \n 0\n
          \n \\(944\\)\n \n 0\n \n 0\n
          \n \\(945\\)\n \n 0\n \n 0\n
          \n \\(946\\)\n \n 0\n \n 0\n
          \n \\(947\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
          \n \\(948\\)\n \n 0\n \n 0\n
          \n \\(949\\)\n \n 0\n \n 0\n
          \n \\(950\\)\n \n 0\n \n 0\n
          \n \\(951\\)\n \n 0\n \n 0\n
          \n \\(952\\)\n \n 0\n \n 0\n
          \n \\(953\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(954\\)\n \n 0\n \n 0\n
          \n \\(955\\)\n \n 0\n \n 0\n
          \n \\(956\\)\n \n 0\n \n 0\n
          \n \\(957\\)\n \n 0\n \n 0\n
          \n \\(958\\)\n \n 0\n \n 0\n
          \n \\(959\\)\n \n 0\n \n 0\n
          \n \\(960\\)\n \n 0\n \n 0\n
          \n \\(961\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
          \n \\(962\\)\n \n 0\n \n 0\n
          \n \\(963\\)\n \n 0\n \n 0\n
          \n \\(964\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
          \n \\(965\\)\n \n 2.00000\n \n 2.00000\n
          \n \\(966\\)\n \n 0\n \n 0\n
          \n \\(967\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
          \n 1.00000i\n \\(0.5\\pi\\)\n
          \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
          \n \\(969\\)\n \n 0\n \n 0\n
          \n \\(970\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
          \n \\(971\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
          \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
          \n \\(972\\)\n \n 0\n \n 0\n
          \n \\(973\\)\n \n 0\n \n 0\n
          \n \\(974\\)\n \n 0\n \n 0\n
          \n \\(975\\)\n \n 0\n \n 0\n
          \n \\(976\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
          \n \\(977\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
          \n \\(978\\)\n \n 0\n \n 0\n
          \n \\(979\\)\n \n 0\n \n 0\n
          \n \\(980\\)\n \n 0\n \n 0\n
          \n \\(981\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
          \n \\(982\\)\n \n 0\n \n 0\n
          \n \\(983\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(984\\)\n \n 0\n \n 0\n
          \n \\(985\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
          \n \\(986\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
          \n \\(987\\)\n \n 0\n \n 0\n
          \n \\(988\\)\n \n 0\n \n 0\n
          \n \\(989\\)\n \n 0\n \n 0\n
          \n \\(990\\)\n \n 0\n \n 0\n
          \n \\(991\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
          \n \\(992\\)\n \n 0\n \n 0\n
          \n \\(993\\)\n \n 0\n \n 0\n
          \n \\(994\\)\n \n 0\n \n 0\n
          \n \\(995\\)\n \n 0\n \n 0\n
          \n \\(996\\)\n \n 0\n \n 0\n
          \n \\(997\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
          \n \\(998\\)\n \n 0\n \n 0\n
          \n \\(999\\)\n \n 0\n \n 0\n
          \u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisting character
          CharParityOrdTypeTwistMinDim
          1.1even1trivial3332.1.bc.b.2027.14
          4.3odd2CM3332.1.bc.b.2027.14
          7.2even3inner3332.1.bc.b.667.14
          7.3odd668.1.f.a.55.1yes2
          7.4even33332.1.m.b.2843.12
          7.5odd63332.1.bc.c.667.14
          7.6odd23332.1.bc.c.2027.14
          17.13even4inner3332.1.bc.b.2223.14
          21.17even6612.1.l.a.55.12
          28.3even668.1.f.a.55.1yes2
          28.11odd63332.1.m.b.2843.12
          28.19even63332.1.bc.c.667.14
          28.23odd6inner3332.1.bc.b.667.14
          28.27even23332.1.bc.c.2027.14
          35.3even121700.1.n.a.599.12
          35.17even121700.1.n.b.599.12
          35.24odd61700.1.p.a.1551.12
          56.3even61088.1.p.a.191.12
          56.45odd61088.1.p.a.191.12
          68.47odd4inner3332.1.bc.b.2223.14
          84.59odd6612.1.l.a.55.12
          119.3even481156.1.g.b.155.18
          119.10even481156.1.g.b.179.18
          119.13odd43332.1.bc.c.2223.14
          119.24even481156.1.g.b.179.28
          119.30even12inner3332.1.bc.b.863.14
          119.31even481156.1.g.b.155.28
          119.38odd121156.1.f.b.251.12
          119.45even481156.1.g.b.399.18
          119.47odd123332.1.bc.c.863.14
          119.59odd241156.1.c.b.579.12
          119.66odd241156.1.d.a.1155.22
          119.73even481156.1.g.b.423.28
          119.80even481156.1.g.b.423.18
          119.81even123332.1.m.b.3039.12
          119.87odd241156.1.d.a.1155.12
          119.94odd241156.1.c.b.579.22
          119.101odd61156.1.f.b.327.12
          119.108even481156.1.g.b.399.28
          119.115odd1268.1.f.a.47.1&check;2
          140.3odd121700.1.n.a.599.12
          140.59even61700.1.p.a.1551.12
          140.87odd121700.1.n.b.599.12
          357.353even12612.1.l.a.523.12
          476.3odd481156.1.g.b.155.18
          476.31odd481156.1.g.b.155.28
          476.47even123332.1.bc.c.863.14
          476.59even241156.1.c.b.579.12
          476.87even241156.1.d.a.1155.12
          476.115even1268.1.f.a.47.1&check;2
          476.143odd481156.1.g.b.179.28
          476.199odd481156.1.g.b.423.18
          476.227odd481156.1.g.b.399.28
          476.251even43332.1.bc.c.2223.14
          476.283odd481156.1.g.b.399.18
          476.311odd481156.1.g.b.423.28
          476.319odd123332.1.m.b.3039.12
          476.339even61156.1.f.b.327.12
          476.367odd481156.1.g.b.179.18
          476.387odd12inner3332.1.bc.b.863.14
          476.395even121156.1.f.b.251.12
          476.423even241156.1.d.a.1155.22
          476.451even241156.1.c.b.579.22
          595.234odd121700.1.p.a.251.12
          595.353even121700.1.n.b.999.12
          595.472even121700.1.n.a.999.12
          952.115even121088.1.p.a.319.12
          952.829odd121088.1.p.a.319.12
          1428.1067odd12612.1.l.a.523.12
          2380.1067odd121700.1.n.a.999.12
          2380.1543odd121700.1.n.b.999.12
          2380.2019even121700.1.p.a.251.12
          \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisted newform
          TwistMinDimCharParityOrdType
          68.1.f.a.47.1&check;2119.115odd12
          68.1.f.a.47.1&check;2476.115even12
          68.1.f.a.55.1yes27.3odd6
          68.1.f.a.55.1yes228.3even6
          612.1.l.a.55.1221.17even6
          612.1.l.a.55.1284.59odd6
          612.1.l.a.523.12357.353even12
          612.1.l.a.523.121428.1067odd12
          1088.1.p.a.191.1256.3even6
          1088.1.p.a.191.1256.45odd6
          1088.1.p.a.319.12952.115even12
          1088.1.p.a.319.12952.829odd12
          1156.1.c.b.579.12119.59odd24
          1156.1.c.b.579.12476.59even24
          1156.1.c.b.579.22119.94odd24
          1156.1.c.b.579.22476.451even24
          1156.1.d.a.1155.12119.87odd24
          1156.1.d.a.1155.12476.87even24
          1156.1.d.a.1155.22119.66odd24
          1156.1.d.a.1155.22476.423even24
          1156.1.f.b.251.12119.38odd12
          1156.1.f.b.251.12476.395even12
          1156.1.f.b.327.12119.101odd6
          1156.1.f.b.327.12476.339even6
          1156.1.g.b.155.18119.3even48
          1156.1.g.b.155.18476.3odd48
          1156.1.g.b.155.28119.31even48
          1156.1.g.b.155.28476.31odd48
          1156.1.g.b.179.18119.10even48
          1156.1.g.b.179.18476.367odd48
          1156.1.g.b.179.28119.24even48
          1156.1.g.b.179.28476.143odd48
          1156.1.g.b.399.18119.45even48
          1156.1.g.b.399.18476.283odd48
          1156.1.g.b.399.28119.108even48
          1156.1.g.b.399.28476.227odd48
          1156.1.g.b.423.18119.80even48
          1156.1.g.b.423.18476.199odd48
          1156.1.g.b.423.28119.73even48
          1156.1.g.b.423.28476.311odd48
          1700.1.n.a.599.1235.3even12
          1700.1.n.a.599.12140.3odd12
          1700.1.n.a.999.12595.472even12
          1700.1.n.a.999.122380.1067odd12
          1700.1.n.b.599.1235.17even12
          1700.1.n.b.599.12140.87odd12
          1700.1.n.b.999.12595.353even12
          1700.1.n.b.999.122380.1543odd12
          1700.1.p.a.251.12595.234odd12
          1700.1.p.a.251.122380.2019even12
          1700.1.p.a.1551.1235.24odd6
          1700.1.p.a.1551.12140.59even6
          3332.1.m.b.2843.127.4even3
          3332.1.m.b.2843.1228.11odd6
          3332.1.m.b.3039.12119.81even12
          3332.1.m.b.3039.12476.319odd12
          3332.1.bc.b.667.147.2even3inner
          3332.1.bc.b.667.1428.23odd6inner
          3332.1.bc.b.863.14119.30even12inner
          3332.1.bc.b.863.14476.387odd12inner
          3332.1.bc.b.2027.141.1even1trivial
          3332.1.bc.b.2027.144.3odd2CM
          3332.1.bc.b.2223.1417.13even4inner
          3332.1.bc.b.2223.1468.47odd4inner
          3332.1.bc.c.667.147.5odd6
          3332.1.bc.c.667.1428.19even6
          3332.1.bc.c.863.14119.47odd12
          3332.1.bc.c.863.14476.47even12
          3332.1.bc.c.2027.147.6odd2
          3332.1.bc.c.2027.1428.27even2
          3332.1.bc.c.2223.14119.13odd4
          3332.1.bc.c.2223.14476.251even4
          ", "statics": {"paragraph": 22, "paragraph.text": 24, "table": 8, "title": 4, "table.complex": 4, "paragraph.equation-inline": 2}, "url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/3332/1/bc/b/2027/1/", "content": "Show commands: Magma/ PariGP/ SageMath\n\n[N,k,chi] = [3332,1,Mod(667,3332)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)\n\nfrom sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3332, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 4, 9])) N = Newforms(chi, 1, names=\"a\")\n\n//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter(\"3332.667\"); S:= CuspForms(chi, 1); N := Newforms(S);\n\n| Level | \\( N \\) | \\(=\\) | \\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\) |\n|---|---|---|---|\n| Weight | \\( k \\) | \\(=\\) | \\( 1 \\) |\n| Character orbit | \\([\\chi]\\) | \\(=\\) | 3332.bc
          order
          degree
          minimal |\n\n## Newform invariants\n\nsage:\u00a0f = N[0] \\# Warning: the index may be different\n\ngp:\u00a0f = lf[1] \\\\ Warning: the index may be different\n\n
          Self dualno
          Analytic conductor\\(1.66288462209\\)
          Analytic rank\\(0\\)
          Dimension\\(4\\)
          Coefficient field\\(\\Q(\\zeta_{12})\\)
          gp:\u00a0f.mod \\\\ as an extension of the character field
          Defining polynomial\\( x^{4} - x^{2} + 1 \\)
          Coefficient ring\\(\\Z[a_1, a_2]\\)
          Coefficient ring index\\( 1 \\)
          Twist minimalno (minimal twist has level 68)
          Projective image\\(D_{4}\\)
          Projective fieldGalois closure of<br>4.2.19652.1
          Artin image$C_4\\wr C_2\\times C_6$
          Artin fieldGalois closure of<br>\\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
          \n\n## Embedding invariants\n\n| Embedding label | | | 2027.1 |\n|---|---|---|---|\n| Root | | | \\(0.866025 - 0.500000i\\) of defining polynomial |\n| Character | \\(\\chi\\) | \\(=\\) | 3332.2027 |\n| Dual form | | | 3332.1.bc.b.863.1 |\n\nsage:\u00a0f.q_expansion() \\# note that sage often uses an isomorphic number field\n\ngp:\u00a0mfcoefs(f, 20)\n\n| \\(f(q)\\) | \\(=\\) | \\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\) |\n|---|---|---|\n| \\(\\operatorname{Tr}(f)(q)\\) | \\(=\\) | \\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\) |\n\n## Character values\n\nWe give the values of $\\chi$ on generators for $\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times$ .\n\n| \\(n\\) | \\(785\\) | \\(885\\) | \\(1667\\) |\n|---|---|---|---|\n| \\(\\chi(n)\\) | \\(e\\left(\\frac{3}{4}\\right)\\) | \\(e\\left(\\frac{2}{3}\\right)\\) | \\(-1\\) |\n\n## Coefficient data\n\nFor each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the Satake parameters\\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).\n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n
          \\(n\\)\\(a_n\\)\\(a_n / n^{(k-1)/2}\\)\\( \\alpha_n \\)\\( \\theta_n \\)
          \\(p\\)\\(a_p\\)\\(a_p / p^{(k-1)/2}\\)\\( \\alpha_p\\)\\( \\theta_p \\)
          \\(2\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(3\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(4\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(5\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(6\\)00
          \\(7\\)00
          \\(8\\)1.00000<br>i1.00000<br>i
          \\(9\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(10\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(11\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(12\\)00
          \\(13\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(14\\)00
          \\(15\\)00
          \\(16\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(17\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(18\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(19\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(20\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(21\\)00
          \\(22\\)00
          \\(23\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(24\\)00
          \\(25\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(26\\)00
          \\(27\\)00
          \\(28\\)00
          \\(29\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(30\\)00
          \\(31\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(32\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(33\\)00
          \\(34\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(35\\)00
          \\(36\\)1.00000<br>i1.00000<br>i
          \\(37\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(38\\)00
          \\(39\\)00
          \\(40\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(41\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(42\\)00
          \\(43\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(44\\)00
          \\(45\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(46\\)00
          \\(47\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(48\\)00
          \\(49\\)00
          \\(50\\)\u22121.00000\u22121.00000
          \\(51\\)00
          \\(52\\)00
          \\(53\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(54\\)00
          \\(55\\)00
          \\(56\\)00
          \\(57\\)00
          \\(58\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(59\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(60\\)00
          \\(61\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(62\\)00
          \\(63\\)00
          \\(64\\)\u22121.00000\u22121.00000
          \\(65\\)00
          \\(66\\)00
          \\(67\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(68\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(69\\)00
          \\(70\\)00
          \\(71\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(72\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(73\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(74\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(75\\)00
          \\(76\\)00
          \\(77\\)00
          \\(78\\)00
          \\(79\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(80\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(81\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(82\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(83\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(84\\)00
          \\(85\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(86\\)00
          \\(87\\)00
          \\(88\\)00
          \\(89\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(90\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(91\\)00
          \\(92\\)00
          \\(93\\)00
          \\(94\\)00
          \\(95\\)00
          \\(96\\)00
          \\(97\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(98\\)00
          \\(99\\)00
          \\(100\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(101\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(102\\)00
          \\(103\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(104\\)00
          \\(105\\)00
          \\(106\\)00
          \\(107\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(108\\)00
          \\(109\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(110\\)00
          \\(111\\)00
          \\(112\\)00
          \\(113\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(114\\)00
          \\(115\\)00
          \\(116\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(117\\)00
          \\(118\\)00
          \\(119\\)00
          \\(120\\)00
          \\(121\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(122\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(123\\)00
          \\(124\\)00
          \\(125\\)00
          \\(126\\)00
          \\(127\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(128\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(129\\)00
          \\(130\\)00
          \\(131\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(132\\)00
          \\(133\\)00
          \\(134\\)00
          \\(135\\)00
          \\(136\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(137\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(138\\)00
          \\(139\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(140\\)00
          \\(141\\)00
          \\(142\\)00
          \\(143\\)00
          \\(144\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(145\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(146\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(147\\)00
          \\(148\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(149\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \\(150\\)00
          \\(151\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(152\\)00
          \\(153\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(154\\)00
          \\(155\\)00
          \\(156\\)00
          \\(157\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(158\\)00
          \\(159\\)00
          \\(160\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(161\\)00
          \\(162\\)1.00000<br>i1.00000<br>i
          \\(163\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(164\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(165\\)00
          \\(166\\)00
          \\(167\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(168\\)00
          \\(169\\)\u22121.00000\u22121.00000
          \\(170\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(171\\)00
          \\(172\\)00
          \\(173\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(174\\)00
          \\(175\\)00
          \\(176\\)00
          \\(177\\)00
          \\(178\\)00
          \\(179\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(180\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(181\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(182\\)00
          \\(183\\)00
          \\(184\\)00
          \\(185\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(186\\)00
          \\(187\\)00
          \\(188\\)00
          \\(189\\)00
          \\(190\\)00
          \\(191\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(192\\)00
          \\(193\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(194\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(195\\)00
          \\(196\\)00
          \\(197\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(198\\)00
          \\(199\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(200\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(201\\)00
          \\(202\\)00
          \\(203\\)00
          \\(204\\)00
          \\(205\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
          \\(206\\)00
          \\(207\\)00
          \\(208\\)00
          \\(209\\)00
          \\(210\\)00
          \\(211\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(212\\)00
          \\(213\\)00
          \\(214\\)00
          \\(215\\)00
          \\(216\\)00
          \\(217\\)00
          \\(218\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(219\\)00
          \\(220\\)00
          \\(221\\)00
          \\(222\\)00
          \\(223\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(224\\)00
          \\(225\\)\u22121.00000\u22121.00000
          \\(226\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(227\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(228\\)00
          \\(229\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(230\\)00
          \\(231\\)00
          \\(232\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(233\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(234\\)00
          \\(235\\)00
          \\(236\\)00
          \\(237\\)00
          \\(238\\)00
          \\(239\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(240\\)00
          \\(241\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(242\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(243\\)00
          \\(244\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(245\\)00
          \\(246\\)00
          \\(247\\)00
          \\(248\\)00
          \\(249\\)00
          \\(250\\)00
          \\(251\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(252\\)00
          \\(253\\)00
          \\(254\\)00
          \\(255\\)00
          \\(256\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(257\\)1.73205\u22121.00000<br>i1.73205\u22121.00000<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(258\\)00
          \\(259\\)00
          \\(260\\)00
          \\(261\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(262\\)00
          \\(263\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(264\\)00
          \\(265\\)00
          \\(266\\)00
          \\(267\\)00
          \\(268\\)00
          \\(269\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(270\\)00
          \\(271\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(272\\)1.000001.00000
          \\(273\\)00
          \\(274\\)00
          \\(275\\)00
          \\(276\\)00
          \\(277\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(278\\)00
          \\(279\\)00
          \\(280\\)00
          \\(281\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(282\\)00
          \\(283\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(284\\)00
          \\(285\\)00
          \\(286\\)00
          \\(287\\)00
          \\(288\\)\u22121.00000\u22121.00000
          \\(289\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(290\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(291\\)00
          \\(292\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(293\\)2.000002.000001.00000\\(0\\)
          1.00000\\(0\\)
          \\(294\\)00
          \\(295\\)00
          \\(296\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(297\\)00
          \\(298\\)1.73205+1.00000<br>i1.73205+1.00000<br>i
          \\(299\\)00
          \\(300\\)00
          \\(301\\)00
          \\(302\\)00
          \\(303\\)00
          \\(304\\)00
          \\(305\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
          \\(306\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(307\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(308\\)00
          \\(309\\)00
          \\(310\\)00
          \\(311\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(312\\)00
          \\(313\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(314\\)2.00000<br>i2.00000<br>i
          \\(315\\)00
          \\(316\\)00
          \\(317\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(318\\)00
          \\(319\\)00
          \\(320\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(321\\)00
          \\(322\\)00
          \\(323\\)00
          \\(324\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(325\\)00
          \\(326\\)00
          \\(327\\)00
          \\(328\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(329\\)00
          \\(330\\)00
          \\(331\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(332\\)00
          \\(333\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(334\\)00
          \\(335\\)00
          \\(336\\)00
          \\(337\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(338\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(339\\)00
          \\(340\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(341\\)00
          \\(342\\)00
          \\(343\\)00
          \\(344\\)00
          \\(345\\)00
          \\(346\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(347\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(348\\)00
          \\(349\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(350\\)00
          \\(351\\)00
          \\(352\\)00
          \\(353\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(354\\)00
          \\(355\\)00
          \\(356\\)00
          \\(357\\)00
          \\(358\\)00
          \\(359\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(360\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(361\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(362\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(363\\)00
          \\(364\\)00
          \\(365\\)2.00000<br>i2.00000<br>i
          \\(366\\)00
          \\(367\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(368\\)00
          \\(369\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(370\\)2.000002.00000
          \\(371\\)00
          \\(372\\)00
          \\(373\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(374\\)00
          \\(375\\)00
          \\(376\\)00
          \\(377\\)00
          \\(378\\)00
          \\(379\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(380\\)00
          \\(381\\)00
          \\(382\\)00
          \\(383\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(384\\)00
          \\(385\\)00
          \\(386\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(387\\)00
          \\(388\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(389\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \\(390\\)00
          \\(391\\)00
          \\(392\\)00
          \\(393\\)00
          \\(394\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(395\\)00
          \\(396\\)00
          \\(397\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(398\\)00
          \\(399\\)00
          \\(400\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(401\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(402\\)00
          \\(403\\)00
          \\(404\\)00
          \\(405\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(406\\)00
          \\(407\\)00
          \\(408\\)00
          \\(409\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(410\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(411\\)00
          \\(412\\)00
          \\(413\\)00
          \\(414\\)00
          \\(415\\)00
          \\(416\\)00
          \\(417\\)00
          \\(418\\)00
          \\(419\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(420\\)00
          \\(421\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(422\\)00
          \\(423\\)00
          \\(424\\)00
          \\(425\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(426\\)00
          \\(427\\)00
          \\(428\\)00
          \\(429\\)00
          \\(430\\)00
          \\(431\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(432\\)00
          \\(433\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(434\\)00
          \\(435\\)00
          \\(436\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(437\\)00
          \\(438\\)00
          \\(439\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(440\\)00
          \\(441\\)00
          \\(442\\)00
          \\(443\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(444\\)00
          \\(445\\)00
          \\(446\\)00
          \\(447\\)00
          \\(448\\)00
          \\(449\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(450\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(451\\)00
          \\(452\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(453\\)00
          \\(454\\)00
          \\(455\\)00
          \\(456\\)00
          \\(457\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(458\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(459\\)00
          \\(460\\)00
          \\(461\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(462\\)00
          \\(463\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(464\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(465\\)00
          \\(466\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(467\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(468\\)00
          \\(469\\)00
          \\(470\\)00
          \\(471\\)00
          \\(472\\)00
          \\(473\\)00
          \\(474\\)00
          \\(475\\)00
          \\(476\\)00
          \\(477\\)00
          \\(478\\)00
          \\(479\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(480\\)00
          \\(481\\)00
          \\(482\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(483\\)00
          \\(484\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(485\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(486\\)00
          \\(487\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(488\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(489\\)00
          \\(490\\)00
          \\(491\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(492\\)00
          \\(493\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(494\\)00
          \\(495\\)00
          \\(496\\)00
          \\(497\\)00
          \\(498\\)00
          \\(499\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(500\\)00
          \\(501\\)00
          \\(502\\)00
          \\(503\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(504\\)00
          \\(505\\)00
          \\(506\\)00
          \\(507\\)00
          \\(508\\)00
          \\(509\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \\(510\\)00
          \\(511\\)00
          \\(512\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(513\\)00
          \\(514\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
          \\(515\\)00
          \\(516\\)00
          \\(517\\)00
          \\(518\\)00
          \\(519\\)00
          \\(520\\)00
          \\(521\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(522\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(523\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(524\\)00
          \\(525\\)00
          \\(526\\)00
          \\(527\\)00
          \\(528\\)00
          \\(529\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(530\\)00
          \\(531\\)00
          \\(532\\)00
          \\(533\\)00
          \\(534\\)00
          \\(535\\)00
          \\(536\\)00
          \\(537\\)00
          \\(538\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(539\\)00
          \\(540\\)00
          \\(541\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(542\\)00
          \\(543\\)00
          \\(544\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(545\\)\u22122.00000\u22122.00000
          \\(546\\)00
          \\(547\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(548\\)00
          \\(549\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(550\\)00
          \\(551\\)00
          \\(552\\)00
          \\(553\\)00
          \\(554\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(555\\)00
          \\(556\\)00
          \\(557\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(558\\)00
          \\(559\\)00
          \\(560\\)00
          \\(561\\)00
          \\(562\\)00
          \\(563\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(564\\)00
          \\(565\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
          \\(566\\)00
          \\(567\\)00
          \\(568\\)00
          \\(569\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(570\\)00
          \\(571\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(572\\)00
          \\(573\\)00
          \\(574\\)00
          \\(575\\)00
          \\(576\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
          \\(577\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(578\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(579\\)00
          \\(580\\)2.00000<br>i2.00000<br>i
          \\(581\\)00
          \\(582\\)00
          \\(583\\)00
          \\(584\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(585\\)00
          \\(586\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(587\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(588\\)00
          \\(589\\)00
          \\(590\\)00
          \\(591\\)00
          \\(592\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(593\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(594\\)00
          \\(595\\)00
          \\(596\\)\u22122.00000\u22122.00000
          \\(597\\)00
          \\(598\\)00
          \\(599\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(600\\)00
          \\(601\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(602\\)00
          \\(603\\)00
          \\(604\\)00
          \\(605\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(606\\)00
          \\(607\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(608\\)00
          \\(609\\)00
          \\(610\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(611\\)00
          \\(612\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
          \\(613\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(614\\)00
          \\(615\\)00
          \\(616\\)00
          \\(617\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(618\\)00
          \\(619\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(620\\)00
          \\(621\\)00
          \\(622\\)00
          \\(623\\)00
          \\(624\\)00
          \\(625\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
          \\(626\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
          \\(627\\)00
          \\(628\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(629\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(630\\)00
          \\(631\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(632\\)00
          \\(633\\)00
          \\(634\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(635\\)00
          \\(636\\)00
          \\(637\\)00
          \\(638\\)00
          \\(639\\)00
          \\(640\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(641\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(642\\)00
          \\(643\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(644\\)00
          \\(645\\)00
          \\(646\\)00
          \\(647\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(648\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
          \\(649\\)00
          \\(650\\)00
          \\(651\\)00
          \\(652\\)00
          \\(653\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \\(654\\)00
          \\(655\\)00
          \\(656\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(657\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(658\\)00
          \\(659\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(660\\)00
          \\(661\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(662\\)00
          \\(663\\)00
          \\(664\\)00
          \\(665\\)00
          \\(666\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(667\\)00
          \\(668\\)00
          \\(669\\)00
          \\(670\\)00
          \\(671\\)00
          \\(672\\)00
          \\(673\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(674\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(675\\)00
          \\(676\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(677\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(678\\)00
          \\(679\\)00
          \\(680\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(681\\)00
          \\(682\\)00
          \\(683\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(684\\)00
          \\(685\\)00
          \\(686\\)00
          \\(687\\)00
          \\(688\\)00
          \\(689\\)00
          \\(690\\)00
          \\(691\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
          \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
          \\(692\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(693\\)00
          \\(694\\)00
          \\(695\\)00
          \\(696\\)00
          \\(697\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(698\\)00
          \\(699\\)00
          \\(700\\)00
          \\(701\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(702\\)00
          \\(703\\)00
          \\(704\\)00
          \\(705\\)00
          \\(706\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(707\\)00
          \\(708\\)00
          \\(709\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(710\\)00
          \\(711\\)00
          \\(712\\)00
          \\(713\\)00
          \\(714\\)00
          \\(715\\)00
          \\(716\\)00
          \\(717\\)00
          \\(718\\)00
          \\(719\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(720\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(721\\)00
          \\(722\\)1.00000<br>i1.00000<br>i
          \\(723\\)00
          \\(724\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(725\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(726\\)00
          \\(727\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(728\\)00
          \\(729\\)1.00000<br>i1.00000<br>i
          \\(730\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(731\\)00
          \\(732\\)00
          \\(733\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(734\\)00
          \\(735\\)00
          \\(736\\)00
          \\(737\\)00
          \\(738\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(739\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(740\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
          \\(741\\)00
          \\(742\\)00
          \\(743\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(744\\)00
          \\(745\\)0.732051+2.73205<br>i0.732051+2.73205<br>i
          \\(746\\)00
          \\(747\\)00
          \\(748\\)00
          \\(749\\)00
          \\(750\\)00
          \\(751\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(752\\)00
          \\(753\\)00
          \\(754\\)00
          \\(755\\)00
          \\(756\\)00
          \\(757\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(758\\)00
          \\(759\\)00
          \\(760\\)00
          \\(761\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(762\\)00
          \\(763\\)00
          \\(764\\)00
          \\(765\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(766\\)00
          \\(767\\)00
          \\(768\\)00
          \\(769\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(770\\)00
          \\(771\\)00
          \\(772\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(773\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
          \\(774\\)00
          \\(775\\)00
          \\(776\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(777\\)00
          \\(778\\)2.000002.00000
          \\(779\\)00
          \\(780\\)00
          \\(781\\)00
          \\(782\\)00
          \\(783\\)00
          \\(784\\)00
          \\(785\\)\u22122.00000+2.00000<br>i\u22122.00000+2.00000<br>i
          \\(786\\)00
          \\(787\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(788\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(789\\)00
          \\(790\\)00
          \\(791\\)00
          \\(792\\)00
          \\(793\\)00
          \\(794\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(795\\)00
          \\(796\\)00
          \\(797\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(798\\)00
          \\(799\\)00
          \\(800\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(801\\)00
          \\(802\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(803\\)00
          \\(804\\)00
          \\(805\\)00
          \\(806\\)00
          \\(807\\)00
          \\(808\\)00
          \\(809\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(810\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
          \\(811\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
          \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
          \\(812\\)00
          \\(813\\)00
          \\(814\\)00
          \\(815\\)00
          \\(816\\)00
          \\(817\\)00
          \\(818\\)\u22122.00000<br>i\u22122.00000<br>i
          \\(819\\)00
          \\(820\\)2.000002.00000
          \\(821\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(822\\)00
          \\(823\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(824\\)00
          \\(825\\)00
          \\(826\\)00
          \\(827\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(828\\)00
          \\(829\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \\(830\\)00
          \\(831\\)00
          \\(832\\)00
          \\(833\\)00
          \\(834\\)00
          \\(835\\)00
          \\(836\\)00
          \\(837\\)00
          \\(838\\)00
          \\(839\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(840\\)00
          \\(841\\)\u22121.00000<br>i\u22121.00000<br>i
          \\(842\\)00
          \\(843\\)00
          \\(844\\)00
          \\(845\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(846\\)00
          \\(847\\)00
          \\(848\\)00
          \\(849\\)00
          \\(850\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
          \\(851\\)00
          \\(852\\)00
          \\(853\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000\\(0\\)
          \\(854\\)00
          \\(855\\)00
          \\(856\\)00
          \\(857\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(858\\)00
          \\(859\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(860\\)00
          \\(861\\)00
          \\(862\\)00
          \\(863\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(864\\)00
          \\(865\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
          \\(866\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(867\\)00
          \\(868\\)00
          \\(869\\)00
          \\(870\\)00
          \\(871\\)00
          \\(872\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
          \\(873\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
          \\(874\\)00
          \\(875\\)00
          \\(876\\)00
          \\(877\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(878\\)00
          \\(879\\)00
          \\(880\\)00
          \\(881\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
          \u22121.00000\\(\\pi\\)
          \\(882\\)00
          \\(883\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(884\\)00
          \\(885\\)00
          \\(886\\)00
          \\(887\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(888\\)00
          \\(889\\)00
          \\(890\\)00
          \\(891\\)00
          \\(892\\)00
          \\(893\\)00
          \\(894\\)00
          \\(895\\)00
          \\(896\\)00
          \\(897\\)00
          \\(898\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
          \\(899\\)00
          \\(900\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
          \\(901\\)00
          \\(902\\)00
          \\(903\\)00
          \\(904\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(905\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(906\\)00
          \\(907\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
          \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
          \\(908\\)00
          \\(909\\)00
          \\(910\\)00
          \\(911\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
          0.707107+0.707107<br>i\\(0.250000\\pi\\)
          \\(912\\)00
          \\(913\\)00
          \\(914\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(915\\)00
          \\(916\\)2.00000<br>i2.00000<br>i
          \\(917\\)00
          \\(918\\)00
          \\(919\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
          0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \\(920\\)00
          \\(921\\)00
          \\(922\\)00
          \\(923\\)00
          \\(924\\)00
          \\(925\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
          \\(926\\)00
          \\(927\\)00
          \\(928\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
          \\(929\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(930\\)00
          \\(931\\)00
          \\(932\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
          \\(933\\)00
          \\(934\\)00
          \\(935\\)00
          \\(936\\)00
          \\(937\\)001.00000\\(0\\)
          \u22121.00000\\(\\pi\\)
          \\(938\\)00
          \\(939\\)00
          \\(940\\)00
          \\(941\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(942\\)00
          \\(943\\)00
          \\(944\\)00
          \\(945\\)00
          \\(946\\)00
          \\(947\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
          0.965926+0.258819<br>i\\(0.0833333\\pi\\)
          \\(948\\)00
          \\(949\\)00
          \\(950\\)00
          \\(951\\)00
          \\(952\\)00
          \\(953\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(954\\)00
          \\(955\\)00
          \\(956\\)00
          \\(957\\)00
          \\(958\\)00
          \\(959\\)00
          \\(960\\)00
          \\(961\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
          \\(962\\)00
          \\(963\\)00
          \\(964\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
          \\(965\\)2.000002.00000
          \\(966\\)00
          \\(967\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
          1.00000<br>i\\(0.5\\pi\\)
          \\(968\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
          \\(969\\)00
          \\(970\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
          \\(971\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
          \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
          \\(972\\)00
          \\(973\\)00
          \\(974\\)00
          \\(975\\)00
          \\(976\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
          \\(977\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
          \\(978\\)00
          \\(979\\)00
          \\(980\\)00
          \\(981\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
          \\(982\\)00
          \\(983\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(984\\)00
          \\(985\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
          \\(986\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
          \\(987\\)00
          \\(988\\)00
          \\(989\\)00
          \\(990\\)00
          \\(991\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
          0.258819+0.965926<br>i\\(0.416667\\pi\\)
          \\(992\\)00
          \\(993\\)00
          \\(994\\)00
          \\(995\\)00
          \\(996\\)00
          \\(997\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
          0.866025+0.500000<br>i\\(0.166667\\pi\\)
          \\(998\\)00
          \\(999\\)00
          \n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n
          By<br>twisting character
          CharParityOrdTypeTwistMinDim
          1.1even1trivial3332.1.bc.b.2027.14
          4.3odd2CM3332.1.bc.b.2027.14
          7.2even3inner3332.1.bc.b.667.14
          7.3odd668.1.f.a.55.1yes2
          7.4even33332.1.m.b.2843.12
          7.5odd63332.1.bc.c.667.14
          7.6odd23332.1.bc.c.2027.14
          17.13even4inner3332.1.bc.b.2223.14
          21.17even6612.1.l.a.55.12
          28.3even668.1.f.a.55.1yes2
          28.11odd63332.1.m.b.2843.12
          28.19even63332.1.bc.c.667.14
          28.23odd6inner3332.1.bc.b.667.14
          28.27even23332.1.bc.c.2027.14
          35.3even121700.1.n.a.599.12
          35.17even121700.1.n.b.599.12
          35.24odd61700.1.p.a.1551.12
          56.3even61088.1.p.a.191.12
          56.45odd61088.1.p.a.191.12
          68.47odd4inner3332.1.bc.b.2223.14
          84.59odd6612.1.l.a.55.12
          119.3even481156.1.g.b.155.18
          119.10even481156.1.g.b.179.18
          119.13odd43332.1.bc.c.2223.14
          119.24even481156.1.g.b.179.28
          119.30even12inner3332.1.bc.b.863.14
          119.31even481156.1.g.b.155.28
          119.38odd121156.1.f.b.251.12
          119.45even481156.1.g.b.399.18
          119.47odd123332.1.bc.c.863.14
          119.59odd241156.1.c.b.579.12
          119.66odd241156.1.d.a.1155.22
          119.73even481156.1.g.b.423.28
          119.80even481156.1.g.b.423.18
          119.81even123332.1.m.b.3039.12
          119.87odd241156.1.d.a.1155.12
          119.94odd241156.1.c.b.579.22
          119.101odd61156.1.f.b.327.12
          119.108even481156.1.g.b.399.28
          119.115odd1268.1.f.a.47.1&check;2
          140.3odd121700.1.n.a.599.12
          140.59even61700.1.p.a.1551.12
          140.87odd121700.1.n.b.599.12
          357.353even12612.1.l.a.523.12
          476.3odd481156.1.g.b.155.18
          476.31odd481156.1.g.b.155.28
          476.47even123332.1.bc.c.863.14
          476.59even241156.1.c.b.579.12
          476.87even241156.1.d.a.1155.12
          476.115even1268.1.f.a.47.1&check;2
          476.143odd481156.1.g.b.179.28
          476.199odd481156.1.g.b.423.18
          476.227odd481156.1.g.b.399.28
          476.251even43332.1.bc.c.2223.14
          476.283odd481156.1.g.b.399.18
          476.311odd481156.1.g.b.423.28
          476.319odd123332.1.m.b.3039.12
          476.339even61156.1.f.b.327.12
          476.367odd481156.1.g.b.179.18
          476.387odd12inner3332.1.bc.b.863.14
          476.395even121156.1.f.b.251.12
          476.423even241156.1.d.a.1155.22
          476.451even241156.1.c.b.579.22
          595.234odd121700.1.p.a.251.12
          595.353even121700.1.n.b.999.12
          595.472even121700.1.n.a.999.12
          952.115even121088.1.p.a.319.12
          952.829odd121088.1.p.a.319.12
          1428.1067odd12612.1.l.a.523.12
          2380.1067odd121700.1.n.a.999.12
          2380.1543odd121700.1.n.b.999.12
          2380.2019even121700.1.p.a.251.12
          \n\n
          By<br>twisted newform
          TwistMinDimCharParityOrdType
          68.1.f.a.47.1&check;2119.115odd12
          68.1.f.a.47.1&check;2476.115even12
          68.1.f.a.55.1yes27.3odd6
          68.1.f.a.55.1yes228.3even6
          612.1.l.a.55.1221.17even6
          612.1.l.a.55.1284.59odd6
          612.1.l.a.523.12357.353even12
          612.1.l.a.523.121428.1067odd12
          1088.1.p.a.191.1256.3even6
          1088.1.p.a.191.1256.45odd6
          1088.1.p.a.319.12952.115even12
          1088.1.p.a.319.12952.829odd12
          1156.1.c.b.579.12119.59odd24
          1156.1.c.b.579.12476.59even24
          1156.1.c.b.579.22119.94odd24
          1156.1.c.b.579.22476.451even24
          1156.1.d.a.1155.12119.87odd24
          1156.1.d.a.1155.12476.87even24
          1156.1.d.a.1155.22119.66odd24
          1156.1.d.a.1155.22476.423even24
          1156.1.f.b.251.12119.38odd12
          1156.1.f.b.251.12476.395even12
          1156.1.f.b.327.12119.101odd6
          1156.1.f.b.327.12476.339even6
          1156.1.g.b.155.18119.3even48
          1156.1.g.b.155.18476.3odd48
          1156.1.g.b.155.28119.31even48
          1156.1.g.b.155.28476.31odd48
          1156.1.g.b.179.18119.10even48
          1156.1.g.b.179.18476.367odd48
          1156.1.g.b.179.28119.24even48
          1156.1.g.b.179.28476.143odd48
          1156.1.g.b.399.18119.45even48
          1156.1.g.b.399.18476.283odd48
          1156.1.g.b.399.28119.108even48
          1156.1.g.b.399.28476.227odd48
          1156.1.g.b.423.18119.80even48
          1156.1.g.b.423.18476.199odd48
          1156.1.g.b.423.28119.73even48
          1156.1.g.b.423.28476.311odd48
          1700.1.n.a.599.1235.3even12
          1700.1.n.a.599.12140.3odd12
          1700.1.n.a.999.12595.472even12
          1700.1.n.a.999.122380.1067odd12
          1700.1.n.b.599.1235.17even12
          1700.1.n.b.599.12140.87odd12
          1700.1.n.b.999.12595.353even12
          1700.1.n.b.999.122380.1543odd12
          1700.1.p.a.251.12595.234odd12
          1700.1.p.a.251.122380.2019even12
          1700.1.p.a.1551.1235.24odd6
          1700.1.p.a.1551.12140.59even6
          3332.1.m.b.2843.127.4even3
          3332.1.m.b.2843.1228.11odd6
          3332.1.m.b.3039.12119.81even12
          3332.1.m.b.3039.12476.319odd12
          3332.1.bc.b.667.147.2even3inner
          3332.1.bc.b.667.1428.23odd6inner
          3332.1.bc.b.863.14119.30even12inner
          3332.1.bc.b.863.14476.387odd12inner
          3332.1.bc.b.2027.141.1even1trivial
          3332.1.bc.b.2027.144.3odd2CM
          3332.1.bc.b.2223.1417.13even4inner
          3332.1.bc.b.2223.1468.47odd4inner
          3332.1.bc.c.667.147.5odd6
          3332.1.bc.c.667.1428.19even6
          3332.1.bc.c.863.14119.47odd12
          3332.1.bc.c.863.14476.47even12
          3332.1.bc.c.2027.147.6odd2
          3332.1.bc.c.2027.1428.27even2
          3332.1.bc.c.2223.14119.13odd4
          3332.1.bc.c.2223.14476.251even4
          \n", "html": "\n\n\n \n \n \n \n LMFDB - Embedded newform 3332.1.bc.b.2027.1 \n \n\n \n \n \n \n \n\n \n \n\n\n\n\n \n\n \n \n\n \n \n\n\n \n \n \n \n \n \n\n\n\n\n\n \n \n\n
          \n \n
          \n
          \n
          \n ⌂\n → Modular forms\n → Classical\n → Level 3332\n → Weight 1\n → Character orbit bc\n → Newform orbit b\n → Embedding 2027.1\n
          \n
          \n \n
          \n Citation\n ·\n Feedback\n ·\n Hide Menu\n \n
          \n
          \n\n
          \n
          Embedded newform 3332.1.bc.b.2027.1
          \n\n
          \n
          \n\n\n\n\n\n
          \n
          ↑
          \n

          Properties

          \n
          \n \n
          Label\n 3332.1.bc.b.2027.1
          \n
          Level\n $3332$
          Weight\n $1$
          Character\n 3332.2027
          Analytic conductor\n $1.663$
          Analytic rank\n $0$
          Dimension\n $4$
          Projective image\n $D_{4}$
          CM discriminant\n -4
          Inner twists\n $8$
          \n
          \n\n\n\n

          Related objects

          \n
          \n \n
          \n\n\n\n

          Downloads

          \n
          \n \n
          \n\n

          Learn more

          \n
          \n \n
          \n\n
          \n
          \n
          \n
          \n
          \n \n
          \n Show commands:\n Magma\n / PariGP\n / SageMath\n
          \n\n\n\n\n

          Newspace parameters

          \n\n
          comment: Compute space of new eigenforms
           
          \n
          [N,k,chi] = [3332,1,Mod(667,3332)]
           
          mf = mfinit([N,k,chi],0)
           
          lf = mfeigenbasis(mf)
           
          \n
          from sage.modular.dirichlet import DirichletCharacter
           
          H = DirichletGroup(3332, base_ring=CyclotomicField(12))
           
          chi = DirichletCharacter(H, H._module([6, 4, 9]))
           
          N = Newforms(chi, 1, names="a")
           
          \n
          //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
           
          chi := DirichletCharacter("3332.667");
           
          S:= CuspForms(chi, 1);
           
          N := Newforms(S);
           
          \n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
          Weight: \\( k \\) \\(=\\)\\( 1 \\)
          Character orbit: \\([\\chi]\\) \\(=\\) 3332.bc (of order \\(12\\), degree \\(4\\), not minimal)
          \n\n

          Newform invariants

          \n\n
          comment: select newform
           
          \n
          sage: f = N[0] # Warning: the index may be different
           
          \n
          gp: f = lf[1] \\\\ Warning: the index may be different
           
          \n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\t\n \n \t\n \t\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Self dual: no
          Analytic conductor: \\(1.66288462209\\)
          Analytic rank: \\(0\\)
          Dimension: \\(4\\)
          Coefficient field: \\(\\Q(\\zeta_{12})\\)
          \n
          comment: defining polynomial
           
          \n
          gp: f.mod \\\\ as an extension of the character field
           
          \n\n
          Defining polynomial: \n\n \\( x^{4} - x^{2} + 1 \\)\n \n\n \n \"Copy\n \n \n \"Toggle\n \n
          Coefficient ring: \\(\\Z[a_1, a_2]\\)
          Coefficient ring index: \\( 1 \\)
          Twist minimal: no (minimal twist has level 68)
          Projective image:\\(D_{4}\\)
          Projective field:Galois closure of 4.2.19652.1
          Artin image:$C_4\\wr C_2\\times C_6$
          Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
          \n\n\n

          Embedding invariants

          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Embedding label 2027.1
          Root\\(0.866025 - 0.500000i\\) of defining polynomial
          Character\\(\\chi\\)\\(=\\)3332.2027
          Dual form 3332.1.bc.b.863.1
          \n\n\n

          $q$-expansion

          \n
          \n
          comment: q-expansion
           
          \n
          sage: f.q_expansion() # note that sage often uses an isomorphic number field
           
          \n
          gp: mfcoefs(f, 20)
           
          \n\n
          \n \n \n \n \n \n \n \n \n \n \n \n
          \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
          \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\n\n \\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\n \n\n \n \"Copy\n \n \n \"Toggle\n \n
          \n
          \n\n
          \n\n\n

          Character values

          \n

          We give the values of \\(\\chi\\) on generators for \\(\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times\\).

          \n\n \n \n \n \n \n \n \n \n \n \n \n \n
          \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
          \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
          \n\n\n

          Coefficient data

          \n\n

          For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the\nSatake parameters \\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).

          \n\n\n\n\n

          \n
          \n \n Display \\(a_p\\) with \\(p\\) up to:\n 50\n 250\n 1000\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n \n Display \\(a_n\\) with \\(n\\) up to:\n 50\n 250\n 1000\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n
          \n\n\n
          \n \n \n \n \n \n \n
          Significant digits:
          \n
          \n\n
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\n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n
          \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
          \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
          \n \\(2\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n\n \n\n \n\n \n\n
          \n\n \n\n \n\n \n\n
          \n \\(3\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(4\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
          \n \\(5\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(6\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(7\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n\n \n\n \n\n
          \n\n \n\n \n\n \n\n
          \n \\(8\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
          \n \\(9\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
          \n \\(10\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(11\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(12\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(13\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(14\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(15\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(16\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
          \n \\(17\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n \n\n \n\n \n\n \n\n
          \n\n \n\n \n\n \n\n
          \n \\(18\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
          \n \\(19\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(20\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(21\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(22\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(23\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
          \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
          \n \\(24\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
          \n \\(26\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(27\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(28\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(29\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n \\(30\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(31\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
          \n \\(33\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(34\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(35\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(36\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
          \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(38\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(39\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(40\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
          \n \\(41\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(42\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(43\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(44\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(45\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(46\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(47\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(48\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(49\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(50\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
          \n \\(51\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(52\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(53\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(54\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(55\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(56\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(57\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(58\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(59\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(60\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(61\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(62\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(63\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(64\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
          \n \\(65\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(66\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(67\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
          \n \\(69\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(70\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(71\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(72\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
          \n \\(73\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n \\(74\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(75\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(76\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(77\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(78\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(79\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
          \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
          \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(81\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
          \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(83\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(84\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(85\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(86\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(87\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(88\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(89\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(90\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(91\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(92\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(93\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(94\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(95\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(96\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(97\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n \\(98\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(99\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(100\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(101\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(102\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(103\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(104\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(105\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(106\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(107\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(108\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(109\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(110\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(111\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(112\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(113\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(114\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(115\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(116\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(117\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(118\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(119\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(120\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(121\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(123\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(124\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(125\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(126\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(127\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(128\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(129\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(130\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(131\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(132\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(133\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(134\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(135\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(136\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
          \n \\(137\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(138\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(139\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(140\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(141\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(142\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(143\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
          \n \\(145\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
          \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(147\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(148\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(149\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n \\(150\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(151\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(152\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(153\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(154\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(155\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(156\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(157\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n \\(158\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(159\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(160\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(161\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(162\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
          \n \\(163\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
          \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
          \n \\(164\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(165\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(166\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(167\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(168\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(169\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
          \n \\(170\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(171\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(172\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(174\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(175\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(176\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(177\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(178\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(179\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(180\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
          \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(182\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(183\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(184\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(185\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
          \n \\(186\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(187\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(188\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(189\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(190\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(191\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(192\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(193\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(194\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(195\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(196\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(198\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(199\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(200\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
          \n \\(201\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(202\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(203\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(204\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n \n 1.00000\n \n +\n \n 1.73205i\n
          \n \\(206\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(207\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(208\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(209\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(210\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(211\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(212\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(213\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(214\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(215\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(216\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(217\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(218\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(219\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(220\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(221\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(222\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(223\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(224\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(225\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
          \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(227\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(228\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(229\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(230\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(231\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(234\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(235\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(236\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(237\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(238\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(239\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(240\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(241\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n \\(242\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
          \n \\(243\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(244\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(245\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(246\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(247\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(248\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(249\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(250\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(251\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(252\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(253\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(254\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(255\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(256\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
          \n \\(257\\)\n \n 1.73205\n \n −\n \n 1.00000i\n \n 1.73205\n \n −\n \n 1.00000i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(258\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(259\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(260\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(261\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(262\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(263\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(264\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(265\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(266\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(267\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(268\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(270\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(271\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(272\\)\n \n 1.00000\n \n\n \n\n \n 1.00000\n \n\n \n\n
          \n \\(273\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(274\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(275\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(276\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(278\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(279\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(280\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(281\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(282\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(283\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(284\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(285\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(286\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(287\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(288\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
          \n \\(289\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
          \n \\(290\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
          \n \\(291\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(292\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
          \n \\(293\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n \\(294\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(295\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(296\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(297\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n \n 1.73205\n \n +\n \n 1.00000i\n
          \n \\(299\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(300\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(301\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(302\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(303\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(304\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(305\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n
          \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
          \n \\(307\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(308\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(309\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(310\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(311\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(312\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(313\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(314\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
          \n \\(315\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(316\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(317\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(318\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(319\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(321\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(322\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(323\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(324\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
          \n \\(325\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(326\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(327\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(328\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(329\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(330\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(331\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(332\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(333\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(334\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(335\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(336\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(337\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n \\(338\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(339\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(340\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(341\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(342\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(343\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(344\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(345\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(346\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(347\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(348\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(349\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(350\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(351\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(352\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(353\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(354\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(355\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(356\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(357\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(358\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(359\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(361\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
          \n \\(362\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
          \n \\(363\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(364\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(365\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
          \n \\(366\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(367\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(368\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(370\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
          \n \\(371\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(372\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(373\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(374\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(375\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(376\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(377\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(378\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(379\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(380\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(381\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(382\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(383\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(384\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(385\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(386\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(387\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(388\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(389\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n \\(390\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(391\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(392\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(393\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(394\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
          \n \\(395\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(396\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(397\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(398\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(399\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(400\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(401\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(402\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(403\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(404\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(405\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(406\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(407\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(408\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(409\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(410\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
          \n \\(411\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(412\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(413\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(414\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(415\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(416\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(417\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(418\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(419\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(420\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(421\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(422\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(423\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(424\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(425\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
          \n \\(426\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(427\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(428\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(429\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(430\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(431\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(432\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(433\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(434\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(435\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(436\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
          \n \\(437\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(438\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(439\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(440\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(441\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(442\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(443\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(444\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(445\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(446\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(447\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(448\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(449\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(450\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(451\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(452\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(453\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(454\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(455\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(456\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(457\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(458\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
          \n \\(459\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(460\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(461\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(462\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(463\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(464\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
          \n \\(465\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(466\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(467\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(468\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(469\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(470\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(471\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(472\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(473\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(474\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(475\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(476\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(477\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(478\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(479\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(480\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(481\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(483\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(484\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(485\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
          \n \\(486\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(487\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(489\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(490\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(491\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(492\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(494\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(495\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(496\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(497\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(498\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(499\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
          \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
          \n \\(500\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(501\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(502\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(503\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(504\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(505\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(506\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(507\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(508\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(509\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n \\(510\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(511\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(512\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(513\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(514\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n
          \n \\(515\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(516\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(517\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(518\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(519\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(520\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(521\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(522\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(523\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(524\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(525\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(526\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(527\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(528\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(529\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
          \n \\(530\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(531\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(532\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(533\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(534\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(535\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(536\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(537\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(538\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(539\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(540\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(542\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(543\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(544\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
          \n \\(545\\)\n \n −2.00000\n \n\n \n\n \n −2.00000\n \n\n \n\n
          \n \\(546\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(547\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(548\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(550\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(551\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(552\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(553\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(554\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(555\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(556\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(557\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(558\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(559\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(560\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(561\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(562\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(563\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(564\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n \n 1.00000\n \n +\n \n 1.73205i\n
          \n \\(566\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(567\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(568\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(569\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(570\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(571\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(572\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(573\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(574\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(575\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(576\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
          \n \\(577\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
          \n \\(579\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(580\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
          \n \\(581\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(582\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(583\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(584\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(585\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(586\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
          \n \\(587\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(588\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(589\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(590\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(591\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(592\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(593\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(594\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(595\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(596\\)\n \n −2.00000\n \n\n \n\n \n −2.00000\n \n\n \n\n
          \n \\(597\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(598\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(599\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(600\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(601\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(602\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(603\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(604\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(605\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(606\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(607\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(608\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(609\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(610\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
          \n \\(611\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(612\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
          \n \\(613\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(614\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(615\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(616\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(617\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(618\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(619\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(620\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(621\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(622\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(623\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(624\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(625\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
          \n \\(626\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
          \n \\(627\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(628\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
          \n \\(629\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(630\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(631\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(632\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(633\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(635\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(636\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(637\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(638\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(639\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(640\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(641\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n \\(642\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(643\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(644\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(645\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(646\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(647\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
          \n \\(649\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(650\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(651\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(652\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(653\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n \\(654\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(655\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(656\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(658\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(659\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(660\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(661\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(662\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(663\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(664\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(665\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(666\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(667\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(668\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(669\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(670\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(671\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(672\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(673\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(674\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(675\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(676\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
          \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(678\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(679\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(681\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(682\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(683\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(684\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(685\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(686\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(687\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(688\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(689\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(690\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(691\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
          \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
          \n \\(692\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(693\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(694\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(695\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(696\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(697\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(698\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(699\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(700\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(701\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(702\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(703\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(704\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(705\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(706\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
          \n \\(707\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(708\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(709\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(710\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(711\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(712\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(713\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(714\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(715\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(716\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(717\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(718\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(719\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(720\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(721\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(722\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
          \n \\(723\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(724\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(725\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(726\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(727\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(728\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(729\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
          \n \\(730\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
          \n \\(731\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(732\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(733\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(734\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(735\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(736\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(737\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(738\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(739\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(740\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
          \n \\(741\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(742\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(743\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(744\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n \n 0.732051\n \n +\n \n 2.73205i\n
          \n \\(746\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(747\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(748\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(749\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(750\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(751\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(752\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(753\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(754\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(755\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(756\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(757\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(758\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(759\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(760\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(761\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(762\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(763\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(764\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(765\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(766\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(767\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(768\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(769\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(770\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(771\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(772\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(773\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
          \n \\(774\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(775\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(777\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(778\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
          \n \\(779\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(780\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(781\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(782\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(783\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(784\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(785\\)\n \n −2.00000\n \n +\n \n 2.00000i\n \n −2.00000\n \n +\n \n 2.00000i\n
          \n \\(786\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(787\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(788\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(789\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(790\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(791\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(792\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(793\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(795\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(796\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(797\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(798\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(799\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
          \n \\(801\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(803\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(804\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(805\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(806\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(807\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(808\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(809\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(810\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
          \n \\(811\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
          \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
          \n \\(812\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(813\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(814\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(815\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(816\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(817\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(818\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
          \n \\(819\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(820\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
          \n \\(821\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(822\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(823\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(824\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(825\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(826\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(827\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(828\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(829\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n \\(830\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(831\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(832\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(833\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(834\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(835\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(836\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(837\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(838\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(839\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(840\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(841\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
          \n \\(842\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(843\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(844\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(846\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(847\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(848\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(849\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(850\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
          \n \\(851\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(852\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(853\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n \\(854\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(855\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(856\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(857\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(858\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(859\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(860\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(861\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(862\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(863\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(864\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(865\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
          \n \\(866\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
          \n \\(867\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(868\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(869\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(870\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(871\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
          \n \\(873\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
          \n \\(874\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(875\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(876\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(877\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(878\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(879\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(880\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(881\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(882\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(883\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(884\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(885\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(886\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(887\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(888\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(889\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(890\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(891\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(892\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(893\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(894\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(895\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(896\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(897\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
          \n \\(899\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(900\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
          \n \\(901\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(902\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(903\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(904\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(905\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
          \n \\(906\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(907\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
          \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
          \n \\(908\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(909\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(910\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(911\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
          \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
          \n \\(912\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(913\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(914\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
          \n \\(915\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(916\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
          \n \\(917\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(918\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(919\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
          \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n \\(920\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(921\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(922\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(923\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(924\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
          \n \\(926\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(927\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(928\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
          \n \\(929\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(930\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(931\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(932\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
          \n \\(933\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(934\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(935\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(936\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(937\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
          \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
          \n \\(938\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(939\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(940\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(941\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(942\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(943\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(944\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(945\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(946\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(947\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
          \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
          \n \\(948\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(949\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(950\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(951\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(952\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(953\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(954\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(955\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(956\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(957\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(958\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(959\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(960\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(961\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
          \n \\(962\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(963\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(964\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
          \n \\(965\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
          \n \\(966\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(967\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
          \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
          \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
          \n \\(969\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(970\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
          \n \\(971\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
          \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
          \n \\(972\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(973\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(974\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(975\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(976\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
          \n \\(977\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
          \n \\(978\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(979\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(980\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(981\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
          \n \\(982\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(983\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(984\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(985\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
          \n \\(986\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
          \n \\(987\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(988\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(989\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(990\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(991\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
          \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
          \n \\(992\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(993\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(994\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(995\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(996\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(997\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
          \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
          \n \\(998\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n \\(999\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
          \n
          \n
          \n \n Display \\(a_p\\) with \\(p\\) up to:\n 50\n 250\n 1000\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n \n Display \\(a_n\\) with \\(n\\) up to:\n 50\n 250\n 1000\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n
          \n\n\n

          Twists

          \n\n\n\n\n \n \n \n \n \n \n \n\n\n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n
                 By twisting character
          CharParityOrdTypeTwistMinDim
          1.1even1trivial3332.1.bc.b.2027.14
          4.3odd2CM3332.1.bc.b.2027.14
          7.2even3inner3332.1.bc.b.667.14
          7.3odd668.1.f.a.55.1yes2
          7.4even33332.1.m.b.2843.12
          7.5odd63332.1.bc.c.667.14
          7.6odd23332.1.bc.c.2027.14
          17.13even4inner3332.1.bc.b.2223.14
          21.17even6612.1.l.a.55.12
          28.3even668.1.f.a.55.1yes2
          28.11odd63332.1.m.b.2843.12
          28.19even63332.1.bc.c.667.14
          28.23odd6inner3332.1.bc.b.667.14
          28.27even23332.1.bc.c.2027.14
          35.3even121700.1.n.a.599.12
          35.17even121700.1.n.b.599.12
          35.24odd61700.1.p.a.1551.12
          56.3even61088.1.p.a.191.12
          56.45odd61088.1.p.a.191.12
          68.47odd4inner3332.1.bc.b.2223.14
          84.59odd6612.1.l.a.55.12
          119.3even481156.1.g.b.155.18
          119.10even481156.1.g.b.179.18
          119.13odd43332.1.bc.c.2223.14
          119.24even481156.1.g.b.179.28
          119.30even12inner3332.1.bc.b.863.14
          119.31even481156.1.g.b.155.28
          119.38odd121156.1.f.b.251.12
          119.45even481156.1.g.b.399.18
          119.47odd123332.1.bc.c.863.14
          119.59odd241156.1.c.b.579.12
          119.66odd241156.1.d.a.1155.22
          119.73even481156.1.g.b.423.28
          119.80even481156.1.g.b.423.18
          119.81even123332.1.m.b.3039.12
          119.87odd241156.1.d.a.1155.12
          119.94odd241156.1.c.b.579.22
          119.101odd61156.1.f.b.327.12
          119.108even481156.1.g.b.399.28
          119.115odd1268.1.f.a.47.1✓2
          140.3odd121700.1.n.a.599.12
          140.59even61700.1.p.a.1551.12
          140.87odd121700.1.n.b.599.12
          357.353even12612.1.l.a.523.12
          476.3odd481156.1.g.b.155.18
          476.31odd481156.1.g.b.155.28
          476.47even123332.1.bc.c.863.14
          476.59even241156.1.c.b.579.12
          476.87even241156.1.d.a.1155.12
          476.115even1268.1.f.a.47.1✓2
          476.143odd481156.1.g.b.179.28
          476.199odd481156.1.g.b.423.18
          476.227odd481156.1.g.b.399.28
          476.251even43332.1.bc.c.2223.14
          476.283odd481156.1.g.b.399.18
          476.311odd481156.1.g.b.423.28
          476.319odd123332.1.m.b.3039.12
          476.339even61156.1.f.b.327.12
          476.367odd481156.1.g.b.179.18
          476.387odd12inner3332.1.bc.b.863.14
          476.395even121156.1.f.b.251.12
          476.423even241156.1.d.a.1155.22
          476.451even241156.1.c.b.579.22
          595.234odd121700.1.p.a.251.12
          595.353even121700.1.n.b.999.12
          595.472even121700.1.n.a.999.12
          952.115even121088.1.p.a.319.12
          952.829odd121088.1.p.a.319.12
          1428.1067odd12612.1.l.a.523.12
          2380.1067odd121700.1.n.a.999.12
          2380.1543odd121700.1.n.b.999.12
          2380.2019even121700.1.p.a.251.12
          \n
              
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          \n
          \n\n\n\n

          \n
          \n
          \n\n\n
          \n\n

          This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

          \n
          \n Contact\n ·\n Citation\n ·\n Acknowledgments\n ·\n Editorial Board\n ·\n Source\n ·\n SageMath version 10.1\n ·\n LMFDB Release 1.2.1\n
          \n
          \n\n"} diff --git a/bench/data/groundtruth/math_mathjax_asciimath_1.jsonl b/bench/data/groundtruth/math_mathjax_asciimath_1.jsonl index b257a2f7..20841ac9 100644 --- a/bench/data/groundtruth/math_mathjax_asciimath_1.jsonl +++ b/bench/data/groundtruth/math_mathjax_asciimath_1.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "title", "raw_content": "

          \n 1. Words and Equations

          ", "content": {"title_content": "1. Words and Equations", "level": "2"}}, {"type": "paragraph", "raw_content": "

          \n ASCIIMath works by transforming the text you type into mathematics. To see how it\n works, type this text

          ", "content": [{"c": "ASCIIMath works by transforming the text you type into mathematics. To see how it\n works, type this text", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Note:\n
          \n The \\` key is normally next to the 1 key on the keyboard.
          ", "content": [{"c": "Note: The \\` key is normally next to the 1 key on the keyboard.", "t": "text"}]}, {"type": "code", "raw_content": "
          \n    The area of a triangle with base `b`\n    and height `h` is `1/2 bh`. The area\n    of a circle with radius `r` is `pi r^2`.
          \n ", "inline": false, "content": {"code_content": " The area of a triangle with base `b`\n and height `h` is `1/2 bh`. The area\n of a circle with radius `r` is `pi r^2`.", "by": "tag_pre"}}, {"type": "paragraph", "raw_content": "

          \n into this text box:

          ", "content": [{"c": "into this text box:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n It is also possible to put the mathematics between dollar signs like this: \\$E=mc^2\\$.\n There is no difference in the output.
          ", "content": [{"c": "It is also possible to put the mathematics between dollar signs like this: \\$E=mc^2\\$.\n There is no difference in the output.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n As you type, the output appears in the grey coloured area next to the text entry\n box. We want to type mathematics and text, so to indicate mathematics we enclose\n it between left-quotes, like this: \\`E=mc^2\\`. Because you type the mathematics\n directly, it is quick and easy to include a lot of mathematics in your text.\n

          ", "content": [{"c": "As you type, the output appears in the grey coloured area next to the text entry\n box. We want to type mathematics and text, so to indicate mathematics we enclose\n it between left-quotes, like this: \\`E=mc^2\\`. Because you type the mathematics\n directly, it is quick and easy to include a lot of mathematics in your text.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n It is easier to read longer formulae if you put them on a separate line, as you\n can see in the text box below (and the output to the right). Take a look at any\n standard mathematics book to see the use of mathematics in the text and on separate\n lines.\n

          ", "content": [{"c": "It is easier to read longer formulae if you put them on a separate line, as you\n can see in the text box below (and the output to the right). Take a look at any\n standard mathematics book to see the use of mathematics in the text and on separate\n lines.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Einstein's famous formula is\n

          \n `E=mc^2`,\n

          \n where `E` is energy, `m` is mass and `c` is the speed of light.\n
          ", "content": [{"c": "Einstein's famous formula is`E=mc^2`, where `E` is energy, `m` is mass and `c` is the speed of light.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Try removing the blank lines above and below the equation in the text box above.\n After doing this, the equation will no longer appear on a separate line in the output.\n This is intentional, as sometimes you will want to type an equation on a separate\n line (for easy editing) without that occurring in the output.\n

          ", "content": [{"c": "Try removing the blank lines above and below the equation in the text box above.\n After doing this, the equation will no longer appear on a separate line in the output.\n This is intentional, as sometimes you will want to type an equation on a separate\n line (for easy editing) without that occurring in the output.", "t": "text"}]}, {"type": "title", "raw_content": "

          \n 2. Symbols

          ", "content": {"title_content": "2. Symbols", "level": "2"}}, {"type": "paragraph", "raw_content": "

          \n If you need to use a mathematical symbol, look it up in the \n symbol list. ASCIIMath has standard functions predefined, so to get `sin\n x` just type \\`sin x\\` and `ln x` type \\`ln x\\`.

          ", "content": [{"c": "If you need to use a mathematical symbol, look it up in the symbol list. ASCIIMath has standard functions predefined, so to get `sin\n x` just type \\`sin x\\` and `ln x` type \\`ln x\\`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Remember left-quotes (i.e. \\`) are used to indicate mathematics.
          ", "content": [{"c": "Remember left-quotes (i.e. \\`) are used to indicate mathematics.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Greek letters are produced by typing the name of the letter, e.g. \\`alpha\\` gives\n `alpha` and \\`beta\\` gives `beta`. To get capital Greek letters, capitalise the\n first letter of the name, e.g. \\`Sigma\\` gives `Sigma`, whereas \\`sigma\\` gives\n `sigma`.

          ", "content": [{"c": "Greek letters are produced by typing the name of the letter, e.g. \\`alpha\\` gives\n `alpha` and \\`beta\\` gives `beta`. To get capital Greek letters, capitalise the\n first letter of the name, e.g. \\`Sigma\\` gives `Sigma`, whereas \\`sigma\\` gives\n `sigma`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n For addition and subtraction, just use the symbols on the keyboard. For division,\n type \\`3 -: 4\\` to get `3-: 4` or \\`3//4\\` to get `3//4`. There are several symbols\n for multiplication and these are shown in the box below.

          ", "content": [{"c": "For addition and subtraction, just use the symbols on the keyboard. For division,\n type \\`3 -: 4\\` to get `3-: 4` or \\`3//4\\` to get `3//4`. There are several symbols\n for multiplication and these are shown in the box below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Typing `2xx7 = 14` and `2 times 7 =14` gives the same output.\n \n You might also want\n to write this as `2*7=14` or even `2**7=14`.
          ", "content": [{"c": "Typing `2xx7 = 14` and `2 times 7 =14` gives the same output.\n \n You might also want\n to write this as `2*7=14` or even `2**7=14`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Type the following expressions, referring to the symbol list\n if necessary.\n

          ", "content": [{"c": "Type the following expressions, referring to the symbol list if necessary.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \n `\\ \\ \\ \\ \\ ` `int cos x dx` `\\ \\ \\ \\ \\ \\ \\ \\ ` `f @ g (x)` `\\ \\ \\ \\ \\ \\ \\ \\ ` `x in RR` `\\ \\\n \\ \\ \\ \\ \\ \\ ` `phi: A -> B`\n

          ", "content": {"math_content": "\\setminus \\setminus \\setminus \\setminus \\setminus \\int \\cos x \\mathrm{dx} \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus f \\circ g \\left(x\\right) \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus x \\in \\mathbb{R} \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\phi : A \\to B", "math_type": "asciimath", "by": "None"}}, {"type": "paragraph", "raw_content": "
          \n To check your answers, hang the mouse pointer over one of the expressions above.\n After a couple of seconds a box will appear showing the ASCIIMath used to type the\n expression. You can always use this to see the ASCIIMath.
          ", "content": [{"c": "To check your answers, hang the mouse pointer over one of the expressions above.\n After a couple of seconds a box will appear showing the ASCIIMath used to type the\n expression. You can always use this to see the ASCIIMath.", "t": "text"}]}, {"type": "title", "raw_content": "

          \n 3. Indices, subscripts, fractions and brackets

          ", "content": {"title_content": "3. Indices, subscripts, fractions and brackets", "level": "2"}}, {"type": "paragraph", "raw_content": "
          \n Notice the brackets in \\`e^(2x)\\` don't appear in the output `e^(2x)`. In the expression\n `f^((3)) (x)` for the third derivative of `f`, we want the brackets to appear so\n we type \\`f^((3))(x)\\`.
          ", "content": [{"c": "Notice the brackets in \\`e^(2x)\\` don't appear in the output `e^(2x)`. In the expression\n `f^((3)) (x)` for the third derivative of `f`, we want the brackets to appear so\n we type \\`f^((3))(x)\\`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n To get an index we use the symbol ^, so for example \\`x^2\\` gives `x^2`. An index\n that is anything other than a number or a single letter requires brackets. So to\n get `e^(2x)` we type \\`e^(2x)\\`, because \\`e^2x\\` gives `e^2x`. However \\`p^32\\`\n gives `p^32` and \\`x^alpha\\` gives `x^alpha`.\n

          ", "content": [{"c": "To get an index we use the symbol ^, so for example \\`x^2\\` gives `x^2`. An index\n that is anything other than a number or a single letter requires brackets. So to\n get `e^(2x)` we type \\`e^(2x)\\`, because \\`e^2x\\` gives `e^2x`. However \\`p^32\\`\n gives `p^32` and \\`x^alpha\\` gives `x^alpha`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n The symbol _ is used for subscripts and they work in the same way as indices. So\n \\`x_10\\` gives `x_10` and it is necessary to use brackets in \\`x_(ij)\\` to get `x_(ij)`.\n

          ", "content": [{"c": "The symbol _ is used for subscripts and they work in the same way as indices. So\n \\`x_10\\` gives `x_10` and it is necessary to use brackets in \\`x_(ij)\\` to get `x_(ij)`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Remember you can check your answers by hanging the mouse over the expression.
          ", "content": [{"c": "Remember you can check your answers by hanging the mouse over the expression.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Use the text box to type the expressions below.\n

          ", "content": [{"c": "Use the text box to type the expressions below.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \n `\\ \\ \\ \\ ` `e^x` `\\ \\ \\ \\ ` `pi^2` `\\ \\ \\ \\ ` `(x + 1)^2` `\\ \\ \\ \\ ` `x^3 + 4x^2\n - 5` `\\ \\ \\ \\ ` `a_(nm)` `\\ \\ \\ \\ ` `x_2^3`\n

          ", "content": {"math_content": "\\setminus \\setminus \\setminus \\setminus {e}^{x} \\setminus \\setminus \\setminus \\setminus {\\pi}^{2} \\setminus \\setminus \\setminus \\setminus {\\left(x + 1\\right)}^{2} \\setminus \\setminus \\setminus \\setminus {x}^{3} + 4 {x}^{2} - 5 \\setminus \\setminus \\setminus \\setminus {a}_{n m} \\setminus \\setminus \\setminus \\setminus {x}_{2}^{3}", "math_type": "asciimath", "by": "None"}}, {"type": "paragraph", "raw_content": "

          \n The index and subscript notation is used to put the upper and lower limits on integrals\n and sums, as in the examples below.

          ", "content": [{"c": "The index and subscript notation is used to put the upper and lower limits on integrals\n and sums, as in the examples below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n `sum_(k=1)^n k = 1+2+ cdots +n=(n(n+1))/2`\n

          \n `int_0^1 x^2 dx`
          ", "content": [{"c": "`sum_(k=1)^n k = 1+2+ cdots +n=(n(n+1))/2``int_0^1 x^2 dx`", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n As you can see above, the symbol \\`/\\` is used for fractions. Unless they are very\n simple expressions, the denominator and numerator must be enclosed in brackets (which\n don't appear in the output). So, for example\n

          ", "content": [{"c": "As you can see above, the symbol \\`/\\` is used for fractions. Unless they are very\n simple expressions, the denominator and numerator must be enclosed in brackets (which\n don't appear in the output). So, for example", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \n `\\ \\ \\ \\ ` `1/(x+1)`\n

          ", "content": {"math_content": "\\setminus \\setminus \\setminus \\setminus \\frac{1}{x + 1}", "math_type": "asciimath", "by": "None"}}, {"type": "paragraph", "raw_content": "

          \n is achieved by typing \\`1/(x+1)\\`. What do you think would be produced by typing\n \\`1/x+1\\`?\n

          ", "content": [{"c": "is achieved by typing \\`1/(x+1)\\`. What do you think would be produced by typing\n \\`1/x+1\\`?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Both \\`x_n/10\\` and \\`(x_n)/(10)\\` give the same output, so sometimes it is not\n necessary to use brackets. Try typing the following expressions and experiment with\n using/missing out brackets to produce the same output.\n

          ", "content": [{"c": "Both \\`x_n/10\\` and \\`(x_n)/(10)\\` give the same output, so sometimes it is not\n necessary to use brackets. Try typing the following expressions and experiment with\n using/missing out brackets to produce the same output.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \n `\\ \\ \\ \\ ` `(x+1)/(x+1)` `\\ \\ \\ \\ ` `(x+1)/x^2` `\\ \\ \\ \\ ` `1/3245` `\\ \\ \\ \\ ` `dy/dx`\n `\\ \\ \\ \\ ` `(ab)/(cd)`\n

          ", "content": {"math_content": "\\setminus \\setminus \\setminus \\setminus \\frac{x + 1}{x + 1} \\setminus \\setminus \\setminus \\setminus \\frac{x + 1}{x} ^ 2 \\setminus \\setminus \\setminus \\setminus \\frac{1}{3245} \\setminus \\setminus \\setminus \\setminus \\frac{\\mathrm{dy}}{\\mathrm{dx}} \\setminus \\setminus \\setminus \\setminus \\frac{a b}{c d}", "math_type": "asciimath", "by": "None"}}, {"type": "paragraph", "raw_content": "

          \n Remember that we can get a one line fraction by using //.

          ", "content": [{"c": "Remember that we can get a one line fraction by using //.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n You may wish to use this for fractions that appear as indices, because it is easier\n to read `e^(x//2)` than `e^(x/2)`.
          ", "content": [{"c": "You may wish to use this for fractions that appear as indices, because it is easier\n to read `e^(x//2)` than `e^(x/2)`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Finally, use \\`sqrt(x)\\` for `\\sqrt(x)` and \\`root(x)(y)\\` for `root(x)(y)`. Use\n the box below to type the formula for the solutions of the equation `ax^2+bx+c=0`,\n where `a !=0`.

          ", "content": [{"c": "Finally, use \\`sqrt(x)\\` for `\\sqrt(x)` and \\`root(x)(y)\\` for `root(x)(y)`. Use\n the box below to type the formula for the solutions of the equation `ax^2+bx+c=0`,\n where `a !=0`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n [Hint: the formula appears at the top of this page and you may need to use the symbol\n list.]

          ", "content": [{"c": "[Hint: the formula appears at the top of this page and you may need to use the symbol\n list.]", "t": "text"}]}, {"type": "title", "raw_content": "

          \n 4. Spaces, multiple lines and text

          ", "content": {"title_content": "4. Spaces, multiple lines and text", "level": "2"}}, {"type": "paragraph", "raw_content": "
          \n In some browsers, the words quad and qquad will produce longer spaces than a backslash\n and a space.
          ", "content": [{"c": "In some browsers, the words quad and qquad will produce longer spaces than a backslash\n and a space.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n ASCIIMath should generate spaces that are about right in the output, but if you\n want to get an extra space in the mathematics you can use a backslash and a space\n i.e. \\`\\ \\`. Extra spaces are especially useful when you want to put more than one\n expression on a line, as in the example below.\n

          ", "content": [{"c": "ASCIIMath should generate spaces that are about right in the output, but if you\n want to get an extra space in the mathematics you can use a backslash and a space\n i.e. \\`\\ \\`. Extra spaces are especially useful when you want to put more than one\n expression on a line, as in the example below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n We set\n

          \n `a=3, \\ \\ \\ b=-3,\\ \\ ` and `\\ \\ c=2`.\n

          \n Thus `(a+b)(c+b)=0`.
          ", "content": [{"c": "We set`a=3, \\ \\ \\ b=-3,\\ \\ ` and `\\ \\ c=2`. Thus `(a+b)(c+b)=0`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Notice that the punctuation of the sentence appears in the middle line even though\n it is mainly mathematical formulae. When beginning to type mathematics it might\n feel strange to include punctuation, but it is important to do so.\n

          ", "content": [{"c": "Notice that the punctuation of the sentence appears in the middle line even though\n it is mainly mathematical formulae. When beginning to type mathematics it might\n feel strange to include punctuation, but it is important to do so.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n When answering questions that require several lines of working you may like to follow\n the format of the example below. Because the second, third and fourth lines have been indented\n it makes it clear that all lines are equal to `x`.\n

          ", "content": [{"c": "When answering questions that require several lines of working you may like to follow\n the format of the example below. Because the second, third and fourth lines have been indented\n it makes it clear that all lines are equal to `x`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Using the quadratic formula, the roots of `x^2-6x+4=0` are\n \n `x = (-6 +- sqrt((-6)^2 - 4 (1)(4)))/(2 xx 1)`\n \n `\\ \\ = (-6 +- sqrt(36 - 16))/2`\n \n `\\ \\ =(-6 +- sqrt(20))/2`\n \n `\\ \\ = -0.8 or 2.2 \\ \\ \\ `to 1 decimal place.
          ", "content": [{"c": "Using the quadratic formula, the roots of `x^2-6x+4=0` are\n \n `x = (-6 +- sqrt((-6)^2 - 4 (1)(4)))/(2 xx 1)`\n \n `\\ \\ = (-6 +- sqrt(36 - 16))/2`\n \n `\\ \\ =(-6 +- sqrt(20))/2`\n \n `\\ \\ = -0.8 or 2.2 \\ \\ \\ `to 1 decimal place.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n There is a full stop at the end of the last line in this example because\n it is at the end of a sentence, albeit a sentence with a lot of mathematics in it.\n
          ", "content": [{"c": "There is a full stop at the end of the last line in this example because\n it is at the end of a sentence, albeit a sentence with a lot of mathematics in it.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n On the last line in the example above we have used the word \"or\". ASCIIMath recognises the words\n \"or\", \"and\" and \"if\". Any other words need to be enclosed in inverted commas or using \\`text(some words)\\`.\n Try removing the inverted commas in the example below to see their effect.

          ", "content": [{"c": "On the last line in the example above we have used the word \"or\". ASCIIMath recognises the words\n \"or\", \"and\" and \"if\". Any other words need to be enclosed in inverted commas or using \\`text(some words)\\`.\n Try removing the inverted commas in the example below to see their effect.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          `sin = \"opp\"/\"hyp\" \\ \\ \\ and \\ \\ \\ cos = text(adj)/text(hyp)`.
          ", "content": [{"c": "`sin = \"opp\"/\"hyp\" \\ \\ \\ and \\ \\ \\ cos = text(adj)/text(hyp)`.", "t": "text"}]}, {"type": "title", "raw_content": "

          \n 5. Matrices and advanced alignment

          ", "content": {"title_content": "5. Matrices and advanced alignment", "level": "2"}}, {"type": "paragraph", "raw_content": "

          \n The ASCIIMath notation for matrices is demonstrated below.\n

          ", "content": [{"c": "The ASCIIMath notation for matrices is demonstrated below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n A `3xx3` matrix,`((1,2,3),(4,5,6),(7,8,9))`, and a `2xx1` matrix, or vector, `((1),(0))`.\n

          \n The outer brackets determine the delimiters e.g. `|(a,b),(c,d)|=ad-bc`.\n

          \n A general `m xx n` matrix `((a_(11), cdots , a_(1n)),(vdots, ddots, vdots),(a_(m1),\n cdots , a_(mn)))`\n
          ", "content": [{"c": "A `3xx3` matrix,`((1,2,3),(4,5,6),(7,8,9))`, and a `2xx1` matrix, or vector, `((1),(0))`. The outer brackets determine the delimiters e.g. `|(a,b),(c,d)|=ad-bc`. A general `m xx n` matrix `((a_(11), cdots , a_(1n)),(vdots, ddots, vdots),(a_(m1),\n cdots , a_(mn)))`", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Alter the text in the box above so that it starts with\n

          ", "content": [{"c": "Alter the text in the box above so that it starts with", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n `\\ \\ \\ \\ \\ `A `3 xx 2` matrix ,`((1,2),(3,4),(5,6))`,...\n

          ", "content": [{"c": "`\\ \\ \\ \\ \\ `A `3 xx 2` matrix ,`((1,2),(3,4),(5,6))`,...", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n In the general `m xx n` matrix, we used \\`cdots\\` for centred dots. This can be\n useful when writing infinite series, for example\n

          ", "content": [{"c": "In the general `m xx n` matrix, we used \\`cdots\\` for centred dots. This can be\n useful when writing infinite series, for example", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n `e^x = 1+x+1/(2!)x^2 + 1/(3!)x^3 + cdots `.
          ", "content": [{"c": "`e^x = 1+x+1/(2!)x^2 + 1/(3!)x^3 + cdots `.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Similarly, for low dots, as in `1, ldots, n` use \\`...\\` or \\`ldots\\`.\n

          ", "content": [{"c": "Similarly, for low dots, as in `1, ldots, n` use \\`...\\` or \\`ldots\\`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n The matrix notation can be used for aligning material. When doing this, we will\n want to draw matrices without delimiters. For this purpose, we can type \\`{:\\` and\n \\`:}\\` to get invisible brackets. This is necessary when defining a function by\n cases.\n

          ", "content": [{"c": "The matrix notation can be used for aligning material. When doing this, we will\n want to draw matrices without delimiters. For this purpose, we can type \\`{:\\` and\n \\`:}\\` to get invisible brackets. This is necessary when defining a function by\n cases.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n `|x|= {(x , if x ge 0 text(,)),(-x , if x <0.):}`
          ", "content": [{"c": "`|x|= {(x , if x ge 0 text(,)),(-x , if x <0.):}`", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Notice the use of \\`text( )\\` round\n the comma after `x ge 0` to indicate that it really is a comma and not part of the\n matrix syntax. Try to write the matrix below and then try to write the function\n definition using a matrix with two invisible brackets.\n

          ", "content": [{"c": "Notice the use of \\`text( )\\` round\n the comma after `x ge 0` to indicate that it really is a comma and not part of the\n matrix syntax. Try to write the matrix below and then try to write the function\n definition using a matrix with two invisible brackets.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n You can look up symbols in the symbol list.
          ", "content": [{"c": "You can look up symbols in the symbol list.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \n `((1,2,3,4),(a,b,c,d))` `\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ` `{:(f:,[0,6],\n ->, RR),( ,x ,|->, x^2+1):}`\n

          ", "content": {"math_content": "\\left(\\begin{matrix}1 & 2 & 3 & 4 \\\\ a & b & c & d\\end{matrix}\\right) \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\left.\\left(f : , \\left[0 , 6\\right] , \\to , \\mathbb{R}\\right) , \\left(, x , \\mapsto , {x}^{2} + 1\\right)\\right.", "math_type": "asciimath", "by": "None"}}]], "main_html": "

          \n 1. Words and Equations

          \n ASCIIMath works by transforming the text you type into mathematics. To see how it\n works, type this text

          \n Note:\n
          \n The \\` key is normally next to the 1 key on the keyboard.
          \n    The area of a triangle with base `b`\n    and height `h` is `1/2 bh`. The area\n    of a circle with radius `r` is `pi r^2`.
          \n

          \n into this text box:

          \n It is also possible to put the mathematics between dollar signs like this: \\$E=mc^2\\$.\n There is no difference in the output.

          \n As you type, the output appears in the grey coloured area next to the text entry\n box. We want to type mathematics and text, so to indicate mathematics we enclose\n it between left-quotes, like this: \\`E=mc^2\\`. Because you type the mathematics\n directly, it is quick and easy to include a lot of mathematics in your text.\n

          \n It is easier to read longer formulae if you put them on a separate line, as you\n can see in the text box below (and the output to the right). Take a look at any\n standard mathematics book to see the use of mathematics in the text and on separate\n lines.\n

          \n Einstein's famous formula is\n

          \n `E=mc^2`,\n

          \n where `E` is energy, `m` is mass and `c` is the speed of light.\n

          \n Try removing the blank lines above and below the equation in the text box above.\n After doing this, the equation will no longer appear on a separate line in the output.\n This is intentional, as sometimes you will want to type an equation on a separate\n line (for easy editing) without that occurring in the output.\n

          \n 2. Symbols

          \n If you need to use a mathematical symbol, look it up in the \n symbol list. ASCIIMath has standard functions predefined, so to get `sin\n x` just type \\`sin x\\` and `ln x` type \\`ln x\\`.

          \n Remember left-quotes (i.e. \\`) are used to indicate mathematics.

          \n Greek letters are produced by typing the name of the letter, e.g. \\`alpha\\` gives\n `alpha` and \\`beta\\` gives `beta`. To get capital Greek letters, capitalise the\n first letter of the name, e.g. \\`Sigma\\` gives `Sigma`, whereas \\`sigma\\` gives\n `sigma`.

          \n For addition and subtraction, just use the symbols on the keyboard. For division,\n type \\`3 -: 4\\` to get `3-: 4` or \\`3//4\\` to get `3//4`. There are several symbols\n for multiplication and these are shown in the box below.

          \n Typing `2xx7 = 14` and `2 times 7 =14` gives the same output.\n \n You might also want\n to write this as `2*7=14` or even `2**7=14`.

          \n Type the following expressions, referring to the symbol list\n if necessary.\n

          \n `\\ \\ \\ \\ \\ ` `int cos x dx` `\\ \\ \\ \\ \\ \\ \\ \\ ` `f @ g (x)` `\\ \\ \\ \\ \\ \\ \\ \\ ` `x in RR` `\\ \\\n \\ \\ \\ \\ \\ \\ ` `phi: A -> B`\n

          \n To check your answers, hang the mouse pointer over one of the expressions above.\n After a couple of seconds a box will appear showing the ASCIIMath used to type the\n expression. You can always use this to see the ASCIIMath.

          \n 3. Indices, subscripts, fractions and brackets

          \n Notice the brackets in \\`e^(2x)\\` don't appear in the output `e^(2x)`. In the expression\n `f^((3)) (x)` for the third derivative of `f`, we want the brackets to appear so\n we type \\`f^((3))(x)\\`.

          \n To get an index we use the symbol ^, so for example \\`x^2\\` gives `x^2`. An index\n that is anything other than a number or a single letter requires brackets. So to\n get `e^(2x)` we type \\`e^(2x)\\`, because \\`e^2x\\` gives `e^2x`. However \\`p^32\\`\n gives `p^32` and \\`x^alpha\\` gives `x^alpha`.\n

          \n The symbol _ is used for subscripts and they work in the same way as indices. So\n \\`x_10\\` gives `x_10` and it is necessary to use brackets in \\`x_(ij)\\` to get `x_(ij)`.\n

          \n Remember you can check your answers by hanging the mouse over the expression.

          \n Use the text box to type the expressions below.\n

          \n `\\ \\ \\ \\ ` `e^x` `\\ \\ \\ \\ ` `pi^2` `\\ \\ \\ \\ ` `(x + 1)^2` `\\ \\ \\ \\ ` `x^3 + 4x^2\n - 5` `\\ \\ \\ \\ ` `a_(nm)` `\\ \\ \\ \\ ` `x_2^3`\n

          \n The index and subscript notation is used to put the upper and lower limits on integrals\n and sums, as in the examples below.

          \n `sum_(k=1)^n k = 1+2+ cdots +n=(n(n+1))/2`\n

          \n `int_0^1 x^2 dx`

          \n As you can see above, the symbol \\`/\\` is used for fractions. Unless they are very\n simple expressions, the denominator and numerator must be enclosed in brackets (which\n don't appear in the output). So, for example\n

          \n `\\ \\ \\ \\ ` `1/(x+1)`\n

          \n is achieved by typing \\`1/(x+1)\\`. What do you think would be produced by typing\n \\`1/x+1\\`?\n

          \n Both \\`x_n/10\\` and \\`(x_n)/(10)\\` give the same output, so sometimes it is not\n necessary to use brackets. Try typing the following expressions and experiment with\n using/missing out brackets to produce the same output.\n

          \n `\\ \\ \\ \\ ` `(x+1)/(x+1)` `\\ \\ \\ \\ ` `(x+1)/x^2` `\\ \\ \\ \\ ` `1/3245` `\\ \\ \\ \\ ` `dy/dx`\n `\\ \\ \\ \\ ` `(ab)/(cd)`\n

          \n Remember that we can get a one line fraction by using //.

          \n You may wish to use this for fractions that appear as indices, because it is easier\n to read `e^(x//2)` than `e^(x/2)`.

          \n Finally, use \\`sqrt(x)\\` for `\\sqrt(x)` and \\`root(x)(y)\\` for `root(x)(y)`. Use\n the box below to type the formula for the solutions of the equation `ax^2+bx+c=0`,\n where `a !=0`.

          \n [Hint: the formula appears at the top of this page and you may need to use the symbol\n list.]

          \n 4. Spaces, multiple lines and text

          \n In some browsers, the words quad and qquad will produce longer spaces than a backslash\n and a space.

          \n ASCIIMath should generate spaces that are about right in the output, but if you\n want to get an extra space in the mathematics you can use a backslash and a space\n i.e. \\`\\ \\`. Extra spaces are especially useful when you want to put more than one\n expression on a line, as in the example below.\n

          \n We set\n

          \n `a=3, \\ \\ \\ b=-3,\\ \\ ` and `\\ \\ c=2`.\n

          \n Thus `(a+b)(c+b)=0`.

          \n Notice that the punctuation of the sentence appears in the middle line even though\n it is mainly mathematical formulae. When beginning to type mathematics it might\n feel strange to include punctuation, but it is important to do so.\n

          \n When answering questions that require several lines of working you may like to follow\n the format of the example below. Because the second, third and fourth lines have been indented\n it makes it clear that all lines are equal to `x`.\n

          \n Using the quadratic formula, the roots of `x^2-6x+4=0` are\n \n `x = (-6 +- sqrt((-6)^2 - 4 (1)(4)))/(2 xx 1)`\n \n `\\ \\ = (-6 +- sqrt(36 - 16))/2`\n \n `\\ \\ =(-6 +- sqrt(20))/2`\n \n `\\ \\ = -0.8 or 2.2 \\ \\ \\ `to 1 decimal place.
          \n There is a full stop at the end of the last line in this example because\n it is at the end of a sentence, albeit a sentence with a lot of mathematics in it.\n

          \n On the last line in the example above we have used the word \"or\". ASCIIMath recognises the words\n \"or\", \"and\" and \"if\". Any other words need to be enclosed in inverted commas or using \\`text(some words)\\`.\n Try removing the inverted commas in the example below to see their effect.

          `sin = \"opp\"/\"hyp\" \\ \\ \\ and \\ \\ \\ cos = text(adj)/text(hyp)`.

          \n 5. Matrices and advanced alignment

          \n The ASCIIMath notation for matrices is demonstrated below.\n

          \n A `3xx3` matrix,`((1,2,3),(4,5,6),(7,8,9))`, and a `2xx1` matrix, or vector, `((1),(0))`.\n

          \n The outer brackets determine the delimiters e.g. `|(a,b),(c,d)|=ad-bc`.\n

          \n A general `m xx n` matrix `((a_(11), cdots , a_(1n)),(vdots, ddots, vdots),(a_(m1),\n cdots , a_(mn)))`\n

          \n Alter the text in the box above so that it starts with\n

          \n `\\ \\ \\ \\ \\ `A `3 xx 2` matrix ,`((1,2),(3,4),(5,6))`,...\n

          \n In the general `m xx n` matrix, we used \\`cdots\\` for centred dots. This can be\n useful when writing infinite series, for example\n

          \n `e^x = 1+x+1/(2!)x^2 + 1/(3!)x^3 + cdots `.

          \n Similarly, for low dots, as in `1, ldots, n` use \\`...\\` or \\`ldots\\`.\n

          \n The matrix notation can be used for aligning material. When doing this, we will\n want to draw matrices without delimiters. For this purpose, we can type \\`{:\\` and\n \\`:}\\` to get invisible brackets. This is necessary when defining a function by\n cases.\n

          \n `|x|= {(x , if x ge 0 text(,)),(-x , if x <0.):}`

          \n Notice the use of \\`text( )\\` round\n the comma after `x ge 0` to indicate that it really is a comma and not part of the\n matrix syntax. Try to write the matrix below and then try to write the function\n definition using a matrix with two invisible brackets.\n

          \n You can look up symbols in the symbol list.

          \n `((1,2,3,4),(a,b,c,d))` `\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ` `{:(f:,[0,6],\n ->, RR),( ,x ,|->, x^2+1):}`\n

          ", "statics": {"title": 5, "paragraph": 49, "paragraph.text": 49, "code": 1, "equation-interline": 5}, "url": "https://www.wjagray.co.uk/maths/ASCIIMathTutorial.html", "content": "## 1. Words and Equations\n\nASCIIMath works by transforming the text you type into mathematics. To see how it\n works, type this text\n\nNote: The \\\\` key is normally next to the 1 key on the keyboard.\n\n```\n The area of a triangle with base `b`\n and height `h` is `1/2 bh`. The area\n of a circle with radius `r` is `pi r^2`.\n```\n\ninto this text box:\n\nIt is also possible to put the mathematics between dollar signs like this: \\$E=mc^2\\$.\n There is no difference in the output.\n\nAs you type, the output appears in the grey coloured area next to the text entry\n box. We want to type mathematics and text, so to indicate mathematics we enclose\n it between left-quotes, like this: \\\\`E=mc^2\\\\`. Because you type the mathematics\n directly, it is quick and easy to include a lot of mathematics in your text.\n\nIt is easier to read longer formulae if you put them on a separate line, as you\n can see in the text box below (and the output to the right). Take a look at any\n standard mathematics book to see the use of mathematics in the text and on separate\n lines.\n\nEinstein's famous formula is\\`E=mc^2\\`, where \\`E\\` is energy, \\`m\\` is mass and \\`c\\` is the speed of light.\n\nTry removing the blank lines above and below the equation in the text box above.\n After doing this, the equation will no longer appear on a separate line in the output.\n This is intentional, as sometimes you will want to type an equation on a separate\n line (for easy editing) without that occurring in the output.\n\n## 2. Symbols\n\nIf you need to use a mathematical symbol, look it up in the symbol list. ASCIIMath has standard functions predefined, so to get \\`sin\n x\\` just type \\\\`sin x\\\\` and \\`ln x\\` type \\\\`ln x\\\\`.\n\nRemember left-quotes (i.e. \\\\`) are used to indicate mathematics.\n\nGreek letters are produced by typing the name of the letter, e.g. \\\\`alpha\\\\` gives\n \\`alpha\\` and \\\\`beta\\\\` gives \\`beta\\`. To get capital Greek letters, capitalise the\n first letter of the name, e.g. \\\\`Sigma\\\\` gives \\`Sigma\\`, whereas \\\\`sigma\\\\` gives\n \\`sigma\\`.\n\nFor addition and subtraction, just use the symbols on the keyboard. For division,\n type \\\\`3 -: 4\\\\` to get \\`3-: 4\\` or \\\\`3//4\\\\` to get \\`3//4\\`. There are several symbols\n for multiplication and these are shown in the box below.\n\nTyping \\`2xx7 = 14\\` and \\`2 times 7 =14\\` gives the same output.\n \n You might also want\n to write this as \\`2*7=14\\` or even \\`2**7=14\\`.\n\nType the following expressions, referring to the symbol list if necessary.\n\n$$\n\\setminus \\setminus \\setminus \\setminus \\setminus \\int \\cos x \\mathrm{dx} \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus f \\circ g \\left(x\\right) \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus x \\in \\mathbb{R} \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\phi : A \\to B\n$$\n\nTo check your answers, hang the mouse pointer over one of the expressions above.\n After a couple of seconds a box will appear showing the ASCIIMath used to type the\n expression. You can always use this to see the ASCIIMath.\n\n## 3. Indices, subscripts, fractions and brackets\n\nNotice the brackets in \\\\`e^(2x)\\\\` don't appear in the output \\`e^(2x)\\`. In the expression\n \\`f^((3)) (x)\\` for the third derivative of \\`f\\`, we want the brackets to appear so\n we type \\\\`f^((3))(x)\\\\`.\n\nTo get an index we use the symbol ^, so for example \\\\`x^2\\\\` gives \\`x^2\\`. An index\n that is anything other than a number or a single letter requires brackets. So to\n get \\`e^(2x)\\` we type \\\\`e^(2x)\\\\`, because \\\\`e^2x\\\\` gives \\`e^2x\\`. However \\\\`p^32\\\\`\n gives \\`p^32\\` and \\\\`x^alpha\\\\` gives \\`x^alpha\\`.\n\nThe symbol _ is used for subscripts and they work in the same way as indices. So\n \\\\`x_10\\\\` gives \\`x_10\\` and it is necessary to use brackets in \\\\`x_(ij)\\\\` to get \\`x_(ij)\\`.\n\nRemember you can check your answers by hanging the mouse over the expression.\n\nUse the text box to type the expressions below.\n\n$$\n\\setminus \\setminus \\setminus \\setminus {e}^{x} \\setminus \\setminus \\setminus \\setminus {\\pi}^{2} \\setminus \\setminus \\setminus \\setminus {\\left(x + 1\\right)}^{2} \\setminus \\setminus \\setminus \\setminus {x}^{3} + 4 {x}^{2} - 5 \\setminus \\setminus \\setminus \\setminus {a}_{n m} \\setminus \\setminus \\setminus \\setminus {x}_{2}^{3}\n$$\n\nThe index and subscript notation is used to put the upper and lower limits on integrals\n and sums, as in the examples below.\n\n\\`sum_(k=1)^n k = 1+2+ cdots +n=(n(n+1))/2\\`\\`int_0^1 x^2 dx\\`\n\nAs you can see above, the symbol \\\\`/\\\\` is used for fractions. Unless they are very\n simple expressions, the denominator and numerator must be enclosed in brackets (which\n don't appear in the output). So, for example\n\n$$\n\\setminus \\setminus \\setminus \\setminus \\frac{1}{x + 1}\n$$\n\nis achieved by typing \\\\`1/(x+1)\\\\`. What do you think would be produced by typing\n \\\\`1/x+1\\\\`?\n\nBoth \\\\`x_n/10\\\\` and \\\\`(x_n)/(10)\\\\` give the same output, so sometimes it is not\n necessary to use brackets. Try typing the following expressions and experiment with\n using/missing out brackets to produce the same output.\n\n$$\n\\setminus \\setminus \\setminus \\setminus \\frac{x + 1}{x + 1} \\setminus \\setminus \\setminus \\setminus \\frac{x + 1}{x} ^ 2 \\setminus \\setminus \\setminus \\setminus \\frac{1}{3245} \\setminus \\setminus \\setminus \\setminus \\frac{\\mathrm{dy}}{\\mathrm{dx}} \\setminus \\setminus \\setminus \\setminus \\frac{a b}{c d}\n$$\n\nRemember that we can get a one line fraction by using //.\n\nYou may wish to use this for fractions that appear as indices, because it is easier\n to read \\`e^(x//2)\\` than \\`e^(x/2)\\`.\n\nFinally, use \\\\`sqrt(x)\\\\` for \\`\\sqrt(x)\\` and \\\\`root(x)(y)\\\\` for \\`root(x)(y)\\`. Use\n the box below to type the formula for the solutions of the equation \\`ax^2+bx+c=0\\`,\n where \\`a !=0\\`.\n\n[Hint: the formula appears at the top of this page and you may need to use the symbol\n list.]\n\n## 4. Spaces, multiple lines and text\n\nIn some browsers, the words quad and qquad will produce longer spaces than a backslash\n and a space.\n\nASCIIMath should generate spaces that are about right in the output, but if you\n want to get an extra space in the mathematics you can use a backslash and a space\n i.e. \\\\`\\ \\\\`. Extra spaces are especially useful when you want to put more than one\n expression on a line, as in the example below.\n\nWe set\\`a=3, \\ \\ \\ b=-3,\\ \\ \\` and \\`\\ \\ c=2\\`. Thus \\`(a+b)(c+b)=0\\`.\n\nNotice that the punctuation of the sentence appears in the middle line even though\n it is mainly mathematical formulae. When beginning to type mathematics it might\n feel strange to include punctuation, but it is important to do so.\n\nWhen answering questions that require several lines of working you may like to follow\n the format of the example below. Because the second, third and fourth lines have been indented\n it makes it clear that all lines are equal to \\`x\\`.\n\nUsing the quadratic formula, the roots of \\`x^2-6x+4=0\\` are\n \n \\`x = (-6 +- sqrt((-6)^2 - 4 (1)(4)))/(2 xx 1)\\`\n \n \\`\\ \\ = (-6 +- sqrt(36 - 16))/2\\`\n \n \\`\\ \\ =(-6 +- sqrt(20))/2\\`\n \n \\`\\ \\ = -0.8 or 2.2 \\ \\ \\ \\`to 1 decimal place.\n\nThere is a full stop at the end of the last line in this example because\n it is at the end of a sentence, albeit a sentence with a lot of mathematics in it.\n\nOn the last line in the example above we have used the word \"or\". ASCIIMath recognises the words\n \"or\", \"and\" and \"if\". Any other words need to be enclosed in inverted commas or using \\\\`text(some words)\\\\`.\n Try removing the inverted commas in the example below to see their effect.\n\n\\`sin = \"opp\"/\"hyp\" \\ \\ \\ and \\ \\ \\ cos = text(adj)/text(hyp)\\`.\n\n## 5. Matrices and advanced alignment\n\nThe ASCIIMath notation for matrices is demonstrated below.\n\nA \\`3xx3\\` matrix,\\`((1,2,3),(4,5,6),(7,8,9))\\`, and a \\`2xx1\\` matrix, or vector, \\`((1),(0))\\`. The outer brackets determine the delimiters e.g. \\`|(a,b),(c,d)|=ad-bc\\`. A general \\`m xx n\\` matrix \\`((a_(11), cdots , a_(1n)),(vdots, ddots, vdots),(a_(m1),\n cdots , a_(mn)))\\`\n\nAlter the text in the box above so that it starts with\n\n\\`\\ \\ \\ \\ \\ \\`A \\`3 xx 2\\` matrix ,\\`((1,2),(3,4),(5,6))\\`,...\n\nIn the general \\`m xx n\\` matrix, we used \\\\`cdots\\\\` for centred dots. This can be\n useful when writing infinite series, for example\n\n\\`e^x = 1+x+1/(2!)x^2 + 1/(3!)x^3 + cdots \\`.\n\nSimilarly, for low dots, as in \\`1, ldots, n\\` use \\\\`...\\\\` or \\\\`ldots\\\\`.\n\nThe matrix notation can be used for aligning material. When doing this, we will\n want to draw matrices without delimiters. For this purpose, we can type \\\\`{:\\\\` and\n \\\\`:}\\\\` to get invisible brackets. This is necessary when defining a function by\n cases.\n\n\\`|x|= {(x , if x ge 0 text(,)),(-x , if x <0.):}\\`\n\nNotice the use of \\\\`text( )\\\\` round\n the comma after \\`x ge 0\\` to indicate that it really is a comma and not part of the\n matrix syntax. Try to write the matrix below and then try to write the function\n definition using a matrix with two invisible brackets.\n\nYou can look up symbols in the symbol list.\n\n$$\n\\left(\\begin{matrix}1 & 2 & 3 & 4 \\\\ a & b & c & d\\end{matrix}\\right) \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\left.\\left(f : , \\left[0 , 6\\right] , \\to , \\mathbb{R}\\right) , \\left(, x , \\mapsto , {x}^{2} + 1\\right)\\right.\n$$\n", "html": "\n\n\n \n \n ASCIIMath Tutorial\n\n \n\n \n\n \n\n \n\n\n
          \n
          \n
          \n

          \n ASCIIMath Tutorial

          \n

          \n James Gray - The Open University

          \n

          \n This tutorial and accompanying symbol list introduce\n the ASCIIMath notation and some points of good practice in writing mathematics.\n To display the mathematics correctly, this page requires Internet Explorer 6 with\n MathPlayer installed,\n or Mozilla/Firefox/Netscape 7+ with MathML fonts installed (e.g. download the MIT MathML font packages).\n If you have these installed you should see a familiar formula here:

          \n

          \n `x=(-b +- sqrt(b^2 - 4ac))/(2a)`

          \n
          \n
          \n
          \n

          \n 1. Words and Equations

          \n
          \n
          \n

          \n ASCIIMath works by transforming the text you type into mathematics. To see how it\n works, type this text

          \n
          \n
          \n Note:\n
          \n The \\` key is normally next to the 1 key on the keyboard.
          \n
          \n 
          \n    The area of a triangle with base `b`\n    and height `h` is `1/2 bh`. The area\n    of a circle with radius `r` is `pi r^2`.
          \n

          \n into this text box:

          \n\t
          \n
          \n
          \n
          \n \n
          \n
          \n It is also possible to put the mathematics between dollar signs like this: \\$E=mc^2\\$.\n There is no difference in the output.
          \n
          \n

          \n As you type, the output appears in the grey coloured area next to the text entry\n box. We want to type mathematics and text, so to indicate mathematics we enclose\n it between left-quotes, like this: \\`E=mc^2\\`. Because you type the mathematics\n directly, it is quick and easy to include a lot of mathematics in your text.\n

          \n

          \n It is easier to read longer formulae if you put them on a separate line, as you\n can see in the text box below (and the output to the right). Take a look at any\n standard mathematics book to see the use of mathematics in the text and on separate\n lines.\n

          \n
          \n
          \n Einstein's famous formula is\n

          \n

          \n `E=mc^2`,\n

          \n

          \n where `E` is energy, `m` is mass and `c` is the speed of light.\n
          \n \n
          \n \n
          \n

          \n Try removing the blank lines above and below the equation in the text box above.\n After doing this, the equation will no longer appear on a separate line in the output.\n This is intentional, as sometimes you will want to type an equation on a separate\n line (for easy editing) without that occurring in the output.\n

          \n
          \n

          \n 2. Symbols

          \n
          \n

          \n If you need to use a mathematical symbol, look it up in the \n symbol list. ASCIIMath has standard functions predefined, so to get `sin\n x` just type \\`sin x\\` and `ln x` type \\`ln x\\`.

          \n
          \n
          \n Remember left-quotes (i.e. \\`) are used to indicate mathematics.
          \n
          \n

          \n Greek letters are produced by typing the name of the letter, e.g. \\`alpha\\` gives\n `alpha` and \\`beta\\` gives `beta`. To get capital Greek letters, capitalise the\n first letter of the name, e.g. \\`Sigma\\` gives `Sigma`, whereas \\`sigma\\` gives\n `sigma`.

          \n

          \n For addition and subtraction, just use the symbols on the keyboard. For division,\n type \\`3 -: 4\\` to get `3-: 4` or \\`3//4\\` to get `3//4`. There are several symbols\n for multiplication and these are shown in the box below.

          \n
          \n
          \n Typing `2xx7 = 14` and `2 times 7 =14` gives the same output.\n

          \n You might also want\n to write this as `2*7=14` or even `2**7=14`.
          \n \n
          \n
          \n

          \n Type the following expressions, referring to the symbol list\n if necessary.\n

          \n

          \n `\\ \\ \\ \\ \\ ` `int cos x dx` `\\ \\ \\ \\ \\ \\ \\ \\ ` `f @ g (x)` `\\ \\ \\ \\ \\ \\ \\ \\ ` `x in RR` `\\ \\\n \\ \\ \\ \\ \\ \\ ` `phi: A -> B`\n

          \n
          \n
          \n
          \n \n
          \n
          \n To check your answers, hang the mouse pointer over one of the expressions above.\n After a couple of seconds a box will appear showing the ASCIIMath used to type the\n expression. You can always use this to see the ASCIIMath.
          \n

          \n 3. Indices, subscripts, fractions and brackets

          \n
          \n Notice the brackets in \\`e^(2x)\\` don't appear in the output `e^(2x)`. In the expression\n `f^((3)) (x)` for the third derivative of `f`, we want the brackets to appear so\n we type \\`f^((3))(x)\\`.
          \n
          \n

          \n To get an index we use the symbol ^, so for example \\`x^2\\` gives `x^2`. An index\n that is anything other than a number or a single letter requires brackets. So to\n get `e^(2x)` we type \\`e^(2x)\\`, because \\`e^2x\\` gives `e^2x`. However \\`p^32\\`\n gives `p^32` and \\`x^alpha\\` gives `x^alpha`.\n

          \n

          \n The symbol _ is used for subscripts and they work in the same way as indices. So\n \\`x_10\\` gives `x_10` and it is necessary to use brackets in \\`x_(ij)\\` to get `x_(ij)`.\n

          \n
          \n\t
          \n
          \n Remember you can check your answers by hanging the mouse over the expression.
          \n
          \n

          \n Use the text box to type the expressions below.\n

          \n

          \n `\\ \\ \\ \\ ` `e^x` `\\ \\ \\ \\ ` `pi^2` `\\ \\ \\ \\ ` `(x + 1)^2` `\\ \\ \\ \\ ` `x^3 + 4x^2\n - 5` `\\ \\ \\ \\ ` `a_(nm)` `\\ \\ \\ \\ ` `x_2^3`\n

          \n
          \n\t
          \n
          \n
          \n \n
          \n

          \n The index and subscript notation is used to put the upper and lower limits on integrals\n and sums, as in the examples below.

          \n
          \n
          \n `sum_(k=1)^n k = 1+2+ cdots +n=(n(n+1))/2`\n

          \n

          \n `int_0^1 x^2 dx`
          \n \n
          \n
          \n

          \n As you can see above, the symbol \\`/\\` is used for fractions. Unless they are very\n simple expressions, the denominator and numerator must be enclosed in brackets (which\n don't appear in the output). So, for example\n

          \n

          \n `\\ \\ \\ \\ ` `1/(x+1)`\n

          \n

          \n is achieved by typing \\`1/(x+1)\\`. What do you think would be produced by typing\n \\`1/x+1\\`?\n

          \n

          \n Both \\`x_n/10\\` and \\`(x_n)/(10)\\` give the same output, so sometimes it is not\n necessary to use brackets. Try typing the following expressions and experiment with\n using/missing out brackets to produce the same output.\n

          \n

          \n `\\ \\ \\ \\ ` `(x+1)/(x+1)` `\\ \\ \\ \\ ` `(x+1)/x^2` `\\ \\ \\ \\ ` `1/3245` `\\ \\ \\ \\ ` `dy/dx`\n `\\ \\ \\ \\ ` `(ab)/(cd)`\n

          \n
          \n
          \n
          \n \n
          \n
          \n

          \n Remember that we can get a one line fraction by using //.

          \n
          \n
          \n You may wish to use this for fractions that appear as indices, because it is easier\n to read `e^(x//2)` than `e^(x/2)`.
          \n \n
          \n
          \n

          \n Finally, use \\`sqrt(x)\\` for `\\sqrt(x)` and \\`root(x)(y)\\` for `root(x)(y)`. Use\n the box below to type the formula for the solutions of the equation `ax^2+bx+c=0`,\n where `a !=0`.

          \n
          \n
          \n
          \n \n
          \n
          \n

          \n [Hint: the formula appears at the top of this page and you may need to use the symbol\n list.]

          \n
          \n

          \n 4. Spaces, multiple lines and text

          \n
          \n In some browsers, the words quad and qquad will produce longer spaces than a backslash\n and a space.
          \n
          \n

          \n ASCIIMath should generate spaces that are about right in the output, but if you\n want to get an extra space in the mathematics you can use a backslash and a space\n i.e. \\`\\ \\`. Extra spaces are especially useful when you want to put more than one\n expression on a line, as in the example below.\n

          \n
          \n
          \n We set\n

          \n

          \n `a=3, \\ \\ \\ b=-3,\\ \\ ` and `\\ \\ c=2`.\n

          \n

          \n Thus `(a+b)(c+b)=0`.
          \n \n
          \n
          \n

          \n Notice that the punctuation of the sentence appears in the middle line even though\n it is mainly mathematical formulae. When beginning to type mathematics it might\n feel strange to include punctuation, but it is important to do so.\n

          \n

          \n When answering questions that require several lines of working you may like to follow\n the format of the example below. Because the second, third and fourth lines have been indented\n it makes it clear that all lines are equal to `x`.\n

          \n
          \n
          \n Using the quadratic formula, the roots of `x^2-6x+4=0` are\n

          \n `x = (-6 +- sqrt((-6)^2 - 4 (1)(4)))/(2 xx 1)`\n

          \n `\\ \\ = (-6 +- sqrt(36 - 16))/2`\n

          \n `\\ \\ =(-6 +- sqrt(20))/2`\n

          \n `\\ \\ = -0.8 or 2.2 \\ \\ \\ `to 1 decimal place.
          \n \n
          \n
          \n There is a full stop at the end of the last line in this example because\n it is at the end of a sentence, albeit a sentence with a lot of mathematics in it.\n
          \n
          \n

          \n On the last line in the example above we have used the word \"or\". ASCIIMath recognises the words\n \"or\", \"and\" and \"if\". Any other words need to be enclosed in inverted commas or using \\`text(some words)\\`.\n Try removing the inverted commas in the example below to see their effect.

          \n
          \n\t
          `sin = \"opp\"/\"hyp\" \\ \\ \\ and \\ \\ \\ cos = text(adj)/text(hyp)`.
          \n \n\t
          \n

          \n 5. Matrices and advanced alignment

          \n
          \n

          \n The ASCIIMath notation for matrices is demonstrated below.\n

          \n
          \n
          \n A `3xx3` matrix,`((1,2,3),(4,5,6),(7,8,9))`, and a `2xx1` matrix, or vector, `((1),(0))`.\n

          \n

          \n The outer brackets determine the delimiters e.g. `|(a,b),(c,d)|=ad-bc`.\n

          \n

          \n A general `m xx n` matrix `((a_(11), cdots , a_(1n)),(vdots, ddots, vdots),(a_(m1),\n cdots , a_(mn)))`\n
          \n \n
          \n
          \n

          \n Alter the text in the box above so that it starts with\n

          \n

          \n `\\ \\ \\ \\ \\ `A `3 xx 2` matrix ,`((1,2),(3,4),(5,6))`,...\n

          \n

          \n In the general `m xx n` matrix, we used \\`cdots\\` for centred dots. This can be\n useful when writing infinite series, for example\n

          \n
          \n
          \n `e^x = 1+x+1/(2!)x^2 + 1/(3!)x^3 + cdots `.
          \n \n
          \n

          \n Similarly, for low dots, as in `1, ldots, n` use \\`...\\` or \\`ldots\\`.\n

          \n

          \n The matrix notation can be used for aligning material. When doing this, we will\n want to draw matrices without delimiters. For this purpose, we can type \\`{:\\` and\n \\`:}\\` to get invisible brackets. This is necessary when defining a function by\n cases.\n

          \n
          \n
          \n `|x|= {(x , if x ge 0 text(,)),(-x , if x <0.):}`
          \n \n
          \n
          \n

          \n Notice the use of \\`text( )\\` round\n the comma after `x ge 0` to indicate that it really is a comma and not part of the\n matrix syntax. Try to write the matrix below and then try to write the function\n definition using a matrix with two invisible brackets.\n

          \n
          \n
          \n You can look up symbols in the symbol list.
          \n
          \n

          \n `((1,2,3,4),(a,b,c,d))` `\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ` `{:(f:,[0,6],\n ->, RR),( ,x ,|->, x^2+1):}`\n

          \n
          \n
          \n
          \n
          \n \n
          \n
          \n

          \n Acknowledgements

          \n

          \n The ASCIIMath syntax and javascripts were developed by Peter Jipsen (Chapman University).\n Further information can be found at the \n ASCIIMath Homepage.\n

          \n

          \n The javascripts for this page were adapted from Peter Jipsen's \n ASCIIMath email editor by Keith Brunton at Edinburgh University.\n

          \n
          \n
          \n
          \n\n\n"} +{"content_list": [[{"type": "title", "raw_content": "

          \n 1. Words and Equations

          ", "content": {"title_content": "1. Words and Equations", "level": "2"}}, {"type": "paragraph", "raw_content": "

          \n ASCIIMath works by transforming the text you type into mathematics. To see how it\n works, type this text

          ", "content": [{"c": "ASCIIMath works by transforming the text you type into mathematics. To see how it\n works, type this text", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Note:\n
          \n The \\` key is normally next to the 1 key on the keyboard.
          ", "content": [{"c": "Note: The \\` key is normally next to the 1 key on the keyboard.", "t": "text"}]}, {"type": "code", "raw_content": "
          \n    The area of a triangle with base `b`\n    and height `h` is `1/2 bh`. The area\n    of a circle with radius `r` is `pi r^2`.
          \n ", "inline": false, "content": {"code_content": "The area of a triangle with base `b`\nand height `h` is `1/2 bh`. The area\nof a circle with radius `r` is `pi r^2`.", "by": "tag_pre"}}, {"type": "paragraph", "raw_content": "

          \n into this text box:

          ", "content": [{"c": "into this text box:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n It is also possible to put the mathematics between dollar signs like this: \\$E=mc^2\\$.\n There is no difference in the output.
          ", "content": [{"c": "It is also possible to put the mathematics between dollar signs like this: \\$E=mc^2\\$.\n There is no difference in the output.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n As you type, the output appears in the grey coloured area next to the text entry\n box. We want to type mathematics and text, so to indicate mathematics we enclose\n it between left-quotes, like this: \\E = m {c}^{2}. Because you type the mathematics\n directly, it is quick and easy to include a lot of mathematics in your text.\n

          ", "content": [{"c": "As you type, the output appears in the grey coloured area next to the text entry\n box. We want to type mathematics and text, so to indicate mathematics we enclose\n it between left-quotes, like this: \\", "t": "text"}, {"c": "E = m {c}^{2}", "t": "equation-inline"}, {"c": ". Because you type the mathematics\n directly, it is quick and easy to include a lot of mathematics in your text.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n It is easier to read longer formulae if you put them on a separate line, as you\n can see in the text box below (and the output to the right). Take a look at any\n standard mathematics book to see the use of mathematics in the text and on separate\n lines.\n

          ", "content": [{"c": "It is easier to read longer formulae if you put them on a separate line, as you\n can see in the text box below (and the output to the right). Take a look at any\n standard mathematics book to see the use of mathematics in the text and on separate\n lines.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Einstein's famous formula is\n

          \n `E=mc^2`,\n

          \n where `E` is energy, `m` is mass and `c` is the speed of light.\n
          ", "content": [{"c": "Einstein's famous formula is`E=mc^2`, where `E` is energy, `m` is mass and `c` is the speed of light.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Try removing the blank lines above and below the equation in the text box above.\n After doing this, the equation will no longer appear on a separate line in the output.\n This is intentional, as sometimes you will want to type an equation on a separate\n line (for easy editing) without that occurring in the output.\n

          ", "content": [{"c": "Try removing the blank lines above and below the equation in the text box above.\n After doing this, the equation will no longer appear on a separate line in the output.\n This is intentional, as sometimes you will want to type an equation on a separate\n line (for easy editing) without that occurring in the output.", "t": "text"}]}, {"type": "title", "raw_content": "

          \n 2. Symbols

          ", "content": {"title_content": "2. Symbols", "level": "2"}}, {"type": "paragraph", "raw_content": "

          \n If you need to use a mathematical symbol, look it up in the \n symbol list. ASCIIMath has standard functions predefined, so to get `sin\n x` just type \\`sin x\\` and `ln x` type \\`ln x\\`.

          ", "content": [{"c": "If you need to use a mathematical symbol, look it up in the symbol list. ASCIIMath has standard functions predefined, so to get `sin\n x` just type \\`sin x\\` and `ln x` type \\`ln x\\`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Remember left-quotes (i.e. \\`) are used to indicate mathematics.
          ", "content": [{"c": "Remember left-quotes (i.e. \\`) are used to indicate mathematics.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Greek letters are produced by typing the name of the letter, e.g. \\\\alpha gives\n \\alpha and \\\\beta gives \\beta. To get capital Greek letters, capitalise the\n first letter of the name, e.g. \\\\Sigma gives \\Sigma, whereas \\\\sigma gives\n \\sigma.

          ", "content": [{"c": "Greek letters are produced by typing the name of the letter, e.g. \\", "t": "text"}, {"c": "\\alpha", "t": "equation-inline"}, {"c": "gives", "t": "text"}, {"c": "\\alpha", "t": "equation-inline"}, {"c": "and \\", "t": "text"}, {"c": "\\beta", "t": "equation-inline"}, {"c": "gives", "t": "text"}, {"c": "\\beta", "t": "equation-inline"}, {"c": ". To get capital Greek letters, capitalise the\n first letter of the name, e.g. \\", "t": "text"}, {"c": "\\Sigma", "t": "equation-inline"}, {"c": "gives", "t": "text"}, {"c": "\\Sigma", "t": "equation-inline"}, {"c": ", whereas \\", "t": "text"}, {"c": "\\sigma", "t": "equation-inline"}, {"c": "gives", "t": "text"}, {"c": "\\sigma", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n For addition and subtraction, just use the symbols on the keyboard. For division,\n type \\3 \\div 4 to get 3 \\div 4 or \\3 / 4 to get 3 / 4. There are several symbols\n for multiplication and these are shown in the box below.

          ", "content": [{"c": "For addition and subtraction, just use the symbols on the keyboard. For division,\n type \\", "t": "text"}, {"c": "3 \\div 4", "t": "equation-inline"}, {"c": "to get", "t": "text"}, {"c": "3 \\div 4", "t": "equation-inline"}, {"c": "or \\", "t": "text"}, {"c": "3 / 4", "t": "equation-inline"}, {"c": "to get", "t": "text"}, {"c": "3 / 4", "t": "equation-inline"}, {"c": ". There are several symbols\n for multiplication and these are shown in the box below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Typing 2 \\times 7 = 14 and 2 \\times 7 = 14 gives the same output.\n \n You might also want\n to write this as 2 \\cdot 7 = 14 or even 2 \\ast 7 = 14.
          ", "content": [{"c": "Typing", "t": "text"}, {"c": "2 \\times 7 = 14", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "2 \\times 7 = 14", "t": "equation-inline"}, {"c": "gives the same output.\n \n You might also want\n to write this as", "t": "text"}, {"c": "2 \\cdot 7 = 14", "t": "equation-inline"}, {"c": "or even", "t": "text"}, {"c": "2 \\ast 7 = 14", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Type the following expressions, referring to the symbol list\n if necessary.\n

          ", "content": [{"c": "Type the following expressions, referring to the symbol list if necessary.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n `\\ \\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\int \\cos x \\mathrm{dx}", "content": {"math_content": "\\int \\cos x \\mathrm{dx}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\ \\ \\ \\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ \\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "f \\circ g \\left(x\\right)", "content": {"math_content": "f \\circ g \\left(x\\right)", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\ \\ \\ \\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ \\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x \\in \\mathbb{R}", "content": {"math_content": "x \\in \\mathbb{R}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\\n \\ \\ \\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\\n \\ \\ \\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\phi : A \\to B", "content": {"math_content": "\\phi : A \\to B", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          \n To check your answers, hang the mouse pointer over one of the expressions above.\n After a couple of seconds a box will appear showing the ASCIIMath used to type the\n expression. You can always use this to see the ASCIIMath.
          ", "content": [{"c": "To check your answers, hang the mouse pointer over one of the expressions above.\n After a couple of seconds a box will appear showing the ASCIIMath used to type the\n expression. You can always use this to see the ASCIIMath.", "t": "text"}]}, {"type": "title", "raw_content": "

          \n 3. Indices, subscripts, fractions and brackets

          ", "content": {"title_content": "3. Indices, subscripts, fractions and brackets", "level": "2"}}, {"type": "paragraph", "raw_content": "
          \n Notice the brackets in \\{e}^{2 x} don't appear in the output {e}^{2 x}. In the expression\n {f}^{\\left(3\\right)} \\left(x\\right) for the third derivative of f, we want the brackets to appear so\n we type \\{f}^{\\left(3\\right)} \\left(x\\right).
          ", "content": [{"c": "Notice the brackets in \\", "t": "text"}, {"c": "{e}^{2 x}", "t": "equation-inline"}, {"c": "don't appear in the output", "t": "text"}, {"c": "{e}^{2 x}", "t": "equation-inline"}, {"c": ". In the expression", "t": "text"}, {"c": "{f}^{\\left(3\\right)} \\left(x\\right)", "t": "equation-inline"}, {"c": "for the third derivative of", "t": "text"}, {"c": "f", "t": "equation-inline"}, {"c": ", we want the brackets to appear so\n we type \\", "t": "text"}, {"c": "{f}^{\\left(3\\right)} \\left(x\\right)", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n To get an index we use the symbol ^, so for example \\{x}^{2} gives {x}^{2}. An index\n that is anything other than a number or a single letter requires brackets. So to\n get {e}^{2 x} we type \\{e}^{2 x}, because \\{e}^{2} x gives {e}^{2} x. However \\{p}^{32}\n gives {p}^{32} and \\{x}^{\\alpha} gives {x}^{\\alpha}.\n

          ", "content": [{"c": "To get an index we use the symbol ^, so for example \\", "t": "text"}, {"c": "{x}^{2}", "t": "equation-inline"}, {"c": "gives", "t": "text"}, {"c": "{x}^{2}", "t": "equation-inline"}, {"c": ". An index\n that is anything other than a number or a single letter requires brackets. So to\n get", "t": "text"}, {"c": "{e}^{2 x}", "t": "equation-inline"}, {"c": "we type \\", "t": "text"}, {"c": "{e}^{2 x}", "t": "equation-inline"}, {"c": ", because \\", "t": "text"}, {"c": "{e}^{2} x", "t": "equation-inline"}, {"c": "gives", "t": "text"}, {"c": "{e}^{2} x", "t": "equation-inline"}, {"c": ". However \\", "t": "text"}, {"c": "{p}^{32}", "t": "equation-inline"}, {"c": "gives", "t": "text"}, {"c": "{p}^{32}", "t": "equation-inline"}, {"c": "and \\", "t": "text"}, {"c": "{x}^{\\alpha}", "t": "equation-inline"}, {"c": "gives", "t": "text"}, {"c": "{x}^{\\alpha}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n The symbol _ is used for subscripts and they work in the same way as indices. So\n \\{x}_{10} gives {x}_{10} and it is necessary to use brackets in \\{x}_{i j} to get {x}_{i j}.\n

          ", "content": [{"c": "The symbol _ is used for subscripts and they work in the same way as indices. So\n \\", "t": "text"}, {"c": "{x}_{10}", "t": "equation-inline"}, {"c": "gives", "t": "text"}, {"c": "{x}_{10}", "t": "equation-inline"}, {"c": "and it is necessary to use brackets in \\", "t": "text"}, {"c": "{x}_{i j}", "t": "equation-inline"}, {"c": "to get", "t": "text"}, {"c": "{x}_{i j}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Remember you can check your answers by hanging the mouse over the expression.
          ", "content": [{"c": "Remember you can check your answers by hanging the mouse over the expression.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Use the text box to type the expressions below.\n

          ", "content": [{"c": "Use the text box to type the expressions below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "{e}^{x}", "content": {"math_content": "{e}^{x}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "{\\pi}^{2}", "content": {"math_content": "{\\pi}^{2}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "{\\left(x + 1\\right)}^{2}", "content": {"math_content": "{\\left(x + 1\\right)}^{2}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "{x}^{3} + 4 {x}^{2} - 5", "content": {"math_content": "{x}^{3} + 4 {x}^{2} - 5", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "{a}_{n m}", "content": {"math_content": "{a}_{n m}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "{x}_{2}^{3}", "content": {"math_content": "{x}_{2}^{3}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          \n The index and subscript notation is used to put the upper and lower limits on integrals\n and sums, as in the examples below.

          ", "content": [{"c": "The index and subscript notation is used to put the upper and lower limits on integrals\n and sums, as in the examples below.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "{\\sum}_{k = 1}^{n} k = 1 + 2 + \\cdots + n = \\frac{n \\left(n + 1\\right)}{2}", "content": {"math_content": "{\\sum}_{k = 1}^{n} k = 1 + 2 + \\cdots + n = \\frac{n \\left(n + 1\\right)}{2}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          \n `int_0^1 x^2 dx`
          ", "content": [{"c": "`int_0^1 x^2 dx`", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n As you can see above, the symbol \\/ is used for fractions. Unless they are very\n simple expressions, the denominator and numerator must be enclosed in brackets (which\n don't appear in the output). So, for example\n

          ", "content": [{"c": "As you can see above, the symbol \\", "t": "text"}, {"c": "/", "t": "equation-inline"}, {"c": "is used for fractions. Unless they are very\n simple expressions, the denominator and numerator must be enclosed in brackets (which\n don't appear in the output). So, for example", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\frac{1}{x + 1}", "content": {"math_content": "\\frac{1}{x + 1}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          \n is achieved by typing \\\\frac{1}{x + 1}. What do you think would be produced by typing\n \\\\frac{1}{x} + 1?\n

          ", "content": [{"c": "is achieved by typing \\", "t": "text"}, {"c": "\\frac{1}{x + 1}", "t": "equation-inline"}, {"c": ". What do you think would be produced by typing\n \\", "t": "text"}, {"c": "\\frac{1}{x} + 1", "t": "equation-inline"}, {"c": "?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Both \\{x}_{n} / 10 and \\\\frac{{x}_{n}}{10} give the same output, so sometimes it is not\n necessary to use brackets. Try typing the following expressions and experiment with\n using/missing out brackets to produce the same output.\n

          ", "content": [{"c": "Both \\", "t": "text"}, {"c": "{x}_{n} / 10", "t": "equation-inline"}, {"c": "and \\", "t": "text"}, {"c": "\\frac{{x}_{n}}{10}", "t": "equation-inline"}, {"c": "give the same output, so sometimes it is not\n necessary to use brackets. Try typing the following expressions and experiment with\n using/missing out brackets to produce the same output.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\frac{x + 1}{x + 1}", "content": {"math_content": "\\frac{x + 1}{x + 1}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\frac{x + 1}{x} ^ 2", "content": {"math_content": "\\frac{x + 1}{x} ^ 2", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\frac{1}{3245}", "content": {"math_content": "\\frac{1}{3245}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\frac{\\mathrm{dy}}{\\mathrm{dx}}", "content": {"math_content": "\\frac{\\mathrm{dy}}{\\mathrm{dx}}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          \n `\\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\frac{a b}{c d}", "content": {"math_content": "\\frac{a b}{c d}", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          \n Remember that we can get a one line fraction by using //.

          ", "content": [{"c": "Remember that we can get a one line fraction by using //.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n You may wish to use this for fractions that appear as indices, because it is easier\n to read {e}^{x / 2} than {e}^{\\frac{x}{2}}.
          ", "content": [{"c": "You may wish to use this for fractions that appear as indices, because it is easier\n to read", "t": "text"}, {"c": "{e}^{x / 2}", "t": "equation-inline"}, {"c": "than", "t": "text"}, {"c": "{e}^{\\frac{x}{2}}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Finally, use \\\\sqrt{x} for \\sqrt{x} and \\\\sqrt[x]{y} for \\sqrt[x]{y}. Use\n the box below to type the formula for the solutions of the equation a {x}^{2} + b x + c = 0,\n where a \\ne 0.

          ", "content": [{"c": "Finally, use \\", "t": "text"}, {"c": "\\sqrt{x}", "t": "equation-inline"}, {"c": "for", "t": "text"}, {"c": "\\sqrt{x}", "t": "equation-inline"}, {"c": "and \\", "t": "text"}, {"c": "\\sqrt[x]{y}", "t": "equation-inline"}, {"c": "for", "t": "text"}, {"c": "\\sqrt[x]{y}", "t": "equation-inline"}, {"c": ". Use\n the box below to type the formula for the solutions of the equation", "t": "text"}, {"c": "a {x}^{2} + b x + c = 0", "t": "equation-inline"}, {"c": ",\n where", "t": "text"}, {"c": "a \\ne 0", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n [Hint: the formula appears at the top of this page and you may need to use the symbol\n list.]

          ", "content": [{"c": "[Hint: the formula appears at the top of this page and you may need to use the symbol\n list.]", "t": "text"}]}, {"type": "title", "raw_content": "

          \n 4. Spaces, multiple lines and text

          ", "content": {"title_content": "4. Spaces, multiple lines and text", "level": "2"}}, {"type": "paragraph", "raw_content": "
          \n In some browsers, the words quad and qquad will produce longer spaces than a backslash\n and a space.
          ", "content": [{"c": "In some browsers, the words quad and qquad will produce longer spaces than a backslash\n and a space.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n ASCIIMath should generate spaces that are about right in the output, but if you\n want to get an extra space in the mathematics you can use a backslash and a space\n i.e. \\`\\ \\`. Extra spaces are especially useful when you want to put more than one\n expression on a line, as in the example below.\n

          ", "content": [{"c": "ASCIIMath should generate spaces that are about right in the output, but if you\n want to get an extra space in the mathematics you can use a backslash and a space\n i.e. \\`\\ \\`. Extra spaces are especially useful when you want to put more than one\n expression on a line, as in the example below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n We set\n

          \n `a=3, \\ \\ \\ b=-3,\\ \\ ` and `\\ \\ c=2`.\n

          \n Thus `(a+b)(c+b)=0`.
          ", "content": [{"c": "We set`a=3, \\ \\ \\ b=-3,\\ \\ ` and `\\ \\ c=2`. Thus `(a+b)(c+b)=0`.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Notice that the punctuation of the sentence appears in the middle line even though\n it is mainly mathematical formulae. When beginning to type mathematics it might\n feel strange to include punctuation, but it is important to do so.\n

          ", "content": [{"c": "Notice that the punctuation of the sentence appears in the middle line even though\n it is mainly mathematical formulae. When beginning to type mathematics it might\n feel strange to include punctuation, but it is important to do so.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n When answering questions that require several lines of working you may like to follow\n the format of the example below. Because the second, third and fourth lines have been indented\n it makes it clear that all lines are equal to x.\n

          ", "content": [{"c": "When answering questions that require several lines of working you may like to follow\n the format of the example below. Because the second, third and fourth lines have been indented\n it makes it clear that all lines are equal to", "t": "text"}, {"c": "x", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n Using the quadratic formula, the roots of {x}^{2} - 6 x + 4 = 0 are\n \n x = \\frac{- 6 \\pm \\sqrt{{\\left(- 6\\right)}^{2} - 4 \\left(1\\right) \\left(4\\right)}}{2 \\times 1}= \\frac{- 6 \\pm \\sqrt{36 - 16}}{2}= \\frac{- 6 \\pm \\sqrt{20}}{2}= - 0.8 \\mathmr{and} 2.2to 1 decimal place.
          ", "content": [{"c": "Using the quadratic formula, the roots of", "t": "text"}, {"c": "{x}^{2} - 6 x + 4 = 0", "t": "equation-inline"}, {"c": "are", "t": "text"}, {"c": "x = \\frac{- 6 \\pm \\sqrt{{\\left(- 6\\right)}^{2} - 4 \\left(1\\right) \\left(4\\right)}}{2 \\times 1}", "t": "equation-inline"}, {"c": "= \\frac{- 6 \\pm \\sqrt{36 - 16}}{2}", "t": "equation-inline"}, {"c": "= \\frac{- 6 \\pm \\sqrt{20}}{2}", "t": "equation-inline"}, {"c": "= - 0.8 \\mathmr{and} 2.2", "t": "equation-inline"}, {"c": "to 1 decimal place.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n There is a full stop at the end of the last line in this example because\n it is at the end of a sentence, albeit a sentence with a lot of mathematics in it.\n
          ", "content": [{"c": "There is a full stop at the end of the last line in this example because\n it is at the end of a sentence, albeit a sentence with a lot of mathematics in it.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n On the last line in the example above we have used the word \"or\". ASCIIMath recognises the words\n \"or\", \"and\" and \"if\". Any other words need to be enclosed in inverted commas or using \\\\textrm{s o m e w \\mathmr{and} \\mathrm{ds}}.\n Try removing the inverted commas in the example below to see their effect.

          ", "content": [{"c": "On the last line in the example above we have used the word \"or\". ASCIIMath recognises the words\n \"or\", \"and\" and \"if\". Any other words need to be enclosed in inverted commas or using \\", "t": "text"}, {"c": "\\textrm{s o m e w \\mathmr{and} \\mathrm{ds}}", "t": "equation-inline"}, {"c": ".\n Try removing the inverted commas in the example below to see their effect.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \\sin = \\text{opp\"/\"hyp} \\mathmr{and} \\cos = \\frac{\\textrm{a \\mathrm{dj}}}{\\textrm{h y p}}.
          ", "content": [{"c": "\\sin = \\text{opp\"/\"hyp} \\mathmr{and} \\cos = \\frac{\\textrm{a \\mathrm{dj}}}{\\textrm{h y p}}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "title", "raw_content": "

          \n 5. Matrices and advanced alignment

          ", "content": {"title_content": "5. Matrices and advanced alignment", "level": "2"}}, {"type": "paragraph", "raw_content": "

          \n The ASCIIMath notation for matrices is demonstrated below.\n

          ", "content": [{"c": "The ASCIIMath notation for matrices is demonstrated below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n A 3 \\times 3 matrix,\\left(\\begin{matrix}1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 7 & 8 & 9\\end{matrix}\\right), and a 2 \\times 1 matrix, or vector, \\left(\\begin{matrix}1 \\\\ 0\\end{matrix}\\right).\n

          \n The outer brackets determine the delimiters e.g. `|(a,b),(c,d)|=ad-bc`.\n

          \n A general `m xx n` matrix `((a_(11), cdots , a_(1n)),(vdots, ddots, vdots),(a_(m1),\n cdots , a_(mn)))`\n
          ", "content": [{"c": "A", "t": "text"}, {"c": "3 \\times 3", "t": "equation-inline"}, {"c": "matrix,", "t": "text"}, {"c": "\\left(\\begin{matrix}1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 7 & 8 & 9\\end{matrix}\\right)", "t": "equation-inline"}, {"c": ", and a", "t": "text"}, {"c": "2 \\times 1", "t": "equation-inline"}, {"c": "matrix, or vector,", "t": "text"}, {"c": "\\left(\\begin{matrix}1 \\\\ 0\\end{matrix}\\right)", "t": "equation-inline"}, {"c": ". The outer brackets determine the delimiters e.g. `|(a,b),(c,d)|=ad-bc`. A general `m xx n` matrix `((a_(11), cdots , a_(1n)),(vdots, ddots, vdots),(a_(m1),\n cdots , a_(mn)))`", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Alter the text in the box above so that it starts with\n

          ", "content": [{"c": "Alter the text in the box above so that it starts with", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n `\\ \\ \\ \\ \\ `A 3 \\times 2 matrix ,\\left(\\begin{matrix}1 & 2 \\\\ 3 & 4 \\\\ 5 & 6\\end{matrix}\\right),...\n

          ", "content": [{"c": "`\\ \\ \\ \\ \\ `A", "t": "text"}, {"c": "3 \\times 2", "t": "equation-inline"}, {"c": "matrix ,", "t": "text"}, {"c": "\\left(\\begin{matrix}1 & 2 \\\\ 3 & 4 \\\\ 5 & 6\\end{matrix}\\right)", "t": "equation-inline"}, {"c": ",...", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n In the general m \\times n matrix, we used \\\\cdots for centred dots. This can be\n useful when writing infinite series, for example\n

          ", "content": [{"c": "In the general", "t": "text"}, {"c": "m \\times n", "t": "equation-inline"}, {"c": "matrix, we used \\", "t": "text"}, {"c": "\\cdots", "t": "equation-inline"}, {"c": "for centred dots. This can be\n useful when writing infinite series, for example", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n `e^x = 1+x+1/(2!)x^2 + 1/(3!)x^3 + cdots `.
          ", "content": [{"c": "`e^x = 1+x+1/(2!)x^2 + 1/(3!)x^3 + cdots `.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n Similarly, for low dots, as in 1 , \\ldots , n use \\\\ldots or \\\\ldots.\n

          ", "content": [{"c": "Similarly, for low dots, as in", "t": "text"}, {"c": "1 , \\ldots , n", "t": "equation-inline"}, {"c": "use \\", "t": "text"}, {"c": "\\ldots", "t": "equation-inline"}, {"c": "or \\", "t": "text"}, {"c": "\\ldots", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          \n The matrix notation can be used for aligning material. When doing this, we will\n want to draw matrices without delimiters. For this purpose, we can type \\`{:\\` and\n \\`:}\\` to get invisible brackets. This is necessary when defining a function by\n cases.\n

          ", "content": [{"c": "The matrix notation can be used for aligning material. When doing this, we will\n want to draw matrices without delimiters. For this purpose, we can type \\`{:\\` and\n \\`:}\\` to get invisible brackets. This is necessary when defining a function by\n cases.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "| x | = \\left\\{\\left(x , \\mathmr{if} x \\ge 0 \\textrm{,}\\right) , \\left(- x , \\mathmr{if} x < 0.\\right)\\right.", "content": {"math_content": "| x | = \\left\\{\\left(x , \\mathmr{if} x \\ge 0 \\textrm{,}\\right) , \\left(- x , \\mathmr{if} x < 0.\\right)\\right.", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          \n Notice the use of \\\\textrm{} round\n the comma after x \\ge 0 to indicate that it really is a comma and not part of the\n matrix syntax. Try to write the matrix below and then try to write the function\n definition using a matrix with two invisible brackets.\n

          ", "content": [{"c": "Notice the use of \\", "t": "text"}, {"c": "\\textrm{}", "t": "equation-inline"}, {"c": "round\n the comma after", "t": "text"}, {"c": "x \\ge 0", "t": "equation-inline"}, {"c": "to indicate that it really is a comma and not part of the\n matrix syntax. Try to write the matrix below and then try to write the function\n definition using a matrix with two invisible brackets.", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n You can look up symbols in the symbol list.
          ", "content": [{"c": "You can look up symbols in the symbol list.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\left(\\begin{matrix}1 & 2 & 3 & 4 \\\\ a & b & c & d\\end{matrix}\\right)", "content": {"math_content": "\\left(\\begin{matrix}1 & 2 & 3 & 4 \\\\ a & b & c & d\\end{matrix}\\right)", "math_type": "asciimath", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          `\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ `

          ", "content": [{"c": "`\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ `", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\left.\\left(f : , \\left[0 , 6\\right] , \\to , \\mathbb{R}\\right) , \\left(, x , \\mapsto , {x}^{2} + 1\\right)\\right.", "content": {"math_content": "\\left.\\left(f : , \\left[0 , 6\\right] , \\to , \\mathbb{R}\\right) , \\left(, x , \\mapsto , {x}^{2} + 1\\right)\\right.", "math_type": "asciimath", "by": "mathjax"}}]], "main_html": "

          \n 1. Words and Equations

          \n ASCIIMath works by transforming the text you type into mathematics. To see how it\n works, type this text

          \n Note:\n
          \n The \\` key is normally next to the 1 key on the keyboard.
          \n    The area of a triangle with base `b`\n    and height `h` is `1/2 bh`. The area\n    of a circle with radius `r` is `pi r^2`.
          \n

          \n into this text box:

          \n It is also possible to put the mathematics between dollar signs like this: \\$E=mc^2\\$.\n There is no difference in the output.

          \n As you type, the output appears in the grey coloured area next to the text entry\n box. We want to type mathematics and text, so to indicate mathematics we enclose\n it between left-quotes, like this: \\E = m {c}^{2}. Because you type the mathematics\n directly, it is quick and easy to include a lot of mathematics in your text.\n

          \n It is easier to read longer formulae if you put them on a separate line, as you\n can see in the text box below (and the output to the right). Take a look at any\n standard mathematics book to see the use of mathematics in the text and on separate\n lines.\n

          \n Einstein's famous formula is\n

          \n `E=mc^2`,\n

          \n where `E` is energy, `m` is mass and `c` is the speed of light.\n

          \n Try removing the blank lines above and below the equation in the text box above.\n After doing this, the equation will no longer appear on a separate line in the output.\n This is intentional, as sometimes you will want to type an equation on a separate\n line (for easy editing) without that occurring in the output.\n

          \n 2. Symbols

          \n If you need to use a mathematical symbol, look it up in the \n symbol list. ASCIIMath has standard functions predefined, so to get `sin\n x` just type \\`sin x\\` and `ln x` type \\`ln x\\`.

          \n Remember left-quotes (i.e. \\`) are used to indicate mathematics.

          \n Greek letters are produced by typing the name of the letter, e.g. \\\\alpha gives\n \\alpha and \\\\beta gives \\beta. To get capital Greek letters, capitalise the\n first letter of the name, e.g. \\\\Sigma gives \\Sigma, whereas \\\\sigma gives\n \\sigma.

          \n For addition and subtraction, just use the symbols on the keyboard. For division,\n type \\3 \\div 4 to get 3 \\div 4 or \\3 / 4 to get 3 / 4. There are several symbols\n for multiplication and these are shown in the box below.

          \n Typing 2 \\times 7 = 14 and 2 \\times 7 = 14 gives the same output.\n \n You might also want\n to write this as 2 \\cdot 7 = 14 or even 2 \\ast 7 = 14.

          \n Type the following expressions, referring to the symbol list\n if necessary.\n

          \n `\\ \\ \\ \\ \\ `

          \\int \\cos x \\mathrm{dx}

          `\\ \\ \\ \\ \\ \\ \\ \\ `

          f \\circ g \\left(x\\right)

          `\\ \\ \\ \\ \\ \\ \\ \\ `

          x \\in \\mathbb{R}

          `\\ \\\n \\ \\ \\ \\ \\ \\ `

          \\phi : A \\to B
          \n To check your answers, hang the mouse pointer over one of the expressions above.\n After a couple of seconds a box will appear showing the ASCIIMath used to type the\n expression. You can always use this to see the ASCIIMath.

          \n 3. Indices, subscripts, fractions and brackets

          \n Notice the brackets in \\{e}^{2 x} don't appear in the output {e}^{2 x}. In the expression\n {f}^{\\left(3\\right)} \\left(x\\right) for the third derivative of f, we want the brackets to appear so\n we type \\{f}^{\\left(3\\right)} \\left(x\\right).

          \n To get an index we use the symbol ^, so for example \\{x}^{2} gives {x}^{2}. An index\n that is anything other than a number or a single letter requires brackets. So to\n get {e}^{2 x} we type \\{e}^{2 x}, because \\{e}^{2} x gives {e}^{2} x. However \\{p}^{32}\n gives {p}^{32} and \\{x}^{\\alpha} gives {x}^{\\alpha}.\n

          \n The symbol _ is used for subscripts and they work in the same way as indices. So\n \\{x}_{10} gives {x}_{10} and it is necessary to use brackets in \\{x}_{i j} to get {x}_{i j}.\n

          \n Remember you can check your answers by hanging the mouse over the expression.

          \n Use the text box to type the expressions below.\n

          \n `\\ \\ \\ \\ `

          {e}^{x}

          `\\ \\ \\ \\ `

          {\\pi}^{2}

          `\\ \\ \\ \\ `

          {\\left(x + 1\\right)}^{2}

          `\\ \\ \\ \\ `

          {x}^{3} + 4 {x}^{2} - 5

          `\\ \\ \\ \\ `

          {a}_{n m}

          `\\ \\ \\ \\ `

          {x}_{2}^{3}

          \n The index and subscript notation is used to put the upper and lower limits on integrals\n and sums, as in the examples below.

          {\\sum}_{k = 1}^{n} k = 1 + 2 + \\cdots + n = \\frac{n \\left(n + 1\\right)}{2}

          \n `int_0^1 x^2 dx`

          \n As you can see above, the symbol \\/ is used for fractions. Unless they are very\n simple expressions, the denominator and numerator must be enclosed in brackets (which\n don't appear in the output). So, for example\n

          \n `\\ \\ \\ \\ `

          \\frac{1}{x + 1}

          \n is achieved by typing \\\\frac{1}{x + 1}. What do you think would be produced by typing\n \\\\frac{1}{x} + 1?\n

          \n Both \\{x}_{n} / 10 and \\\\frac{{x}_{n}}{10} give the same output, so sometimes it is not\n necessary to use brackets. Try typing the following expressions and experiment with\n using/missing out brackets to produce the same output.\n

          \n `\\ \\ \\ \\ `

          \\frac{x + 1}{x + 1}

          `\\ \\ \\ \\ `

          \\frac{x + 1}{x} ^ 2

          `\\ \\ \\ \\ `

          \\frac{1}{3245}

          `\\ \\ \\ \\ `

          \\frac{\\mathrm{dy}}{\\mathrm{dx}}

          \n `\\ \\ \\ \\ `

          \\frac{a b}{c d}

          \n Remember that we can get a one line fraction by using //.

          \n You may wish to use this for fractions that appear as indices, because it is easier\n to read {e}^{x / 2} than {e}^{\\frac{x}{2}}.

          \n Finally, use \\\\sqrt{x} for \\sqrt{x} and \\\\sqrt[x]{y} for \\sqrt[x]{y}. Use\n the box below to type the formula for the solutions of the equation a {x}^{2} + b x + c = 0,\n where a \\ne 0.

          \n [Hint: the formula appears at the top of this page and you may need to use the symbol\n list.]

          \n 4. Spaces, multiple lines and text

          \n In some browsers, the words quad and qquad will produce longer spaces than a backslash\n and a space.

          \n ASCIIMath should generate spaces that are about right in the output, but if you\n want to get an extra space in the mathematics you can use a backslash and a space\n i.e. \\`\\ \\`. Extra spaces are especially useful when you want to put more than one\n expression on a line, as in the example below.\n

          \n We set\n

          \n `a=3, \\ \\ \\ b=-3,\\ \\ ` and `\\ \\ c=2`.\n

          \n Thus `(a+b)(c+b)=0`.

          \n Notice that the punctuation of the sentence appears in the middle line even though\n it is mainly mathematical formulae. When beginning to type mathematics it might\n feel strange to include punctuation, but it is important to do so.\n

          \n When answering questions that require several lines of working you may like to follow\n the format of the example below. Because the second, third and fourth lines have been indented\n it makes it clear that all lines are equal to x.\n

          \n Using the quadratic formula, the roots of {x}^{2} - 6 x + 4 = 0 are\n \n x = \\frac{- 6 \\pm \\sqrt{{\\left(- 6\\right)}^{2} - 4 \\left(1\\right) \\left(4\\right)}}{2 \\times 1}= \\frac{- 6 \\pm \\sqrt{36 - 16}}{2}= \\frac{- 6 \\pm \\sqrt{20}}{2}= - 0.8 \\mathmr{and} 2.2to 1 decimal place.
          \n There is a full stop at the end of the last line in this example because\n it is at the end of a sentence, albeit a sentence with a lot of mathematics in it.\n

          \n On the last line in the example above we have used the word \"or\". ASCIIMath recognises the words\n \"or\", \"and\" and \"if\". Any other words need to be enclosed in inverted commas or using \\\\textrm{s o m e w \\mathmr{and} \\mathrm{ds}}.\n Try removing the inverted commas in the example below to see their effect.

          \\sin = \\text{opp\"/\"hyp} \\mathmr{and} \\cos = \\frac{\\textrm{a \\mathrm{dj}}}{\\textrm{h y p}}.

          \n 5. Matrices and advanced alignment

          \n The ASCIIMath notation for matrices is demonstrated below.\n

          \n A 3 \\times 3 matrix,\\left(\\begin{matrix}1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 7 & 8 & 9\\end{matrix}\\right), and a 2 \\times 1 matrix, or vector, \\left(\\begin{matrix}1 \\\\ 0\\end{matrix}\\right).\n

          \n The outer brackets determine the delimiters e.g. `|(a,b),(c,d)|=ad-bc`.\n

          \n A general `m xx n` matrix `((a_(11), cdots , a_(1n)),(vdots, ddots, vdots),(a_(m1),\n cdots , a_(mn)))`\n

          \n Alter the text in the box above so that it starts with\n

          \n `\\ \\ \\ \\ \\ `A 3 \\times 2 matrix ,\\left(\\begin{matrix}1 & 2 \\\\ 3 & 4 \\\\ 5 & 6\\end{matrix}\\right),...\n

          \n In the general m \\times n matrix, we used \\\\cdots for centred dots. This can be\n useful when writing infinite series, for example\n

          \n `e^x = 1+x+1/(2!)x^2 + 1/(3!)x^3 + cdots `.

          \n Similarly, for low dots, as in 1 , \\ldots , n use \\\\ldots or \\\\ldots.\n

          \n The matrix notation can be used for aligning material. When doing this, we will\n want to draw matrices without delimiters. For this purpose, we can type \\`{:\\` and\n \\`:}\\` to get invisible brackets. This is necessary when defining a function by\n cases.\n

          | x | = \\left\\{\\left(x , \\mathmr{if} x \\ge 0 \\textrm{,}\\right) , \\left(- x , \\mathmr{if} x < 0.\\right)\\right.

          \n Notice the use of \\\\textrm{} round\n the comma after x \\ge 0 to indicate that it really is a comma and not part of the\n matrix syntax. Try to write the matrix below and then try to write the function\n definition using a matrix with two invisible brackets.\n

          \n You can look up symbols in the symbol list.
          \\left(\\begin{matrix}1 & 2 & 3 & 4 \\\\ a & b & c & d\\end{matrix}\\right)

          `\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ `

          \\left.\\left(f : , \\left[0 , 6\\right] , \\to , \\mathbb{R}\\right) , \\left(, x , \\mapsto , {x}^{2} + 1\\right)\\right.", "statics": {"title": 5, "paragraph": 65, "paragraph.text": 131, "code": 1, "paragraph.equation-inline": 70, "equation-interline": 20}, "url": "https://www.wjagray.co.uk/maths/ASCIIMathTutorial.html", "content": "## 1. Words and Equations\n\nASCIIMath works by transforming the text you type into mathematics. To see how it\n works, type this text\n\nNote: The \\\\` key is normally next to the 1 key on the keyboard.\n\n```\nThe area of a triangle with base `b`\nand height `h` is `1/2 bh`. The area\nof a circle with radius `r` is `pi r^2`.\n```\n\ninto this text box:\n\nIt is also possible to put the mathematics between dollar signs like this: \\$E=mc^2\\$.\n There is no difference in the output.\n\nAs you type, the output appears in the grey coloured area next to the text entry\n box. We want to type mathematics and text, so to indicate mathematics we enclose\n it between left-quotes, like this: \\ $E = m {c}^{2}$ . Because you type the mathematics\n directly, it is quick and easy to include a lot of mathematics in your text.\n\nIt is easier to read longer formulae if you put them on a separate line, as you\n can see in the text box below (and the output to the right). Take a look at any\n standard mathematics book to see the use of mathematics in the text and on separate\n lines.\n\nEinstein's famous formula is\\`E=mc^2\\`, where \\`E\\` is energy, \\`m\\` is mass and \\`c\\` is the speed of light.\n\nTry removing the blank lines above and below the equation in the text box above.\n After doing this, the equation will no longer appear on a separate line in the output.\n This is intentional, as sometimes you will want to type an equation on a separate\n line (for easy editing) without that occurring in the output.\n\n## 2. Symbols\n\nIf you need to use a mathematical symbol, look it up in the symbol list. ASCIIMath has standard functions predefined, so to get \\`sin\n x\\` just type \\\\`sin x\\\\` and \\`ln x\\` type \\\\`ln x\\\\`.\n\nRemember left-quotes (i.e. \\\\`) are used to indicate mathematics.\n\nGreek letters are produced by typing the name of the letter, e.g. \\ $\\alpha$ gives $\\alpha$ and \\ $\\beta$ gives $\\beta$ . To get capital Greek letters, capitalise the\n first letter of the name, e.g. \\ $\\Sigma$ gives $\\Sigma$ , whereas \\ $\\sigma$ gives $\\sigma$ .\n\nFor addition and subtraction, just use the symbols on the keyboard. For division,\n type \\ $3 \\div 4$ to get $3 \\div 4$ or \\ $3 / 4$ to get $3 / 4$ . There are several symbols\n for multiplication and these are shown in the box below.\n\nTyping $2 \\times 7 = 14$ and $2 \\times 7 = 14$ gives the same output.\n \n You might also want\n to write this as $2 \\cdot 7 = 14$ or even $2 \\ast 7 = 14$ .\n\nType the following expressions, referring to the symbol list if necessary.\n\n\\`\\ \\ \\ \\ \\ \\`\n\n$$\n\\int \\cos x \\mathrm{dx}\n$$\n\n\\`\\ \\ \\ \\ \\ \\ \\ \\ \\`\n\n$$\nf \\circ g \\left(x\\right)\n$$\n\n\\`\\ \\ \\ \\ \\ \\ \\ \\ \\`\n\n$$\nx \\in \\mathbb{R}\n$$\n\n\\`\\ \\\n \\ \\ \\ \\ \\ \\ \\`\n\n$$\n\\phi : A \\to B\n$$\n\nTo check your answers, hang the mouse pointer over one of the expressions above.\n After a couple of seconds a box will appear showing the ASCIIMath used to type the\n expression. You can always use this to see the ASCIIMath.\n\n## 3. Indices, subscripts, fractions and brackets\n\nNotice the brackets in \\ ${e}^{2 x}$ don't appear in the output ${e}^{2 x}$ . In the expression ${f}^{\\left(3\\right)} \\left(x\\right)$ for the third derivative of $f$ , we want the brackets to appear so\n we type \\ ${f}^{\\left(3\\right)} \\left(x\\right)$ .\n\nTo get an index we use the symbol ^, so for example \\ ${x}^{2}$ gives ${x}^{2}$ . An index\n that is anything other than a number or a single letter requires brackets. So to\n get ${e}^{2 x}$ we type \\ ${e}^{2 x}$ , because \\ ${e}^{2} x$ gives ${e}^{2} x$ . However \\ ${p}^{32}$ gives ${p}^{32}$ and \\ ${x}^{\\alpha}$ gives ${x}^{\\alpha}$ .\n\nThe symbol _ is used for subscripts and they work in the same way as indices. So\n \\ ${x}_{10}$ gives ${x}_{10}$ and it is necessary to use brackets in \\ ${x}_{i j}$ to get ${x}_{i j}$ .\n\nRemember you can check your answers by hanging the mouse over the expression.\n\nUse the text box to type the expressions below.\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n{e}^{x}\n$$\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n{\\pi}^{2}\n$$\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n{\\left(x + 1\\right)}^{2}\n$$\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n{x}^{3} + 4 {x}^{2} - 5\n$$\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n{a}_{n m}\n$$\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n{x}_{2}^{3}\n$$\n\nThe index and subscript notation is used to put the upper and lower limits on integrals\n and sums, as in the examples below.\n\n$$\n{\\sum}_{k = 1}^{n} k = 1 + 2 + \\cdots + n = \\frac{n \\left(n + 1\\right)}{2}\n$$\n\n\\`int_0^1 x^2 dx\\`\n\nAs you can see above, the symbol \\ $/$ is used for fractions. Unless they are very\n simple expressions, the denominator and numerator must be enclosed in brackets (which\n don't appear in the output). So, for example\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n\\frac{1}{x + 1}\n$$\n\nis achieved by typing \\ $\\frac{1}{x + 1}$ . What do you think would be produced by typing\n \\ $\\frac{1}{x} + 1$ ?\n\nBoth \\ ${x}_{n} / 10$ and \\ $\\frac{{x}_{n}}{10}$ give the same output, so sometimes it is not\n necessary to use brackets. Try typing the following expressions and experiment with\n using/missing out brackets to produce the same output.\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n\\frac{x + 1}{x + 1}\n$$\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n\\frac{x + 1}{x} ^ 2\n$$\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n\\frac{1}{3245}\n$$\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n\\frac{\\mathrm{dy}}{\\mathrm{dx}}\n$$\n\n\\`\\ \\ \\ \\ \\`\n\n$$\n\\frac{a b}{c d}\n$$\n\nRemember that we can get a one line fraction by using //.\n\nYou may wish to use this for fractions that appear as indices, because it is easier\n to read ${e}^{x / 2}$ than ${e}^{\\frac{x}{2}}$ .\n\nFinally, use \\ $\\sqrt{x}$ for $\\sqrt{x}$ and \\ $\\sqrt[x]{y}$ for $\\sqrt[x]{y}$ . Use\n the box below to type the formula for the solutions of the equation $a {x}^{2} + b x + c = 0$ ,\n where $a \\ne 0$ .\n\n[Hint: the formula appears at the top of this page and you may need to use the symbol\n list.]\n\n## 4. Spaces, multiple lines and text\n\nIn some browsers, the words quad and qquad will produce longer spaces than a backslash\n and a space.\n\nASCIIMath should generate spaces that are about right in the output, but if you\n want to get an extra space in the mathematics you can use a backslash and a space\n i.e. \\\\`\\ \\\\`. Extra spaces are especially useful when you want to put more than one\n expression on a line, as in the example below.\n\nWe set\\`a=3, \\ \\ \\ b=-3,\\ \\ \\` and \\`\\ \\ c=2\\`. Thus \\`(a+b)(c+b)=0\\`.\n\nNotice that the punctuation of the sentence appears in the middle line even though\n it is mainly mathematical formulae. When beginning to type mathematics it might\n feel strange to include punctuation, but it is important to do so.\n\nWhen answering questions that require several lines of working you may like to follow\n the format of the example below. Because the second, third and fourth lines have been indented\n it makes it clear that all lines are equal to $x$ .\n\nUsing the quadratic formula, the roots of ${x}^{2} - 6 x + 4 = 0$ are $x = \\frac{- 6 \\pm \\sqrt{{\\left(- 6\\right)}^{2} - 4 \\left(1\\right) \\left(4\\right)}}{2 \\times 1}$ $= \\frac{- 6 \\pm \\sqrt{36 - 16}}{2}$ $= \\frac{- 6 \\pm \\sqrt{20}}{2}$ $= - 0.8 \\mathmr{and} 2.2$ to 1 decimal place.\n\nThere is a full stop at the end of the last line in this example because\n it is at the end of a sentence, albeit a sentence with a lot of mathematics in it.\n\nOn the last line in the example above we have used the word \"or\". ASCIIMath recognises the words\n \"or\", \"and\" and \"if\". Any other words need to be enclosed in inverted commas or using \\ $\\textrm{s o m e w \\mathmr{and} \\mathrm{ds}}$ .\n Try removing the inverted commas in the example below to see their effect.\n\n$\\sin = \\text{opp\"/\"hyp} \\mathmr{and} \\cos = \\frac{\\textrm{a \\mathrm{dj}}}{\\textrm{h y p}}$ .\n\n## 5. Matrices and advanced alignment\n\nThe ASCIIMath notation for matrices is demonstrated below.\n\nA $3 \\times 3$ matrix, $\\left(\\begin{matrix}1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 7 & 8 & 9\\end{matrix}\\right)$ , and a $2 \\times 1$ matrix, or vector, $\\left(\\begin{matrix}1 \\\\ 0\\end{matrix}\\right)$ . The outer brackets determine the delimiters e.g. \\`|(a,b),(c,d)|=ad-bc\\`. A general \\`m xx n\\` matrix \\`((a_(11), cdots , a_(1n)),(vdots, ddots, vdots),(a_(m1),\n cdots , a_(mn)))\\`\n\nAlter the text in the box above so that it starts with\n\n\\`\\ \\ \\ \\ \\ \\`A $3 \\times 2$ matrix , $\\left(\\begin{matrix}1 & 2 \\\\ 3 & 4 \\\\ 5 & 6\\end{matrix}\\right)$ ,...\n\nIn the general $m \\times n$ matrix, we used \\ $\\cdots$ for centred dots. This can be\n useful when writing infinite series, for example\n\n\\`e^x = 1+x+1/(2!)x^2 + 1/(3!)x^3 + cdots \\`.\n\nSimilarly, for low dots, as in $1 , \\ldots , n$ use \\ $\\ldots$ or \\ $\\ldots$ .\n\nThe matrix notation can be used for aligning material. When doing this, we will\n want to draw matrices without delimiters. For this purpose, we can type \\\\`{:\\\\` and\n \\\\`:}\\\\` to get invisible brackets. This is necessary when defining a function by\n cases.\n\n$$\n| x | = \\left\\{\\left(x , \\mathmr{if} x \\ge 0 \\textrm{,}\\right) , \\left(- x , \\mathmr{if} x < 0.\\right)\\right.\n$$\n\nNotice the use of \\ $\\textrm{}$ round\n the comma after $x \\ge 0$ to indicate that it really is a comma and not part of the\n matrix syntax. Try to write the matrix below and then try to write the function\n definition using a matrix with two invisible brackets.\n\nYou can look up symbols in the symbol list.\n\n$$\n\\left(\\begin{matrix}1 & 2 & 3 & 4 \\\\ a & b & c & d\\end{matrix}\\right)\n$$\n\n\\`\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\`\n\n$$\n\\left.\\left(f : , \\left[0 , 6\\right] , \\to , \\mathbb{R}\\right) , \\left(, x , \\mapsto , {x}^{2} + 1\\right)\\right.\n$$\n", "html": "\n\n\n \n \n ASCIIMath Tutorial\n\n \n\n \n\n \n\n \n\n\n
          \n
          \n
          \n

          \n ASCIIMath Tutorial

          \n

          \n James Gray - The Open University

          \n

          \n This tutorial and accompanying symbol list introduce\n the ASCIIMath notation and some points of good practice in writing mathematics.\n To display the mathematics correctly, this page requires Internet Explorer 6 with\n MathPlayer installed,\n or Mozilla/Firefox/Netscape 7+ with MathML fonts installed (e.g. download the MIT MathML font packages).\n If you have these installed you should see a familiar formula here:

          \n

          \n `x=(-b +- sqrt(b^2 - 4ac))/(2a)`

          \n
          \n
          \n
          \n

          \n 1. Words and Equations

          \n
          \n
          \n

          \n ASCIIMath works by transforming the text you type into mathematics. To see how it\n works, type this text

          \n
          \n
          \n Note:\n
          \n The \\` key is normally next to the 1 key on the keyboard.
          \n
          \n 
          \n    The area of a triangle with base `b`\n    and height `h` is `1/2 bh`. The area\n    of a circle with radius `r` is `pi r^2`.
          \n

          \n into this text box:

          \n\t
          \n
          \n
          \n
          \n \n
          \n
          \n It is also possible to put the mathematics between dollar signs like this: \\$E=mc^2\\$.\n There is no difference in the output.
          \n
          \n

          \n As you type, the output appears in the grey coloured area next to the text entry\n box. We want to type mathematics and text, so to indicate mathematics we enclose\n it between left-quotes, like this: \\`E=mc^2\\`. Because you type the mathematics\n directly, it is quick and easy to include a lot of mathematics in your text.\n

          \n

          \n It is easier to read longer formulae if you put them on a separate line, as you\n can see in the text box below (and the output to the right). Take a look at any\n standard mathematics book to see the use of mathematics in the text and on separate\n lines.\n

          \n
          \n
          \n Einstein's famous formula is\n

          \n

          \n `E=mc^2`,\n

          \n

          \n where `E` is energy, `m` is mass and `c` is the speed of light.\n
          \n \n
          \n \n
          \n

          \n Try removing the blank lines above and below the equation in the text box above.\n After doing this, the equation will no longer appear on a separate line in the output.\n This is intentional, as sometimes you will want to type an equation on a separate\n line (for easy editing) without that occurring in the output.\n

          \n
          \n

          \n 2. Symbols

          \n
          \n

          \n If you need to use a mathematical symbol, look it up in the \n symbol list. ASCIIMath has standard functions predefined, so to get `sin\n x` just type \\`sin x\\` and `ln x` type \\`ln x\\`.

          \n
          \n
          \n Remember left-quotes (i.e. \\`) are used to indicate mathematics.
          \n
          \n

          \n Greek letters are produced by typing the name of the letter, e.g. \\`alpha\\` gives\n `alpha` and \\`beta\\` gives `beta`. To get capital Greek letters, capitalise the\n first letter of the name, e.g. \\`Sigma\\` gives `Sigma`, whereas \\`sigma\\` gives\n `sigma`.

          \n

          \n For addition and subtraction, just use the symbols on the keyboard. For division,\n type \\`3 -: 4\\` to get `3-: 4` or \\`3//4\\` to get `3//4`. There are several symbols\n for multiplication and these are shown in the box below.

          \n
          \n
          \n Typing `2xx7 = 14` and `2 times 7 =14` gives the same output.\n

          \n You might also want\n to write this as `2*7=14` or even `2**7=14`.
          \n \n
          \n
          \n

          \n Type the following expressions, referring to the symbol list\n if necessary.\n

          \n

          \n `\\ \\ \\ \\ \\ ` `int cos x dx` `\\ \\ \\ \\ \\ \\ \\ \\ ` `f @ g (x)` `\\ \\ \\ \\ \\ \\ \\ \\ ` `x in RR` `\\ \\\n \\ \\ \\ \\ \\ \\ ` `phi: A -> B`\n

          \n
          \n
          \n
          \n \n
          \n
          \n To check your answers, hang the mouse pointer over one of the expressions above.\n After a couple of seconds a box will appear showing the ASCIIMath used to type the\n expression. You can always use this to see the ASCIIMath.
          \n

          \n 3. Indices, subscripts, fractions and brackets

          \n
          \n Notice the brackets in \\`e^(2x)\\` don't appear in the output `e^(2x)`. In the expression\n `f^((3)) (x)` for the third derivative of `f`, we want the brackets to appear so\n we type \\`f^((3))(x)\\`.
          \n
          \n

          \n To get an index we use the symbol ^, so for example \\`x^2\\` gives `x^2`. An index\n that is anything other than a number or a single letter requires brackets. So to\n get `e^(2x)` we type \\`e^(2x)\\`, because \\`e^2x\\` gives `e^2x`. However \\`p^32\\`\n gives `p^32` and \\`x^alpha\\` gives `x^alpha`.\n

          \n

          \n The symbol _ is used for subscripts and they work in the same way as indices. So\n \\`x_10\\` gives `x_10` and it is necessary to use brackets in \\`x_(ij)\\` to get `x_(ij)`.\n

          \n
          \n\t
          \n
          \n Remember you can check your answers by hanging the mouse over the expression.
          \n
          \n

          \n Use the text box to type the expressions below.\n

          \n

          \n `\\ \\ \\ \\ ` `e^x` `\\ \\ \\ \\ ` `pi^2` `\\ \\ \\ \\ ` `(x + 1)^2` `\\ \\ \\ \\ ` `x^3 + 4x^2\n - 5` `\\ \\ \\ \\ ` `a_(nm)` `\\ \\ \\ \\ ` `x_2^3`\n

          \n
          \n\t
          \n
          \n
          \n \n
          \n

          \n The index and subscript notation is used to put the upper and lower limits on integrals\n and sums, as in the examples below.

          \n
          \n
          \n `sum_(k=1)^n k = 1+2+ cdots +n=(n(n+1))/2`\n

          \n

          \n `int_0^1 x^2 dx`
          \n \n
          \n
          \n

          \n As you can see above, the symbol \\`/\\` is used for fractions. Unless they are very\n simple expressions, the denominator and numerator must be enclosed in brackets (which\n don't appear in the output). So, for example\n

          \n

          \n `\\ \\ \\ \\ ` `1/(x+1)`\n

          \n

          \n is achieved by typing \\`1/(x+1)\\`. What do you think would be produced by typing\n \\`1/x+1\\`?\n

          \n

          \n Both \\`x_n/10\\` and \\`(x_n)/(10)\\` give the same output, so sometimes it is not\n necessary to use brackets. Try typing the following expressions and experiment with\n using/missing out brackets to produce the same output.\n

          \n

          \n `\\ \\ \\ \\ ` `(x+1)/(x+1)` `\\ \\ \\ \\ ` `(x+1)/x^2` `\\ \\ \\ \\ ` `1/3245` `\\ \\ \\ \\ ` `dy/dx`\n `\\ \\ \\ \\ ` `(ab)/(cd)`\n

          \n
          \n
          \n
          \n \n
          \n
          \n

          \n Remember that we can get a one line fraction by using //.

          \n
          \n
          \n You may wish to use this for fractions that appear as indices, because it is easier\n to read `e^(x//2)` than `e^(x/2)`.
          \n \n
          \n
          \n

          \n Finally, use \\`sqrt(x)\\` for `\\sqrt(x)` and \\`root(x)(y)\\` for `root(x)(y)`. Use\n the box below to type the formula for the solutions of the equation `ax^2+bx+c=0`,\n where `a !=0`.

          \n
          \n
          \n
          \n \n
          \n
          \n

          \n [Hint: the formula appears at the top of this page and you may need to use the symbol\n list.]

          \n
          \n

          \n 4. Spaces, multiple lines and text

          \n
          \n In some browsers, the words quad and qquad will produce longer spaces than a backslash\n and a space.
          \n
          \n

          \n ASCIIMath should generate spaces that are about right in the output, but if you\n want to get an extra space in the mathematics you can use a backslash and a space\n i.e. \\`\\ \\`. Extra spaces are especially useful when you want to put more than one\n expression on a line, as in the example below.\n

          \n
          \n
          \n We set\n

          \n

          \n `a=3, \\ \\ \\ b=-3,\\ \\ ` and `\\ \\ c=2`.\n

          \n

          \n Thus `(a+b)(c+b)=0`.
          \n \n
          \n
          \n

          \n Notice that the punctuation of the sentence appears in the middle line even though\n it is mainly mathematical formulae. When beginning to type mathematics it might\n feel strange to include punctuation, but it is important to do so.\n

          \n

          \n When answering questions that require several lines of working you may like to follow\n the format of the example below. Because the second, third and fourth lines have been indented\n it makes it clear that all lines are equal to `x`.\n

          \n
          \n
          \n Using the quadratic formula, the roots of `x^2-6x+4=0` are\n

          \n `x = (-6 +- sqrt((-6)^2 - 4 (1)(4)))/(2 xx 1)`\n

          \n `\\ \\ = (-6 +- sqrt(36 - 16))/2`\n

          \n `\\ \\ =(-6 +- sqrt(20))/2`\n

          \n `\\ \\ = -0.8 or 2.2 \\ \\ \\ `to 1 decimal place.
          \n \n
          \n
          \n There is a full stop at the end of the last line in this example because\n it is at the end of a sentence, albeit a sentence with a lot of mathematics in it.\n
          \n
          \n

          \n On the last line in the example above we have used the word \"or\". ASCIIMath recognises the words\n \"or\", \"and\" and \"if\". Any other words need to be enclosed in inverted commas or using \\`text(some words)\\`.\n Try removing the inverted commas in the example below to see their effect.

          \n
          \n\t
          `sin = \"opp\"/\"hyp\" \\ \\ \\ and \\ \\ \\ cos = text(adj)/text(hyp)`.
          \n \n\t
          \n

          \n 5. Matrices and advanced alignment

          \n
          \n

          \n The ASCIIMath notation for matrices is demonstrated below.\n

          \n
          \n
          \n A `3xx3` matrix,`((1,2,3),(4,5,6),(7,8,9))`, and a `2xx1` matrix, or vector, `((1),(0))`.\n

          \n

          \n The outer brackets determine the delimiters e.g. `|(a,b),(c,d)|=ad-bc`.\n

          \n

          \n A general `m xx n` matrix `((a_(11), cdots , a_(1n)),(vdots, ddots, vdots),(a_(m1),\n cdots , a_(mn)))`\n
          \n \n
          \n
          \n

          \n Alter the text in the box above so that it starts with\n

          \n

          \n `\\ \\ \\ \\ \\ `A `3 xx 2` matrix ,`((1,2),(3,4),(5,6))`,...\n

          \n

          \n In the general `m xx n` matrix, we used \\`cdots\\` for centred dots. This can be\n useful when writing infinite series, for example\n

          \n
          \n
          \n `e^x = 1+x+1/(2!)x^2 + 1/(3!)x^3 + cdots `.
          \n \n
          \n

          \n Similarly, for low dots, as in `1, ldots, n` use \\`...\\` or \\`ldots\\`.\n

          \n

          \n The matrix notation can be used for aligning material. When doing this, we will\n want to draw matrices without delimiters. For this purpose, we can type \\`{:\\` and\n \\`:}\\` to get invisible brackets. This is necessary when defining a function by\n cases.\n

          \n
          \n
          \n `|x|= {(x , if x ge 0 text(,)),(-x , if x <0.):}`
          \n \n
          \n
          \n

          \n Notice the use of \\`text( )\\` round\n the comma after `x ge 0` to indicate that it really is a comma and not part of the\n matrix syntax. Try to write the matrix below and then try to write the function\n definition using a matrix with two invisible brackets.\n

          \n
          \n
          \n You can look up symbols in the symbol list.
          \n
          \n

          \n `((1,2,3,4),(a,b,c,d))` `\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ` `{:(f:,[0,6],\n ->, RR),( ,x ,|->, x^2+1):}`\n

          \n
          \n
          \n
          \n
          \n \n
          \n
          \n

          \n Acknowledgements

          \n

          \n The ASCIIMath syntax and javascripts were developed by Peter Jipsen (Chapman University).\n Further information can be found at the \n ASCIIMath Homepage.\n

          \n

          \n The javascripts for this page were adapted from Peter Jipsen's \n ASCIIMath email editor by Keith Brunton at Edinburgh University.\n

          \n
          \n
          \n
          \n\n\n"} diff --git a/bench/data/groundtruth/math_mathjax_latex_1.jsonl b/bench/data/groundtruth/math_mathjax_latex_1.jsonl index 3471ae7c..b93fdb78 100644 --- a/bench/data/groundtruth/math_mathjax_latex_1.jsonl +++ b/bench/data/groundtruth/math_mathjax_latex_1.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "title", "raw_content": "

          MathJax v3 beta: TeX input, HTML output test

          ", "content": {"title_content": "MathJax v3 beta: TeX input, HTML output test", "level": "1"}}, {"type": "paragraph", "raw_content": "

          \n When $a \\ne 0$, there are two solutions to \\(ax^2 + bx + c = 0\\) and they are\n $$x = {-b \\pm \\sqrt{b^2-4ac} \\over 2a}.$$\n

          ", "content": [{"c": "When $a \\ne 0$, there are two solutions to \\(ax^2 + bx + c = 0\\) and they are\n $$x = {-b \\pm \\sqrt{b^2-4ac} \\over 2a}.$$", "t": "text"}]}, {"type": "title", "raw_content": "

          The Lorenz Equations

          ", "content": {"title_content": "The Lorenz Equations", "level": "2"}}, {"type": "equation-interline", "raw_content": "

          \n \\begin{align}\n \\dot{x} & = \\sigma(y-x) \\\\\n \\dot{y} & = \\rho x - y - xz \\\\\n \\dot{z} & = -\\beta z + xy\n \\end{align}\n

          ", "content": {"math_content": "\\begin{align}\n \\dot{x} & = \\sigma(y-x) \\\\\n \\dot{y} & = \\rho x - y - xz \\\\\n \\dot{z} & = -\\beta z + xy\n \\end{align}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          The Cauchy-Schwarz Inequality

          ", "content": {"title_content": "The Cauchy-Schwarz Inequality", "level": "2"}}, {"type": "equation-interline", "raw_content": "

          \\[\n \\left( \\sum_{k=1}^n a_k b_k \\right)^{\\!\\!2} \\leq\n \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)\n \\]

          ", "content": {"math_content": "\\left( \\sum_{k=1}^n a_k b_k \\right)^{\\!\\!2} \\leq\n \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          A Cross Product Formula

          ", "content": {"title_content": "A Cross Product Formula", "level": "2"}}, {"type": "equation-interline", "raw_content": "

          \\[\n \\mathbf{V}_1 \\times \\mathbf{V}_2 =\n \\begin{vmatrix}\n \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n \\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n \\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0 \\\\\n \\end{vmatrix}\n \\]

          ", "content": {"math_content": "\\mathbf{V}_1 \\times \\mathbf{V}_2 =\n \\begin{vmatrix}\n \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n \\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n \\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0 \\\\\n \\end{vmatrix}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          The probability of getting \\(k\\) heads when flipping \\(n\\) coins is:

          ", "content": {"title_content": "The probability of getting \\(k\\) heads when flipping \\(n\\) coins is:", "level": "2"}}, {"type": "equation-interline", "raw_content": "

          \\[P(E) = {n \\choose k} p^k (1-p)^{ n-k} \\]

          ", "content": {"math_content": "P(E) = {n \\choose k} p^k (1-p)^{ n-k}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          An Identity of Ramanujan

          ", "content": {"title_content": "An Identity of Ramanujan", "level": "2"}}, {"type": "equation-interline", "raw_content": "

          \\[\n \\frac{1}{(\\sqrt{\\phi \\sqrt{5}}-\\phi) e^{\\frac25 \\pi}} =\n 1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n {1+\\frac{e^{-8\\pi}} {1+\\ldots} } } }\n \\]

          ", "content": {"math_content": "\\frac{1}{(\\sqrt{\\phi \\sqrt{5}}-\\phi) e^{\\frac25 \\pi}} =\n 1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n {1+\\frac{e^{-8\\pi}} {1+\\ldots} } } }", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          A Rogers-Ramanujan Identity

          ", "content": {"title_content": "A Rogers-Ramanujan Identity", "level": "2"}}, {"type": "equation-interline", "raw_content": "

          \\[\n 1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n \\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n \\quad\\quad \\text{for $|q| < 1$}.\n \\]

          ", "content": {"math_content": "1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n \\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n \\quad\\quad \\text{for $|q| < 1$}.", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          Maxwell's Equations

          ", "content": {"title_content": "Maxwell's Equations", "level": "2"}}, {"type": "equation-interline", "raw_content": "

          \n \\begin{align}\n \\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n \\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{B}} & = 0\n \\end{align}\n

          ", "content": {"math_content": "\\begin{align}\n \\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n \\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{B}} & = 0\n \\end{align}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          In-line Mathematics

          ", "content": {"title_content": "In-line Mathematics", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression $\\sqrt{3x-1}+(1+x)^2$ is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

          ", "content": [{"c": "Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression $\\sqrt{3x-1}+(1+x)^2$ is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.", "t": "text"}]}]], "main_html": "

          MathJax v3 beta: TeX input, HTML output test

          \n When $a \\ne 0$, there are two solutions to \\(ax^2 + bx + c = 0\\) and they are\n $$x = {-b \\pm \\sqrt{b^2-4ac} \\over 2a}.$$\n

          The Lorenz Equations

          \n \\begin{align}\n \\dot{x} & = \\sigma(y-x) \\\\\n \\dot{y} & = \\rho x - y - xz \\\\\n \\dot{z} & = -\\beta z + xy\n \\end{align}\n

          The Cauchy-Schwarz Inequality

          \\[\n \\left( \\sum_{k=1}^n a_k b_k \\right)^{\\!\\!2} \\leq\n \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)\n \\]

          A Cross Product Formula

          \\[\n \\mathbf{V}_1 \\times \\mathbf{V}_2 =\n \\begin{vmatrix}\n \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n \\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n \\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0 \\\\\n \\end{vmatrix}\n \\]

          The probability of getting \\(k\\) heads when flipping \\(n\\) coins is:

          \\[P(E) = {n \\choose k} p^k (1-p)^{ n-k} \\]

          An Identity of Ramanujan

          \\[\n \\frac{1}{(\\sqrt{\\phi \\sqrt{5}}-\\phi) e^{\\frac25 \\pi}} =\n 1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n {1+\\frac{e^{-8\\pi}} {1+\\ldots} } } }\n \\]

          A Rogers-Ramanujan Identity

          \\[\n 1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n \\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n \\quad\\quad \\text{for $|q| < 1$}.\n \\]

          Maxwell's Equations

          \n \\begin{align}\n \\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n \\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{B}} & = 0\n \\end{align}\n

          In-line Mathematics

          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression $\\sqrt{3x-1}+(1+x)^2$ is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

          ", "statics": {"title": 9, "paragraph": 2, "paragraph.text": 2, "equation-interline": 7}, "url": "https://mathjax.github.io/MathJax-demos-web/tex-chtml.html", "content": "# MathJax v3 beta: TeX input, HTML output test\n\nWhen $a \\ne 0$, there are two solutions to \\(ax^2 + bx + c = 0\\) and they are\n $$x = {-b \\pm \\sqrt{b^2-4ac} \\over 2a}.$$\n\n## The Lorenz Equations\n\n$$\n\\begin{align}\n \\dot{x} & = \\sigma(y-x) \\\\\n \\dot{y} & = \\rho x - y - xz \\\\\n \\dot{z} & = -\\beta z + xy\n \\end{align}\n$$\n\n## The Cauchy-Schwarz Inequality\n\n$$\n\\left( \\sum_{k=1}^n a_k b_k \\right)^{\\!\\!2} \\leq\n \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)\n$$\n\n## A Cross Product Formula\n\n$$\n\\mathbf{V}_1 \\times \\mathbf{V}_2 =\n \\begin{vmatrix}\n \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n \\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n \\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0 \\\\\n \\end{vmatrix}\n$$\n\n## The probability of getting \\(k\\) heads when flipping \\(n\\) coins is:\n\n$$\nP(E) = {n \\choose k} p^k (1-p)^{ n-k}\n$$\n\n## An Identity of Ramanujan\n\n$$\n\\frac{1}{(\\sqrt{\\phi \\sqrt{5}}-\\phi) e^{\\frac25 \\pi}} =\n 1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n {1+\\frac{e^{-8\\pi}} {1+\\ldots} } } }\n$$\n\n## A Rogers-Ramanujan Identity\n\n$$\n1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n \\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n \\quad\\quad \\text{for $|q| < 1$}.\n$$\n\n## Maxwell's Equations\n\n$$\n\\begin{align}\n \\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n \\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{B}} & = 0\n \\end{align}\n$$\n\n## In-line Mathematics\n\nFinally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression $\\sqrt{3x-1}+(1+x)^2$ is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.\n", "html": "\n\n\n\n \n \n \n MathJax v3 with TeX input and HTML output\n \n \n\n\n

          MathJax v3 beta: TeX input, HTML output test

          \n\n

          \n When $a \\ne 0$, there are two solutions to \\(ax^2 + bx + c = 0\\) and they are\n $$x = {-b \\pm \\sqrt{b^2-4ac} \\over 2a}.$$\n

          \n\n

          The Lorenz Equations

          \n\n

          \n \\begin{align}\n \\dot{x} & = \\sigma(y-x) \\\\\n \\dot{y} & = \\rho x - y - xz \\\\\n \\dot{z} & = -\\beta z + xy\n \\end{align}\n

          \n\n

          The Cauchy-Schwarz Inequality

          \n\n

          \\[\n \\left( \\sum_{k=1}^n a_k b_k \\right)^{\\!\\!2} \\leq\n \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)\n \\]

          \n\n

          A Cross Product Formula

          \n\n

          \\[\n \\mathbf{V}_1 \\times \\mathbf{V}_2 =\n \\begin{vmatrix}\n \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n \\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n \\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0 \\\\\n \\end{vmatrix}\n \\]

          \n\n

          The probability of getting \\(k\\) heads when flipping \\(n\\) coins is:

          \n\n

          \\[P(E) = {n \\choose k} p^k (1-p)^{ n-k} \\]

          \n\n

          An Identity of Ramanujan

          \n\n

          \\[\n \\frac{1}{(\\sqrt{\\phi \\sqrt{5}}-\\phi) e^{\\frac25 \\pi}} =\n 1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n {1+\\frac{e^{-8\\pi}} {1+\\ldots} } } }\n \\]

          \n\n

          A Rogers-Ramanujan Identity

          \n\n

          \\[\n 1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n \\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n \\quad\\quad \\text{for $|q| < 1$}.\n \\]

          \n\n

          Maxwell's Equations

          \n\n

          \n \\begin{align}\n \\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n \\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{B}} & = 0\n \\end{align}\n

          \n\n

          In-line Mathematics

          \n\n

          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression $\\sqrt{3x-1}+(1+x)^2$ is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

          \n\n\n\n"} +{"content_list": [[{"type": "title", "raw_content": "

          MathJax v3 beta: TeX input, HTML output test

          ", "content": {"title_content": "MathJax v3 beta: TeX input, HTML output test", "level": "1"}}, {"type": "paragraph", "raw_content": "

          \n When a \\ne 0, there are two solutions to ax^2 + bx + c = 0 and they are\n

          ", "content": [{"c": "When", "t": "text"}, {"c": "a \\ne 0", "t": "equation-inline"}, {"c": ", there are two solutions to", "t": "text"}, {"c": "ax^2 + bx + c = 0", "t": "equation-inline"}, {"c": "and they are", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x = {-b \\pm \\sqrt{b^2-4ac} \\over 2a}.", "content": {"math_content": "x = {-b \\pm \\sqrt{b^2-4ac} \\over 2a}.", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          The Lorenz Equations

          ", "content": {"title_content": "The Lorenz Equations", "level": "2"}}, {"type": "equation-interline", "raw_content": "\\begin{align}\n \\dot{x} & = \\sigma(y-x) \\\\\n \\dot{y} & = \\rho x - y - xz \\\\\n \\dot{z} & = -\\beta z + xy\n \\end{align}", "content": {"math_content": "\\begin{align}\n \\dot{x} & = \\sigma(y-x) \\\\\n \\dot{y} & = \\rho x - y - xz \\\\\n \\dot{z} & = -\\beta z + xy\n \\end{align}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          The Cauchy-Schwarz Inequality

          ", "content": {"title_content": "The Cauchy-Schwarz Inequality", "level": "2"}}, {"type": "equation-interline", "raw_content": "\n \\left( \\sum_{k=1}^n a_k b_k \\right)^{\\!\\!2} \\leq\n \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)\n ", "content": {"math_content": "\\left( \\sum_{k=1}^n a_k b_k \\right)^{\\!\\!2} \\leq\n \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          A Cross Product Formula

          ", "content": {"title_content": "A Cross Product Formula", "level": "2"}}, {"type": "equation-interline", "raw_content": "\n \\mathbf{V}_1 \\times \\mathbf{V}_2 =\n \\begin{vmatrix}\n \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n \\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n \\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0 \\\\\n \\end{vmatrix}\n ", "content": {"math_content": "\\mathbf{V}_1 \\times \\mathbf{V}_2 =\n \\begin{vmatrix}\n \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n \\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n \\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0 \\\\\n \\end{vmatrix}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          The probability of getting \\(k\\) heads when flipping \\(n\\) coins is:

          ", "content": {"title_content": "The probability of getting \\(k\\) heads when flipping \\(n\\) coins is:", "level": "2"}}, {"type": "equation-interline", "raw_content": "P(E) = {n \\choose k} p^k (1-p)^{ n-k} ", "content": {"math_content": "P(E) = {n \\choose k} p^k (1-p)^{ n-k}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          An Identity of Ramanujan

          ", "content": {"title_content": "An Identity of Ramanujan", "level": "2"}}, {"type": "equation-interline", "raw_content": "\n \\frac{1}{(\\sqrt{\\phi \\sqrt{5}}-\\phi) e^{\\frac25 \\pi}} =\n 1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n {1+\\frac{e^{-8\\pi}} {1+\\ldots} } } }\n ", "content": {"math_content": "\\frac{1}{(\\sqrt{\\phi \\sqrt{5}}-\\phi) e^{\\frac25 \\pi}} =\n 1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n {1+\\frac{e^{-8\\pi}} {1+\\ldots} } } }", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          A Rogers-Ramanujan Identity

          ", "content": {"title_content": "A Rogers-Ramanujan Identity", "level": "2"}}, {"type": "equation-interline", "raw_content": "\n 1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n \\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n \\quad\\quad \\text{for $|q| < 1$}.\n ", "content": {"math_content": "1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n \\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n \\quad\\quad \\text{for $|q| < 1$}.", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          Maxwell's Equations

          ", "content": {"title_content": "Maxwell's Equations", "level": "2"}}, {"type": "equation-interline", "raw_content": "\\begin{align}\n \\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n \\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{B}} & = 0\n \\end{align}", "content": {"math_content": "\\begin{align}\n \\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n \\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{B}} & = 0\n \\end{align}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "

          In-line Mathematics

          ", "content": {"title_content": "In-line Mathematics", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression \\sqrt{3x-1}+(1+x)^2 is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

          ", "content": [{"c": "Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression", "t": "text"}, {"c": "\\sqrt{3x-1}+(1+x)^2", "t": "equation-inline"}, {"c": "is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.", "t": "text"}]}]], "main_html": "

          MathJax v3 beta: TeX input, HTML output test

          \n When a \\ne 0, there are two solutions to ax^2 + bx + c = 0 and they are\n

          x = {-b \\pm \\sqrt{b^2-4ac} \\over 2a}.

          The Lorenz Equations

          \\begin{align}\n \\dot{x} & = \\sigma(y-x) \\\\\n \\dot{y} & = \\rho x - y - xz \\\\\n \\dot{z} & = -\\beta z + xy\n \\end{align}

          The Cauchy-Schwarz Inequality

          \n \\left( \\sum_{k=1}^n a_k b_k \\right)^{\\!\\!2} \\leq\n \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)\n

          A Cross Product Formula

          \n \\mathbf{V}_1 \\times \\mathbf{V}_2 =\n \\begin{vmatrix}\n \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n \\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n \\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0 \\\\\n \\end{vmatrix}\n

          The probability of getting \\(k\\) heads when flipping \\(n\\) coins is:

          P(E) = {n \\choose k} p^k (1-p)^{ n-k}

          An Identity of Ramanujan

          \n \\frac{1}{(\\sqrt{\\phi \\sqrt{5}}-\\phi) e^{\\frac25 \\pi}} =\n 1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n {1+\\frac{e^{-8\\pi}} {1+\\ldots} } } }\n

          A Rogers-Ramanujan Identity

          \n 1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n \\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n \\quad\\quad \\text{for $|q| < 1$}.\n

          Maxwell's Equations

          \\begin{align}\n \\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n \\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{B}} & = 0\n \\end{align}

          In-line Mathematics

          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression \\sqrt{3x-1}+(1+x)^2 is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

          ", "statics": {"title": 9, "paragraph": 2, "paragraph.text": 5, "paragraph.equation-inline": 3, "equation-interline": 8}, "url": "https://mathjax.github.io/MathJax-demos-web/tex-chtml.html", "content": "# MathJax v3 beta: TeX input, HTML output test\n\nWhen $a \\ne 0$ , there are two solutions to $ax^2 + bx + c = 0$ and they are\n\n$$\nx = {-b \\pm \\sqrt{b^2-4ac} \\over 2a}.\n$$\n\n## The Lorenz Equations\n\n$$\n\\begin{align}\n \\dot{x} & = \\sigma(y-x) \\\\\n \\dot{y} & = \\rho x - y - xz \\\\\n \\dot{z} & = -\\beta z + xy\n \\end{align}\n$$\n\n## The Cauchy-Schwarz Inequality\n\n$$\n\\left( \\sum_{k=1}^n a_k b_k \\right)^{\\!\\!2} \\leq\n \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)\n$$\n\n## A Cross Product Formula\n\n$$\n\\mathbf{V}_1 \\times \\mathbf{V}_2 =\n \\begin{vmatrix}\n \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n \\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n \\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0 \\\\\n \\end{vmatrix}\n$$\n\n## The probability of getting \\(k\\) heads when flipping \\(n\\) coins is:\n\n$$\nP(E) = {n \\choose k} p^k (1-p)^{ n-k}\n$$\n\n## An Identity of Ramanujan\n\n$$\n\\frac{1}{(\\sqrt{\\phi \\sqrt{5}}-\\phi) e^{\\frac25 \\pi}} =\n 1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n {1+\\frac{e^{-8\\pi}} {1+\\ldots} } } }\n$$\n\n## A Rogers-Ramanujan Identity\n\n$$\n1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n \\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n \\quad\\quad \\text{for $|q| < 1$}.\n$$\n\n## Maxwell's Equations\n\n$$\n\\begin{align}\n \\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n \\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{B}} & = 0\n \\end{align}\n$$\n\n## In-line Mathematics\n\nFinally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression $\\sqrt{3x-1}+(1+x)^2$ is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.\n", "html": "\n\n\n\n \n \n \n MathJax v3 with TeX input and HTML output\n \n \n\n\n

          MathJax v3 beta: TeX input, HTML output test

          \n\n

          \n When $a \\ne 0$, there are two solutions to \\(ax^2 + bx + c = 0\\) and they are\n $$x = {-b \\pm \\sqrt{b^2-4ac} \\over 2a}.$$\n

          \n\n

          The Lorenz Equations

          \n\n

          \n \\begin{align}\n \\dot{x} & = \\sigma(y-x) \\\\\n \\dot{y} & = \\rho x - y - xz \\\\\n \\dot{z} & = -\\beta z + xy\n \\end{align}\n

          \n\n

          The Cauchy-Schwarz Inequality

          \n\n

          \\[\n \\left( \\sum_{k=1}^n a_k b_k \\right)^{\\!\\!2} \\leq\n \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)\n \\]

          \n\n

          A Cross Product Formula

          \n\n

          \\[\n \\mathbf{V}_1 \\times \\mathbf{V}_2 =\n \\begin{vmatrix}\n \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n \\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n \\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0 \\\\\n \\end{vmatrix}\n \\]

          \n\n

          The probability of getting \\(k\\) heads when flipping \\(n\\) coins is:

          \n\n

          \\[P(E) = {n \\choose k} p^k (1-p)^{ n-k} \\]

          \n\n

          An Identity of Ramanujan

          \n\n

          \\[\n \\frac{1}{(\\sqrt{\\phi \\sqrt{5}}-\\phi) e^{\\frac25 \\pi}} =\n 1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n {1+\\frac{e^{-8\\pi}} {1+\\ldots} } } }\n \\]

          \n\n

          A Rogers-Ramanujan Identity

          \n\n

          \\[\n 1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n \\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n \\quad\\quad \\text{for $|q| < 1$}.\n \\]

          \n\n

          Maxwell's Equations

          \n\n

          \n \\begin{align}\n \\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n \\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n \\nabla \\cdot \\vec{\\mathbf{B}} & = 0\n \\end{align}\n

          \n\n

          In-line Mathematics

          \n\n

          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression $\\sqrt{3x-1}+(1+x)^2$ is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

          \n\n\n\n"} diff --git a/bench/data/groundtruth/math_mathjax_latex_3.jsonl b/bench/data/groundtruth/math_mathjax_latex_3.jsonl index c8eb9204..cb6376dd 100644 --- a/bench/data/groundtruth/math_mathjax_latex_3.jsonl +++ b/bench/data/groundtruth/math_mathjax_latex_3.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "paragraph", "raw_content": "

          For conductors, we propose that the free current density is proportional to the applied Electric field and the constant of proportionality is defined as conductivity.

          ", "content": [{"c": "For conductors, we propose that the free current density is proportional to the applied Electric field and the constant of proportionality is defined as conductivity.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{equation}\n\\textbf{J}_\\textbf{f} = \\sigma\\,\\textbf{E}\n\\end{equation}", "content": {"math_content": "\\begin{equation}\n\\textbf{J}_\\textbf{f} = \\sigma\\,\\textbf{E}\n\\end{equation}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Maxwell's equation in the conducting material (assuming linear media) take the form,\n

          ", "content": [{"c": "Maxwell's equation in the conducting material (assuming linear media) take the form,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{equation}\n\\vec{\\nabla} \\cdot \\textbf{E} = \\frac{\\rho_{f}}{\\epsilon} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\vec{\\nabla} \\cdot \\textbf{B} = 0\n\\end{equation}", "content": {"math_content": "\\begin{equation}\n\\vec{\\nabla} \\cdot \\textbf{E} = \\frac{\\rho_{f}}{\\epsilon} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\vec{\\nabla} \\cdot \\textbf{B} = 0\n\\end{equation}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "\\begin{equation}\n\\vec{\\nabla} \\times \\textbf{E} = -\\partial_t \\,\\textbf{B} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ \\vec{\\nabla} \\times \\textbf{B} = \\mu\\sigma \\,\\textbf{E} \\,+ \\mu\\epsilon \\, \\partial_t \\,\\textbf{E}\n\\end{equation}", "content": {"math_content": "\\begin{equation}\n\\vec{\\nabla} \\times \\textbf{E} = -\\partial_t \\,\\textbf{B} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ \\vec{\\nabla} \\times \\textbf{B} = \\mu\\sigma \\,\\textbf{E} \\,+ \\mu\\epsilon \\, \\partial_t \\,\\textbf{E}\n\\end{equation}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Taking the divergence of the \\vec{\\nabla} \\times \\textbf{B}equation and substituting divergence of the electric field with charge density gives,

          ", "content": [{"c": "Taking the divergence of the", "t": "text"}, {"c": "\\vec{\\nabla} \\times \\textbf{B}", "t": "equation-inline"}, {"c": "equation and substituting divergence of the electric field with charge density gives,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{equation}\n\\frac{\\partial\\rho_f}{\\partial t} = - \\frac{\\sigma}{\\epsilon}\\rho_f \\implies \\rho_f =\\rho_f(0) \\exp(-\\frac{\\sigma}{\\epsilon} \\;t)\n\\end{equation}", "content": {"math_content": "\\begin{equation}\n\\frac{\\partial\\rho_f}{\\partial t} = - \\frac{\\sigma}{\\epsilon}\\rho_f \\implies \\rho_f =\\rho_f(0) \\exp(-\\frac{\\sigma}{\\epsilon} \\;t)\n\\end{equation}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          I don't understand what this equation is supposed to mean. If I take some conducting wire and connect it to two ends of a battery the wire still retains its free charges. So how can \\rho_fbe an exponentially decreasing function of time?

          ", "content": [{"c": "I don't understand what this equation is supposed to mean. If I take some conducting wire and connect it to two ends of a battery the wire still retains its free charges. So how can", "t": "text"}, {"c": "\\rho_f", "t": "equation-inline"}, {"c": "be an exponentially decreasing function of time?", "t": "text"}]}]], "main_html": "

          For conductors, we propose that the free current density is proportional to the applied Electric field and the constant of proportionality is defined as conductivity.

          \\begin{equation}\n\\textbf{J}_\\textbf{f} = \\sigma\\,\\textbf{E}\n\\end{equation}

          Maxwell's equation in the conducting material (assuming linear media) take the form,\n

          \\begin{equation}\n\\vec{\\nabla} \\cdot \\textbf{E} = \\frac{\\rho_{f}}{\\epsilon} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\vec{\\nabla} \\cdot \\textbf{B} = 0\n\\end{equation}\\begin{equation}\n\\vec{\\nabla} \\times \\textbf{E} = -\\partial_t \\,\\textbf{B} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ \\vec{\\nabla} \\times \\textbf{B} = \\mu\\sigma \\,\\textbf{E} \\,+ \\mu\\epsilon \\, \\partial_t \\,\\textbf{E}\n\\end{equation}

          Taking the divergence of the \\vec{\\nabla} \\times \\textbf{B}equation and substituting divergence of the electric field with charge density gives,

          \\begin{equation}\n\\frac{\\partial\\rho_f}{\\partial t} = - \\frac{\\sigma}{\\epsilon}\\rho_f \\implies \\rho_f =\\rho_f(0) \\exp(-\\frac{\\sigma}{\\epsilon} \\;t)\n\\end{equation}

          I don't understand what this equation is supposed to mean. If I take some conducting wire and connect it to two ends of a battery the wire still retains its free charges. So how can \\rho_fbe an exponentially decreasing function of time?

          ", "statics": {"paragraph": 4, "paragraph.text": 6, "equation-interline": 4, "paragraph.equation-inline": 2}, "url": "https://physics.stackexchange.com/questions/621479/how-to-interpret-charge-continuity-equation-for-conductors-that-obey-ohms-law", "content": "For conductors, we propose that the free current density is proportional to the applied Electric field and the constant of proportionality is defined as conductivity.\n\n$$\n\\begin{equation}\n\\textbf{J}_\\textbf{f} = \\sigma\\,\\textbf{E}\n\\end{equation}\n$$\n\nMaxwell's equation in the conducting material (assuming linear media) take the form,\n\n$$\n\\begin{equation}\n\\vec{\\nabla} \\cdot \\textbf{E} = \\frac{\\rho_{f}}{\\epsilon} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\vec{\\nabla} \\cdot \\textbf{B} = 0\n\\end{equation}\n$$\n\n$$\n\\begin{equation}\n\\vec{\\nabla} \\times \\textbf{E} = -\\partial_t \\,\\textbf{B} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ \\vec{\\nabla} \\times \\textbf{B} = \\mu\\sigma \\,\\textbf{E} \\,+ \\mu\\epsilon \\, \\partial_t \\,\\textbf{E}\n\\end{equation}\n$$\n\nTaking the divergence of the $\\vec{\\nabla} \\times \\textbf{B}$ equation and substituting divergence of the electric field with charge density gives,\n\n$$\n\\begin{equation}\n\\frac{\\partial\\rho_f}{\\partial t} = - \\frac{\\sigma}{\\epsilon}\\rho_f \\implies \\rho_f =\\rho_f(0) \\exp(-\\frac{\\sigma}{\\epsilon} \\;t)\n\\end{equation}\n$$\n\nI don't understand what this equation is supposed to mean. If I take some conducting wire and connect it to two ends of a battery the wire still retains its free charges. So how can $\\rho_f$ be an exponentially decreasing function of time?\n", "html": "\n\n\n\n\n \n\n \n\n electromagnetism - How to interpret charge continuity equation for conductors that obey Ohm's law? - Physics Stack Exchange\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n\n\n\n \n \n\n\n\n\n \n\n\n\n\n\n \n \n\n\n \n \n\n\n\n \n \n \n \n\n\n
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          Stack Exchange Network

          \n\t\t\t\t\t

          \n\t\t\t\t\t\tStack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.\n\t\t\t\t\t

          \n\t\t\t\t\tVisit Stack Exchange\n\t\t\t\t\t\n\t\t\t\t
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          For conductors, we propose that the free current density is proportional to the applied Electric field and the constant of proportionality is defined as conductivity.

          \n

          \\begin{equation}\n\\textbf{J}_\\textbf{f} = \\sigma\\,\\textbf{E}\n\\end{equation}

          \n

          Maxwell's equation in the conducting material (assuming linear media) take the form,\n\\begin{equation}\n\\vec{\\nabla} \\cdot \\textbf{E} = \\frac{\\rho_{f}}{\\epsilon} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\vec{\\nabla} \\cdot \\textbf{B} = 0\n\\end{equation}

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          \\begin{equation}\n\\vec{\\nabla} \\times \\textbf{E} = -\\partial_t \\,\\textbf{B} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ \\vec{\\nabla} \\times \\textbf{B} = \\mu\\sigma \\,\\textbf{E} \\,+ \\mu\\epsilon \\, \\partial_t \\,\\textbf{E}\n\\end{equation}

          \n

          Taking the divergence of the $\\vec{\\nabla} \\times \\textbf{B}$ equation and substituting divergence of the electric field with charge density gives,

          \n

          \\begin{equation}\n\\frac{\\partial\\rho_f}{\\partial t} = - \\frac{\\sigma}{\\epsilon}\\rho_f \\implies \\rho_f =\\rho_f(0) \\exp(-\\frac{\\sigma}{\\epsilon} \\;t)\n\\end{equation}

          \n

          I don't understand what this equation is supposed to mean. If I take some conducting wire and connect it to two ends of a battery the wire still retains its free charges. So how can $\\rho_f$ be an exponentially decreasing function of time?

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          \n asked Mar 16, 2021 at 8:42\n
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          \"a_point_particle's
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          You are considering differential, i.e., local form of the Maxwell equations, and all your quantities are local, i.e., referring to a specific point in space. There is nothing wrong with the charge at a certain point decreasing - it flows away from this point, as the continuity equation tells us:\n$$\\frac{\\partial \\rho}{\\partial t}=-\\nabla\\cdot\\mathbf{J}$$\nLet me also point out that $\\rho_f$ in your equations is not the total charge, but the net charge at this point, i.e., the difference between the amounts of positive and negative charge.

          \n

          Now, if we take a wire or a whole circuit, we are assuming a global picture, which is better described by the integral form of the Maxwell equations. Thus, if we consider a surface enclosing the circuit, the amount of charge inside this surface is not going to change, even though the shape of the distribution of the charge may vary.

          \n

          A more subtle point is that talking about circuits one implies lumped element model, where all the electromagnetic phenomena a reduced to basic parameters, such as resistance, current, voltage bias and capacitor's charge. Note also that simple circuit "battery+wire" does not really sustain charge oscillations.

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          \n answered Mar 16, 2021 at 9:07\n
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          \"Roger
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          \n Roger V.Roger V.\n
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            \n $\\begingroup$\n Thank you, this sort of clears some doubts I had. However, I don't really get how the net charge at a point flows from one point to another when $\\rho_f$ decreases everywhere. Does it get distributed in all directions on average?\n $\\endgroup$\n
            \n– a_point_particle\n
            \n \n Commented\n Mar 16, 2021 at 9:27\n \n
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            \n $\\begingroup$\n @a_point_particle your material equation (Ohms law) is valid only within the wire. The solution says that, if the wire were not electrically neutral, the extra charge will flow away from it.\n $\\endgroup$\n
            \n– Roger V.\n
            \n \n Commented\n Mar 16, 2021 at 10:00\n \n
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          \n\n\n\n\n \n\n\n \n\n\n\n\n \n \n \n \n \n\n \n\n\n\n \n \n"} +{"content_list": [[{"type": "paragraph", "raw_content": "

          For conductors, we propose that the free current density is proportional to the applied Electric field and the constant of proportionality is defined as conductivity.

          ", "content": [{"c": "For conductors, we propose that the free current density is proportional to the applied Electric field and the constant of proportionality is defined as conductivity.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{equation}\n\\textbf{J}_\\textbf{f} = \\sigma\\,\\textbf{E}\n\\end{equation}", "content": {"math_content": "\\begin{equation}\n\\textbf{J}_\\textbf{f} = \\sigma\\,\\textbf{E}\n\\end{equation}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Maxwell's equation in the conducting material (assuming linear media) take the form,\n

          ", "content": [{"c": "Maxwell's equation in the conducting material (assuming linear media) take the form,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{equation}\n\\vec{\\nabla} \\cdot \\textbf{E} = \\frac{\\rho_{f}}{\\epsilon} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\vec{\\nabla} \\cdot \\textbf{B} = 0\n\\end{equation}", "content": {"math_content": "\\begin{equation}\n\\vec{\\nabla} \\cdot \\textbf{E} = \\frac{\\rho_{f}}{\\epsilon} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\vec{\\nabla} \\cdot \\textbf{B} = 0\n\\end{equation}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "\\begin{equation}\n\\vec{\\nabla} \\times \\textbf{E} = -\\partial_t \\,\\textbf{B} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ \\vec{\\nabla} \\times \\textbf{B} = \\mu\\sigma \\,\\textbf{E} \\,+ \\mu\\epsilon \\, \\partial_t \\,\\textbf{E}\n\\end{equation}", "content": {"math_content": "\\begin{equation}\n\\vec{\\nabla} \\times \\textbf{E} = -\\partial_t \\,\\textbf{B} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ \\vec{\\nabla} \\times \\textbf{B} = \\mu\\sigma \\,\\textbf{E} \\,+ \\mu\\epsilon \\, \\partial_t \\,\\textbf{E}\n\\end{equation}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Taking the divergence of the \\vec{\\nabla} \\times \\textbf{B} equation and substituting divergence of the electric field with charge density gives,

          ", "content": [{"c": "Taking the divergence of the", "t": "text"}, {"c": "\\vec{\\nabla} \\times \\textbf{B}", "t": "equation-inline"}, {"c": "equation and substituting divergence of the electric field with charge density gives,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{equation}\n\\frac{\\partial\\rho_f}{\\partial t} = - \\frac{\\sigma}{\\epsilon}\\rho_f \\implies \\rho_f =\\rho_f(0) \\exp(-\\frac{\\sigma}{\\epsilon} \\;t)\n\\end{equation}", "content": {"math_content": "\\begin{equation}\n\\frac{\\partial\\rho_f}{\\partial t} = - \\frac{\\sigma}{\\epsilon}\\rho_f \\implies \\rho_f =\\rho_f(0) \\exp(-\\frac{\\sigma}{\\epsilon} \\;t)\n\\end{equation}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          I don't understand what this equation is supposed to mean. If I take some conducting wire and connect it to two ends of a battery the wire still retains its free charges. So how can \\rho_f be an exponentially decreasing function of time?

          ", "content": [{"c": "I don't understand what this equation is supposed to mean. If I take some conducting wire and connect it to two ends of a battery the wire still retains its free charges. So how can", "t": "text"}, {"c": "\\rho_f", "t": "equation-inline"}, {"c": "be an exponentially decreasing function of time?", "t": "text"}]}, {"type": "list", "raw_content": "
            ", "content": {"items": [], "ordered": false}}, {"type": "list", "raw_content": "
              ", "content": {"items": [], "ordered": false}}, {"type": "title", "raw_content": "

              \n 1 Answer\n \n

              ", "content": {"title_content": "1 Answer", "level": "2"}}, {"type": "paragraph", "raw_content": "
              \n1
              ", "content": [{"c": "1", "t": "text"}]}, {"type": "paragraph", "raw_content": "

              You are considering differential, i.e., local form of the Maxwell equations, and all your quantities are local, i.e., referring to a specific point in space. There is nothing wrong with the charge at a certain point decreasing - it flows away from this point, as the continuity equation tells us:\n

              ", "content": [{"c": "You are considering differential, i.e., local form of the Maxwell equations, and all your quantities are local, i.e., referring to a specific point in space. There is nothing wrong with the charge at a certain point decreasing - it flows away from this point, as the continuity equation tells us:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\frac{\\partial \\rho}{\\partial t}=-\\nabla\\cdot\\mathbf{J}", "content": {"math_content": "\\frac{\\partial \\rho}{\\partial t}=-\\nabla\\cdot\\mathbf{J}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

              \nLet me also point out that \\rho_f in your equations is not the total charge, but the net charge at this point, i.e., the difference between the amounts of positive and negative charge.

              ", "content": [{"c": "Let me also point out that", "t": "text"}, {"c": "\\rho_f", "t": "equation-inline"}, {"c": "in your equations is not the total charge, but the net charge at this point, i.e., the difference between the amounts of positive and negative charge.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

              Now, if we take a wire or a whole circuit, we are assuming a global picture, which is better described by the integral form of the Maxwell equations. Thus, if we consider a surface enclosing the circuit, the amount of charge inside this surface is not going to change, even though the shape of the distribution of the charge may vary.

              ", "content": [{"c": "Now, if we take a wire or a whole circuit, we are assuming a global picture, which is better described by the integral form of the Maxwell equations. Thus, if we consider a surface enclosing the circuit, the amount of charge inside this surface is not going to change, even though the shape of the distribution of the charge may vary.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

              A more subtle point is that talking about circuits one implies lumped element model, where all the electromagnetic phenomena a reduced to basic parameters, such as resistance, current, voltage bias and capacitor's charge. Note also that simple circuit \"battery+wire\" does not really sustain charge oscillations.

              ", "content": [{"c": "A more subtle point is that talking about circuits one implies lumped element model, where all the electromagnetic phenomena a reduced to basic parameters, such as resistance, current, voltage bias and capacitor's charge. Note also that simple circuit \"battery+wire\" does not really sustain charge oscillations.", "t": "text"}]}, {"type": "list", "raw_content": "
              • Thank you, this sort of clears some doubts I had. However, I don't really get how the net charge at a point flows from one point to another when $\\rho_f$ decreases everywhere. Does it get distributed in all directions on average?
                \n\u2013\u00a0a_point_particle
                CommentedMar 16, 2021 at 9:27
              • 1
                @a_point_particle your material equation (Ohms law) is valid only within the wire. The solution says that, if the wire were not electrically neutral, the extra charge will flow away from it.
                \n\u2013\u00a0Roger V.
                CommentedMar 16, 2021 at 10:00
              ", "content": {"items": [[[{"c": "Thank you, this sort of clears some doubts I had. However, I don't really get how the net charge at a point flows from one point to another when $\\rho_f$ decreases everywhere. Does it get distributed in all directions on average?", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "a_point_particle", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 16, 2021 at 9:27", "t": "text"}]], [[{"c": "1", "t": "text"}, {"c": "@a_point_particle your material equation (Ohms law) is valid only within the wire. The solution says that, if the wire were not electrically neutral, the extra charge will flow away from it.", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "Roger V.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 16, 2021 at 10:00", "t": "text"}]]], "ordered": false}}, {"type": "title", "raw_content": "

              \nNot the answer you're looking for? Browse other questions tagged

              ", "content": {"title_content": "Not the answer you're looking for? Browse other questions tagged", "level": "2"}}, {"type": "list", "raw_content": "
                or ", "content": {"items": [], "ordered": false}}, {"type": "title", "raw_content": "

                ", "content": {"title_content": "or ask your own question .", "level": "2"}}]], "main_html": "

                For conductors, we propose that the free current density is proportional to the applied Electric field and the constant of proportionality is defined as conductivity.

                \\begin{equation}\n\\textbf{J}_\\textbf{f} = \\sigma\\,\\textbf{E}\n\\end{equation}

                Maxwell's equation in the conducting material (assuming linear media) take the form,\n

                \\begin{equation}\n\\vec{\\nabla} \\cdot \\textbf{E} = \\frac{\\rho_{f}}{\\epsilon} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\vec{\\nabla} \\cdot \\textbf{B} = 0\n\\end{equation}\\begin{equation}\n\\vec{\\nabla} \\times \\textbf{E} = -\\partial_t \\,\\textbf{B} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ \\vec{\\nabla} \\times \\textbf{B} = \\mu\\sigma \\,\\textbf{E} \\,+ \\mu\\epsilon \\, \\partial_t \\,\\textbf{E}\n\\end{equation}

                Taking the divergence of the \\vec{\\nabla} \\times \\textbf{B} equation and substituting divergence of the electric field with charge density gives,

                \\begin{equation}\n\\frac{\\partial\\rho_f}{\\partial t} = - \\frac{\\sigma}{\\epsilon}\\rho_f \\implies \\rho_f =\\rho_f(0) \\exp(-\\frac{\\sigma}{\\epsilon} \\;t)\n\\end{equation}

                I don't understand what this equation is supposed to mean. If I take some conducting wire and connect it to two ends of a battery the wire still retains its free charges. So how can \\rho_f be an exponentially decreasing function of time?

                    \n 1 Answer\n \n

                    \n1

                    You are considering differential, i.e., local form of the Maxwell equations, and all your quantities are local, i.e., referring to a specific point in space. There is nothing wrong with the charge at a certain point decreasing - it flows away from this point, as the continuity equation tells us:\n

                    \\frac{\\partial \\rho}{\\partial t}=-\\nabla\\cdot\\mathbf{J}

                    \nLet me also point out that \\rho_f in your equations is not the total charge, but the net charge at this point, i.e., the difference between the amounts of positive and negative charge.

                    Now, if we take a wire or a whole circuit, we are assuming a global picture, which is better described by the integral form of the Maxwell equations. Thus, if we consider a surface enclosing the circuit, the amount of charge inside this surface is not going to change, even though the shape of the distribution of the charge may vary.

                    A more subtle point is that talking about circuits one implies lumped element model, where all the electromagnetic phenomena a reduced to basic parameters, such as resistance, current, voltage bias and capacitor's charge. Note also that simple circuit \"battery+wire\" does not really sustain charge oscillations.

                    • Thank you, this sort of clears some doubts I had. However, I don't really get how the net charge at a point flows from one point to another when $\\rho_f$ decreases everywhere. Does it get distributed in all directions on average?
                      \n\u2013\u00a0a_point_particle
                      CommentedMar 16, 2021 at 9:27
                    • 1
                      @a_point_particle your material equation (Ohms law) is valid only within the wire. The solution says that, if the wire were not electrically neutral, the extra charge will flow away from it.
                      \n\u2013\u00a0Roger V.
                      CommentedMar 16, 2021 at 10:00

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                      ", "statics": {"paragraph": 9, "paragraph.text": 12, "equation-interline": 5, "paragraph.equation-inline": 3, "list": 4, "title": 3, "list.text": 11}, "url": "https://physics.stackexchange.com/questions/621479/how-to-interpret-charge-continuity-equation-for-conductors-that-obey-ohms-law", "content": "For conductors, we propose that the free current density is proportional to the applied Electric field and the constant of proportionality is defined as conductivity.\n\n$$\n\\begin{equation}\n\\textbf{J}_\\textbf{f} = \\sigma\\,\\textbf{E}\n\\end{equation}\n$$\n\nMaxwell's equation in the conducting material (assuming linear media) take the form,\n\n$$\n\\begin{equation}\n\\vec{\\nabla} \\cdot \\textbf{E} = \\frac{\\rho_{f}}{\\epsilon} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\vec{\\nabla} \\cdot \\textbf{B} = 0\n\\end{equation}\n$$\n\n$$\n\\begin{equation}\n\\vec{\\nabla} \\times \\textbf{E} = -\\partial_t \\,\\textbf{B} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ \\vec{\\nabla} \\times \\textbf{B} = \\mu\\sigma \\,\\textbf{E} \\,+ \\mu\\epsilon \\, \\partial_t \\,\\textbf{E}\n\\end{equation}\n$$\n\nTaking the divergence of the $\\vec{\\nabla} \\times \\textbf{B}$ equation and substituting divergence of the electric field with charge density gives,\n\n$$\n\\begin{equation}\n\\frac{\\partial\\rho_f}{\\partial t} = - \\frac{\\sigma}{\\epsilon}\\rho_f \\implies \\rho_f =\\rho_f(0) \\exp(-\\frac{\\sigma}{\\epsilon} \\;t)\n\\end{equation}\n$$\n\nI don't understand what this equation is supposed to mean. If I take some conducting wire and connect it to two ends of a battery the wire still retains its free charges. So how can $\\rho_f$ be an exponentially decreasing function of time?\n\n## 1 Answer\n\n1\n\nYou are considering differential, i.e., local form of the Maxwell equations, and all your quantities are local, i.e., referring to a specific point in space. There is nothing wrong with the charge at a certain point decreasing - it flows away from this point, as the continuity equation tells us:\n\n$$\n\\frac{\\partial \\rho}{\\partial t}=-\\nabla\\cdot\\mathbf{J}\n$$\n\nLet me also point out that $\\rho_f$ in your equations is not the total charge, but the net charge at this point, i.e., the difference between the amounts of positive and negative charge.\n\nNow, if we take a wire or a whole circuit, we are assuming a global picture, which is better described by the integral form of the Maxwell equations. Thus, if we consider a surface enclosing the circuit, the amount of charge inside this surface is not going to change, even though the shape of the distribution of the charge may vary.\n\nA more subtle point is that talking about circuits one implies lumped element model, where all the electromagnetic phenomena a reduced to basic parameters, such as resistance, current, voltage bias and capacitor's charge. Note also that simple circuit \"battery+wire\" does not really sustain charge oscillations.\n\n- Thank you, this sort of clears some doubts I had. However, I don't really get how the net charge at a point flows from one point to another when $\\rho_f$ decreases everywhere. Does it get distributed in all directions on average? \u2013 a_point_particle Commented Mar 16, 2021 at 9:27\n- 1 @a_point_particle your material equation (Ohms law) is valid only within the wire. The solution says that, if the wire were not electrically neutral, the extra charge will flow away from it. \u2013 Roger V. Commented Mar 16, 2021 at 10:00\n\n## Not the answer you're looking for? Browse other questions tagged\n\n## or ask your own question .\n", "html": "\n\n\n\n\n \n\n \n\n electromagnetism - How to interpret charge continuity equation for conductors that obey Ohm's law? - Physics Stack Exchange\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n\n\n\n \n \n\n\n\n\n \n\n\n\n\n\n \n \n\n\n \n \n\n\n\n \n \n \n \n\n\n
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                      For conductors, we propose that the free current density is proportional to the applied Electric field and the constant of proportionality is defined as conductivity.

                      \n

                      \\begin{equation}\n\\textbf{J}_\\textbf{f} = \\sigma\\,\\textbf{E}\n\\end{equation}

                      \n

                      Maxwell's equation in the conducting material (assuming linear media) take the form,\n\\begin{equation}\n\\vec{\\nabla} \\cdot \\textbf{E} = \\frac{\\rho_{f}}{\\epsilon} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\vec{\\nabla} \\cdot \\textbf{B} = 0\n\\end{equation}

                      \n

                      \\begin{equation}\n\\vec{\\nabla} \\times \\textbf{E} = -\\partial_t \\,\\textbf{B} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ \\vec{\\nabla} \\times \\textbf{B} = \\mu\\sigma \\,\\textbf{E} \\,+ \\mu\\epsilon \\, \\partial_t \\,\\textbf{E}\n\\end{equation}

                      \n

                      Taking the divergence of the $\\vec{\\nabla} \\times \\textbf{B}$ equation and substituting divergence of the electric field with charge density gives,

                      \n

                      \\begin{equation}\n\\frac{\\partial\\rho_f}{\\partial t} = - \\frac{\\sigma}{\\epsilon}\\rho_f \\implies \\rho_f =\\rho_f(0) \\exp(-\\frac{\\sigma}{\\epsilon} \\;t)\n\\end{equation}

                      \n

                      I don't understand what this equation is supposed to mean. If I take some conducting wire and connect it to two ends of a battery the wire still retains its free charges. So how can $\\rho_f$ be an exponentially decreasing function of time?

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                      \n asked Mar 16, 2021 at 8:42\n
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                      \"a_point_particle's
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                      You are considering differential, i.e., local form of the Maxwell equations, and all your quantities are local, i.e., referring to a specific point in space. There is nothing wrong with the charge at a certain point decreasing - it flows away from this point, as the continuity equation tells us:\n$$\\frac{\\partial \\rho}{\\partial t}=-\\nabla\\cdot\\mathbf{J}$$\nLet me also point out that $\\rho_f$ in your equations is not the total charge, but the net charge at this point, i.e., the difference between the amounts of positive and negative charge.

                      \n

                      Now, if we take a wire or a whole circuit, we are assuming a global picture, which is better described by the integral form of the Maxwell equations. Thus, if we consider a surface enclosing the circuit, the amount of charge inside this surface is not going to change, even though the shape of the distribution of the charge may vary.

                      \n

                      A more subtle point is that talking about circuits one implies lumped element model, where all the electromagnetic phenomena a reduced to basic parameters, such as resistance, current, voltage bias and capacitor's charge. Note also that simple circuit "battery+wire" does not really sustain charge oscillations.

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                      \n answered Mar 16, 2021 at 9:07\n
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                      \"Roger
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                      \n Roger V.Roger V.\n
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                        \n $\\begingroup$\n Thank you, this sort of clears some doubts I had. However, I don't really get how the net charge at a point flows from one point to another when $\\rho_f$ decreases everywhere. Does it get distributed in all directions on average?\n $\\endgroup$\n
                        \n– a_point_particle\n
                        \n \n Commented\n Mar 16, 2021 at 9:27\n \n
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                        \n $\\begingroup$\n @a_point_particle your material equation (Ohms law) is valid only within the wire. The solution says that, if the wire were not electrically neutral, the extra charge will flow away from it.\n $\\endgroup$\n
                        \n– Roger V.\n
                        \n \n Commented\n Mar 16, 2021 at 10:00\n \n
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                      Consider the following system of equations:

                      ", "content": [{"c": "Consider the following system of equations:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      where \\mathbf{x},\\mathbf{y},\\mathbf{z}, and\\mathbf{u}are1\\times nvectors andD_{\\mathbf{x}},D_{\\mathbf{y}}, andD_{\\mathbf{z}}are diagonaln\\times nmatrices with\\mathbf{x},\\mathbf{y}, and\\mathbf{z}, respectively, along their diagonals (i.e.,D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})).

                      ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{z}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "\\mathbf{u}", "t": "equation-inline"}, {"c": "are", "t": "text"}, {"c": "1\\times n", "t": "equation-inline"}, {"c": "vectors and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "D_{\\mathbf{z}}", "t": "equation-inline"}, {"c": "are diagonal", "t": "text"}, {"c": "n\\times n", "t": "equation-inline"}, {"c": "matrices with", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "\\mathbf{z}", "t": "equation-inline"}, {"c": ", respectively, along their diagonals (i.e.,", "t": "text"}, {"c": "D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})", "t": "equation-inline"}, {"c": ").", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                      My question is whether it is possible to solve for \\mathbf{y}here \u2013 both the\\mathbf{y}as a vector and the\\mathbf{y}along the diagonal ofD_{\\mathbf{y}}. The problem is that I do not know of operations to pull\\mathbf{y}out ofD_{\\mathbf{y}}. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussedhere. But I am not sure how this would work in this case, as it would be necessary to distribute \\mathbf{y}out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

                      ", "content": [{"c": "My question is whether it is possible to solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "here \u2013 both the", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as a vector and the", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "along the diagonal of", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". The problem is that I do not know of operations to pull", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "out of", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$", "content": {"math_content": "(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      where \\odotis the Hadamard product ande^T = (1,1,\\ldots)\\in\\mathbb R^n. So, would it be possible to distribute out\\mathbf{y}and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to\\mathbf{y}, notD_{\\mathbf{y}}:

                      ", "content": [{"c": "where", "t": "text"}, {"c": "\\odot", "t": "equation-inline"}, {"c": "is the Hadamard product and", "t": "text"}, {"c": "e^T = (1,1,\\ldots)\\in\\mathbb R^n", "t": "equation-inline"}, {"c": ". So, would it be possible to distribute out", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ", not", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ":", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$", "content": {"math_content": "\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      Edit: Oh, it seems that extracting \\mathbf{y}in this case would be a simple as rewriting the equation above as

                      ", "content": [{"c": "Edit: Oh, it seems that extracting", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in this case would be a simple as rewriting the equation above as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      because rewriting the equation this way would not change the terms along the diagonal of \\mathbf{z}D_{\\mathbf{y}}. Then we can write

                      ", "content": [{"c": "because rewriting the equation this way would not change the terms along the diagonal of", "t": "text"}, {"c": "\\mathbf{z}D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". Then we can write", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$", "content": {"math_content": "\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      But then how would solve for \\mathbf{y}in the following?

                      ", "content": [{"c": "But then how would solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in the following?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      I think that in this case, it would not be possible to solve for a single vector \\mathbf{y}as in the previous case. Instead, we would have a system of polynomials:

                      ", "content": [{"c": "I think that in this case, it would not be possible to solve for a single vector", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as in the previous case. Instead, we would have a system of polynomials:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$", "content": {"math_content": "\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      where \\mathbf{y^2}is a vector where the elements are the squares of the corresponding elements of\\mathbf{y}\u2013 that is,\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}.

                      ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{y^2}", "t": "equation-inline"}, {"c": "is a vector where the elements are the squares of the corresponding elements of", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "\u2013 that is,", "t": "text"}, {"c": "\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                      Next, what if we had an equation as follows?

                      ", "content": [{"c": "Next, what if we had an equation as follows?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      where \\mathbf{M}is ann \\times nmatrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal ofD_{\\mathbf{yM}}, we have some linear combination.

                      ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "is an", "t": "text"}, {"c": "n \\times n", "t": "equation-inline"}, {"c": "matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of", "t": "text"}, {"c": "D_{\\mathbf{yM}}", "t": "equation-inline"}, {"c": ", we have some linear combination.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                      I think in this case, the polynomial system of equations would be rather complicated. We would have the vector \\mathbf{y^2}again, but this time multiplied by some diagonal matrix based on the elements of\\mathbf{M}andD_{\\mathbf{x}}that is multiplied by a scalar (n, I believe). Then, we would have a set of vectors (n-1, I believe) that each have as elements different products of the elements in\\mathbf{y}(e.g.,y_1 y_3) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in\\mathbf{M}andD_{\\mathbf{x}}.

                      ", "content": [{"c": "I think in this case, the polynomial system of equations would be rather complicated. We would have the vector", "t": "text"}, {"c": "\\mathbf{y^2}", "t": "equation-inline"}, {"c": "again, but this time multiplied by some diagonal matrix based on the elements of", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": "that is multiplied by a scalar (", "t": "text"}, {"c": "n", "t": "equation-inline"}, {"c": ", I believe). Then, we would have a set of vectors (", "t": "text"}, {"c": "n-1", "t": "equation-inline"}, {"c": ", I believe) that each have as elements different products of the elements in", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "(e.g.,", "t": "text"}, {"c": "y_1 y_3", "t": "equation-inline"}, {"c": ") and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                      So, I have two questions here:

                      ", "content": [{"c": "So, I have two questions here:", "t": "text"}]}, {"type": "list", "raw_content": "
                      1. Is my general intuition about what this equation would look like correct?
                      2. Are there techniques to solve for the elements of \\mathbf{y}in this system?
                      ", "content": {"items": [[[{"c": "Is my general intuition about what this equation would look like correct?", "t": "text"}]], [[{"c": "Are there techniques to solve for the elements of ", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in this system?", "t": "text"}]]], "ordered": true}}]], "main_html": "

                      Consider the following system of equations:

                      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

                      where \\mathbf{x},\\mathbf{y},\\mathbf{z}, and\\mathbf{u}are1\\times nvectors andD_{\\mathbf{x}},D_{\\mathbf{y}}, andD_{\\mathbf{z}}are diagonaln\\times nmatrices with\\mathbf{x},\\mathbf{y}, and\\mathbf{z}, respectively, along their diagonals (i.e.,D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})).

                      My question is whether it is possible to solve for \\mathbf{y}here \u2013 both the\\mathbf{y}as a vector and the\\mathbf{y}along the diagonal ofD_{\\mathbf{y}}. The problem is that I do not know of operations to pull\\mathbf{y}out ofD_{\\mathbf{y}}. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussedhere. But I am not sure how this would work in this case, as it would be necessary to distribute \\mathbf{y}out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

                      $$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$

                      where \\odotis the Hadamard product ande^T = (1,1,\\ldots)\\in\\mathbb R^n. So, would it be possible to distribute out\\mathbf{y}and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to\\mathbf{y}, notD_{\\mathbf{y}}:

                      $$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$

                      Edit: Oh, it seems that extracting \\mathbf{y}in this case would be a simple as rewriting the equation above as

                      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$

                      because rewriting the equation this way would not change the terms along the diagonal of \\mathbf{z}D_{\\mathbf{y}}. Then we can write

                      $$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$

                      But then how would solve for \\mathbf{y}in the following?

                      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

                      I think that in this case, it would not be possible to solve for a single vector \\mathbf{y}as in the previous case. Instead, we would have a system of polynomials:

                      $$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$

                      where \\mathbf{y^2}is a vector where the elements are the squares of the corresponding elements of\\mathbf{y}\u2013 that is,\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}.

                      Next, what if we had an equation as follows?

                      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$

                      where \\mathbf{M}is ann \\times nmatrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal ofD_{\\mathbf{yM}}, we have some linear combination.

                      I think in this case, the polynomial system of equations would be rather complicated. We would have the vector \\mathbf{y^2}again, but this time multiplied by some diagonal matrix based on the elements of\\mathbf{M}andD_{\\mathbf{x}}that is multiplied by a scalar (n, I believe). Then, we would have a set of vectors (n-1, I believe) that each have as elements different products of the elements in\\mathbf{y}(e.g.,y_1 y_3) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in\\mathbf{M}andD_{\\mathbf{x}}.

                      So, I have two questions here:

                      1. Is my general intuition about what this equation would look like correct?
                      2. Are there techniques to solve for the elements of \\mathbf{y}in this system?
                      ", "statics": {"paragraph": 13, "paragraph.text": 57, "equation-interline": 8, "paragraph.equation-inline": 44, "list": 1, "list.text": 3, "list.equation-inline": 1}, "url": "https://math.stackexchange.com/questions/4082284/solving-for-vector-contained-in-a-diagonal-matrix", "content": "Consider the following system of equations:\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{x}$ , $\\mathbf{y}$ , $\\mathbf{z}$ , and $\\mathbf{u}$ are $1\\times n$ vectors and $D_{\\mathbf{x}}$ , $D_{\\mathbf{y}}$ , and $D_{\\mathbf{z}}$ are diagonal $n\\times n$ matrices with $\\mathbf{x}$ , $\\mathbf{y}$ , and $\\mathbf{z}$ , respectively, along their diagonals (i.e., $D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})$ ).\n\nMy question is whether it is possible to solve for $\\mathbf{y}$ here \u2013 both the $\\mathbf{y}$ as a vector and the $\\mathbf{y}$ along the diagonal of $D_{\\mathbf{y}}$ . The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$ . The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write\n\n$$\n(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}\n$$\n\nwhere $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$ . So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$ , not $D_{\\mathbf{y}}$ :\n\n$$\n\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}\n$$\n\nEdit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nbecause rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$ . Then we can write\n\n$$\n\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}\n$$\n\nBut then how would solve for $\\mathbf{y}$ in the following?\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nI think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:\n\n$$\n\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ \u2013 that is, $\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}$ .\n\nNext, what if we had an equation as follows?\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$ , we have some linear combination.\n\nI think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ( $n$ , I believe). Then, we would have a set of vectors ( $n-1$ , I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$ ) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$ .\n\nSo, I have two questions here:\n\n1. Is my general intuition about what this equation would look like correct?\n2. Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?\n", "html": "\n\n\n\n\n \n\n \n\n linear algebra - Solving for vector contained in a diagonal matrix - Mathematics Stack Exchange\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n\n\n\n \n \n\n\n\n\n \n\n\n \n\n\n \n \n\n\n \n \n\n\n\n \n \n \n \n\n\n
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                      Consider the following system of equations:

                      \n

                      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

                      \n

                      where $\\mathbf{x}$, $\\mathbf{y}$, $\\mathbf{z}$, and $\\mathbf{u}$ are $1\\times n$ vectors and $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$ are diagonal $n\\times n$ matrices with $\\mathbf{x}$, $\\mathbf{y}$, and $\\mathbf{z}$, respectively, along their diagonals (i.e., $D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})$).

                      \n

                      My question is whether it is possible to solve for $\\mathbf{y}$ here \u2013 both the $\\mathbf{y}$ as a vector and the $\\mathbf{y}$ along the diagonal of $D_{\\mathbf{y}}$. The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

                      \n

                      $$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$

                      \n

                      where $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$. So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$, not $D_{\\mathbf{y}}$:

                      \n

                      $$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$

                      \n

                      Edit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as

                      \n

                      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$

                      \n

                      because rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$. Then we can write

                      \n

                      $$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$

                      \n

                      But then how would solve for $\\mathbf{y}$ in the following?

                      \n

                      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

                      \n

                      I think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:

                      \n

                      $$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$

                      \n

                      where $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ \u2013 that is, $\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}$.

                      \n

                      Next, what if we had an equation as follows?

                      \n

                      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$

                      \n

                      where $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$, we have some linear combination.

                      \n

                      I think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ($n$, I believe). Then, we would have a set of vectors ($n-1$, I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$.

                      \n

                      So, I have two questions here:

                      \n
                        \n
                      1. Is my general intuition about what this equation would look like correct?
                        2. \n
                      2. Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?
                        3. \n
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                      \n edited Mar 31, 2021 at 17:37\n
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                      \n Ryan da Silva\n
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                      \n asked Mar 29, 2021 at 23:19\n
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                      \"Ryan
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                      \n Ryan da SilvaRyan da Silva\n
                      \n 6511 silver badge77 bronze badges\n
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                      \n 1 Answer\n 1\n

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                      I am not sure what the notation $D_\\mathbf{x}$, $D_\\mathbf{y}$, and $D_\\mathbf{z}$ represent and if the importance is significant, but assuming $D_{\\mathbf{x}}$ is invertible, you may solve for $\\mathbf{y}$ as follows given the original equation:

                      \n

                      $$ \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}} $$\n$$ \\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} $$\n$$ \\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1} $$

                      \n

                      Note: I can't verify that your edit is correct because I don't understand what $D_{\\mathbf{x} + \\mathbf{z}}$ represents.

                      \n

                      Edit 1:

                      \n

                      Thank you for the response as I didn't realize the meaning of the matrices $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$. Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.

                      \n

                      Since $D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})$ and similarly for the other matrices, we have

                      \n

                      $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

                      \n

                      and multiplying through we have

                      \n

                      $$\n\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}\n$$

                      \n

                      Therefore, this can be written in matrix form as

                      \n

                      $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

                      \n

                      or more concisely as

                      \n

                      $$ \\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}} $$

                      \n

                      which is exactly what is given in your edit:

                      \n

                      $$ \\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}$$

                      \n

                      Edit 2:

                      \n

                      As for the case where you have $\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}$ where $\\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}$, you would not be able to solve for $\\mathbf{y}$ as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):

                      \n

                      $$\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

                      \n

                      where we cannot solve for the matrix containing the $y$ variables because we cannot eliminate $\\mathbf{x}$ from the left side of the equation.

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                      \n edited Mar 31, 2021 at 18:44\n
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                      \n answered Mar 30, 2021 at 8:19\n
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                      \"Ralff's
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                      \n RalffRalff\n
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                        \n $\\begingroup$\n My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$. Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$. Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$. My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$.\n $\\endgroup$\n
                        \n– Ryan da Silva\n
                        \n \n Commented\n Mar 31, 2021 at 5:45\n \n \n \n \n
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                        \n $\\begingroup$\n @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck!\n $\\endgroup$\n
                        \n– Ralff\n
                        \n \n Commented\n Mar 31, 2021 at 6:15\n \n \n \n \n
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                        \n $\\begingroup$\n Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on.\n $\\endgroup$\n
                        \n– Ryan da Silva\n
                        \n \n Commented\n Mar 31, 2021 at 17:38\n \n
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                        \n $\\begingroup$\n @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result.\n $\\endgroup$\n
                        \n– Ralff\n
                        \n \n Commented\n Mar 31, 2021 at 18:47\n \n
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                      \n\n\n\n\n \n\n\n \n\n\n\n\n \n \n \n \n \n\n \n\n\n\n \n \n"} +{"content_list": [[{"type": "paragraph", "raw_content": "

                      Consider the following system of equations:

                      ", "content": [{"c": "Consider the following system of equations:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      where \\mathbf{x}, \\mathbf{y}, \\mathbf{z}, and \\mathbf{u} are 1\\times n vectors and D_{\\mathbf{x}}, D_{\\mathbf{y}}, and D_{\\mathbf{z}} are diagonal n\\times n matrices with \\mathbf{x}, \\mathbf{y}, and \\mathbf{z}, respectively, along their diagonals (i.e., D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})).

                      ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{z}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "\\mathbf{u}", "t": "equation-inline"}, {"c": "are", "t": "text"}, {"c": "1\\times n", "t": "equation-inline"}, {"c": "vectors and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "D_{\\mathbf{z}}", "t": "equation-inline"}, {"c": "are diagonal", "t": "text"}, {"c": "n\\times n", "t": "equation-inline"}, {"c": "matrices with", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "\\mathbf{z}", "t": "equation-inline"}, {"c": ", respectively, along their diagonals (i.e.,", "t": "text"}, {"c": "D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})", "t": "equation-inline"}, {"c": ").", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                      My question is whether it is possible to solve for \\mathbf{y} here \u2013 both the \\mathbf{y} as a vector and the \\mathbf{y} along the diagonal of D_{\\mathbf{y}}. The problem is that I do not know of operations to pull \\mathbf{y} out of D_{\\mathbf{y}}. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute \\mathbf{y} out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

                      ", "content": [{"c": "My question is whether it is possible to solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "here \u2013 both the", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as a vector and the", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "along the diagonal of", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". The problem is that I do not know of operations to pull", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "out of", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write", "t": "text"}]}, {"type": "equation-interline", "raw_content": "(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}", "content": {"math_content": "(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      where \\odot is the Hadamard product and e^T = (1,1,\\ldots)\\in\\mathbb R^n. So, would it be possible to distribute out \\mathbf{y} and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to \\mathbf{y}, not D_{\\mathbf{y}}:

                      ", "content": [{"c": "where", "t": "text"}, {"c": "\\odot", "t": "equation-inline"}, {"c": "is the Hadamard product and", "t": "text"}, {"c": "e^T = (1,1,\\ldots)\\in\\mathbb R^n", "t": "equation-inline"}, {"c": ". So, would it be possible to distribute out", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ", not", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ":", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}", "content": {"math_content": "\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      Edit: Oh, it seems that extracting \\mathbf{y} in this case would be a simple as rewriting the equation above as

                      ", "content": [{"c": "Edit: Oh, it seems that extracting", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in this case would be a simple as rewriting the equation above as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      because rewriting the equation this way would not change the terms along the diagonal of \\mathbf{z}D_{\\mathbf{y}}. Then we can write

                      ", "content": [{"c": "because rewriting the equation this way would not change the terms along the diagonal of", "t": "text"}, {"c": "\\mathbf{z}D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". Then we can write", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}", "content": {"math_content": "\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      But then how would solve for \\mathbf{y} in the following?

                      ", "content": [{"c": "But then how would solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in the following?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      I think that in this case, it would not be possible to solve for a single vector \\mathbf{y} as in the previous case. Instead, we would have a system of polynomials:

                      ", "content": [{"c": "I think that in this case, it would not be possible to solve for a single vector", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as in the previous case. Instead, we would have a system of polynomials:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}", "content": {"math_content": "\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      where \\mathbf{y^2} is a vector where the elements are the squares of the corresponding elements of \\mathbf{y} \u2013 that is, \\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}.

                      ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{y^2}", "t": "equation-inline"}, {"c": "is a vector where the elements are the squares of the corresponding elements of", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "\u2013 that is,", "t": "text"}, {"c": "\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                      Next, what if we had an equation as follows?

                      ", "content": [{"c": "Next, what if we had an equation as follows?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                      where \\mathbf{M} is an n \\times n matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of D_{\\mathbf{yM}}, we have some linear combination.

                      ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "is an", "t": "text"}, {"c": "n \\times n", "t": "equation-inline"}, {"c": "matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of", "t": "text"}, {"c": "D_{\\mathbf{yM}}", "t": "equation-inline"}, {"c": ", we have some linear combination.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                      I think in this case, the polynomial system of equations would be rather complicated. We would have the vector \\mathbf{y^2} again, but this time multiplied by some diagonal matrix based on the elements of \\mathbf{M} and D_{\\mathbf{x}} that is multiplied by a scalar (n, I believe). Then, we would have a set of vectors (n-1, I believe) that each have as elements different products of the elements in \\mathbf{y} (e.g., y_1 y_3) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in \\mathbf{M} and D_{\\mathbf{x}}.

                      ", "content": [{"c": "I think in this case, the polynomial system of equations would be rather complicated. We would have the vector", "t": "text"}, {"c": "\\mathbf{y^2}", "t": "equation-inline"}, {"c": "again, but this time multiplied by some diagonal matrix based on the elements of", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": "that is multiplied by a scalar (", "t": "text"}, {"c": "n", "t": "equation-inline"}, {"c": ", I believe). Then, we would have a set of vectors (", "t": "text"}, {"c": "n-1", "t": "equation-inline"}, {"c": ", I believe) that each have as elements different products of the elements in", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "(e.g.,", "t": "text"}, {"c": "y_1 y_3", "t": "equation-inline"}, {"c": ") and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                      So, I have two questions here:

                      ", "content": [{"c": "So, I have two questions here:", "t": "text"}]}, {"type": "list", "raw_content": "
                      1. Is my general intuition about what this equation would look like correct?
                      2. Are there techniques to solve for the elements of \\mathbf{y} in this system?
                      ", "content": {"items": [[[{"c": "Is my general intuition about what this equation would look like correct?", "t": "text"}]], [[{"c": "Are there techniques to solve for the elements of ", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": " in this system?", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                        ", "content": {"items": [], "ordered": false}}, {"type": "list", "raw_content": "
                          ", "content": {"items": [], "ordered": false}}, {"type": "title", "raw_content": "

                          \n 1 Answer\n \n

                          ", "content": {"title_content": "1 Answer", "level": "2"}}, {"type": "paragraph", "raw_content": "
                          \n1
                          ", "content": [{"c": "1", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                          I am not sure what the notation D_\\mathbf{x}, D_\\mathbf{y}, and D_\\mathbf{z} represent and if the importance is significant, but assuming D_{\\mathbf{x}} is invertible, you may solve for \\mathbf{y} as follows given the original equation:

                          ", "content": [{"c": "I am not sure what the notation", "t": "text"}, {"c": "D_\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "D_\\mathbf{y}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "D_\\mathbf{z}", "t": "equation-inline"}, {"c": "represent and if the importance is significant, but assuming", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": "is invertible, you may solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as follows given the original equation:", "t": "text"}]}, {"type": "equation-interline", "raw_content": " \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}} ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": " \\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": " \\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1} ", "content": {"math_content": "\\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                          Note: I can't verify that your edit is correct because I don't understand what D_{\\mathbf{x} + \\mathbf{z}} represents.

                          ", "content": [{"c": "Note: I can't verify that your edit is correct because I don't understand what", "t": "text"}, {"c": "D_{\\mathbf{x} + \\mathbf{z}}", "t": "equation-inline"}, {"c": "represents.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                          Edit 1:

                          ", "content": [{"c": "Edit 1:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                          Thank you for the response as I didn't realize the meaning of the matrices D_{\\mathbf{x}}, D_{\\mathbf{y}}, and D_{\\mathbf{z}}. Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.

                          ", "content": [{"c": "Thank you for the response as I didn't realize the meaning of the matrices", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "D_{\\mathbf{z}}", "t": "equation-inline"}, {"c": ". Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                          Since D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x}) and similarly for the other matrices, we have

                          ", "content": [{"c": "Since", "t": "text"}, {"c": "D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})", "t": "equation-inline"}, {"c": "and similarly for the other matrices, we have", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n", "content": {"math_content": "\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                          and multiplying through we have

                          ", "content": [{"c": "and multiplying through we have", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\n\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}\n", "content": {"math_content": "\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                          Therefore, this can be written in matrix form as

                          ", "content": [{"c": "Therefore, this can be written in matrix form as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n", "content": {"math_content": "\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                          or more concisely as

                          ", "content": [{"c": "or more concisely as", "t": "text"}]}, {"type": "equation-interline", "raw_content": " \\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}} ", "content": {"math_content": "\\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                          which is exactly what is given in your edit:

                          ", "content": [{"c": "which is exactly what is given in your edit:", "t": "text"}]}, {"type": "equation-interline", "raw_content": " \\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}", "content": {"math_content": "\\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                          Edit 2:

                          ", "content": [{"c": "Edit 2:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                          As for the case where you have \\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}} where \\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}, you would not be able to solve for \\mathbf{y} as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):

                          ", "content": [{"c": "As for the case where you have", "t": "text"}, {"c": "\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}", "t": "equation-inline"}, {"c": "where", "t": "text"}, {"c": "\\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}", "t": "equation-inline"}, {"c": ", you would not be able to solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n", "content": {"math_content": "\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                          where we cannot solve for the matrix containing the y variables because we cannot eliminate \\mathbf{x} from the left side of the equation.

                          ", "content": [{"c": "where we cannot solve for the matrix containing the", "t": "text"}, {"c": "y", "t": "equation-inline"}, {"c": "variables because we cannot eliminate", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": "from the left side of the equation.", "t": "text"}]}, {"type": "list", "raw_content": "
                          • My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$. Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$. Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$. My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$.CommentedMar 31, 2021 at 5:45
                          • 1
                            @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck!
                            \n\u2013\u00a0Ralff
                            CommentedMar 31, 2021 at 6:15
                          • Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on.CommentedMar 31, 2021 at 17:38
                          • @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result.
                            \n\u2013\u00a0Ralff
                            CommentedMar 31, 2021 at 18:47
                          ", "content": {"items": [[[{"c": "My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$. Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$. Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$. My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 31, 2021 at 5:45", "t": "text"}]], [[{"c": "1", "t": "text"}, {"c": "@RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck!", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "Ralff", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 31, 2021 at 6:15", "t": "text"}]], [[{"c": "Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 31, 2021 at 17:38", "t": "text"}]], [[{"c": "@RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result.", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "Ralff", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 31, 2021 at 18:47", "t": "text"}]]], "ordered": false}}, {"type": "title", "raw_content": "

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                            .

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                            Consider the following system of equations:

                            \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}

                            where \\mathbf{x}, \\mathbf{y}, \\mathbf{z}, and \\mathbf{u} are 1\\times n vectors and D_{\\mathbf{x}}, D_{\\mathbf{y}}, and D_{\\mathbf{z}} are diagonal n\\times n matrices with \\mathbf{x}, \\mathbf{y}, and \\mathbf{z}, respectively, along their diagonals (i.e., D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})).

                            My question is whether it is possible to solve for \\mathbf{y} here \u2013 both the \\mathbf{y} as a vector and the \\mathbf{y} along the diagonal of D_{\\mathbf{y}}. The problem is that I do not know of operations to pull \\mathbf{y} out of D_{\\mathbf{y}}. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute \\mathbf{y} out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

                            (\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}

                            where \\odot is the Hadamard product and e^T = (1,1,\\ldots)\\in\\mathbb R^n. So, would it be possible to distribute out \\mathbf{y} and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to \\mathbf{y}, not D_{\\mathbf{y}}:

                            \\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}

                            Edit: Oh, it seems that extracting \\mathbf{y} in this case would be a simple as rewriting the equation above as

                            \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}

                            because rewriting the equation this way would not change the terms along the diagonal of \\mathbf{z}D_{\\mathbf{y}}. Then we can write

                            \\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}

                            But then how would solve for \\mathbf{y} in the following?

                            \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}

                            I think that in this case, it would not be possible to solve for a single vector \\mathbf{y} as in the previous case. Instead, we would have a system of polynomials:

                            \\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}

                            where \\mathbf{y^2} is a vector where the elements are the squares of the corresponding elements of \\mathbf{y} \u2013 that is, \\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}.

                            Next, what if we had an equation as follows?

                            \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}

                            where \\mathbf{M} is an n \\times n matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of D_{\\mathbf{yM}}, we have some linear combination.

                            I think in this case, the polynomial system of equations would be rather complicated. We would have the vector \\mathbf{y^2} again, but this time multiplied by some diagonal matrix based on the elements of \\mathbf{M} and D_{\\mathbf{x}} that is multiplied by a scalar (n, I believe). Then, we would have a set of vectors (n-1, I believe) that each have as elements different products of the elements in \\mathbf{y} (e.g., y_1 y_3) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in \\mathbf{M} and D_{\\mathbf{x}}.

                            So, I have two questions here:

                            1. Is my general intuition about what this equation would look like correct?
                            2. Are there techniques to solve for the elements of \\mathbf{y} in this system?

                                \n 1 Answer\n \n

                                \n1

                                I am not sure what the notation D_\\mathbf{x}, D_\\mathbf{y}, and D_\\mathbf{z} represent and if the importance is significant, but assuming D_{\\mathbf{x}} is invertible, you may solve for \\mathbf{y} as follows given the original equation:

                                \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}} \\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1}

                                Note: I can't verify that your edit is correct because I don't understand what D_{\\mathbf{x} + \\mathbf{z}} represents.

                                Edit 1:

                                Thank you for the response as I didn't realize the meaning of the matrices D_{\\mathbf{x}}, D_{\\mathbf{y}}, and D_{\\mathbf{z}}. Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.

                                Since D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x}) and similarly for the other matrices, we have

                                \n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n

                                and multiplying through we have

                                \n\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}\n

                                Therefore, this can be written in matrix form as

                                \n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n

                                or more concisely as

                                \\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}}

                                which is exactly what is given in your edit:

                                \\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}

                                Edit 2:

                                As for the case where you have \\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}} where \\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}, you would not be able to solve for \\mathbf{y} as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):

                                \n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n

                                where we cannot solve for the matrix containing the y variables because we cannot eliminate \\mathbf{x} from the left side of the equation.

                                • My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$. Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$. Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$. My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$.CommentedMar 31, 2021 at 5:45
                                • 1
                                  @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck!
                                  \n\u2013\u00a0Ralff
                                  CommentedMar 31, 2021 at 6:15
                                • Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on.CommentedMar 31, 2021 at 17:38
                                • @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result.
                                  \n\u2013\u00a0Ralff
                                  CommentedMar 31, 2021 at 18:47

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                                  ", "statics": {"paragraph": 26, "paragraph.text": 85, "equation-interline": 17, "paragraph.equation-inline": 59, "list": 5, "list.text": 20, "list.equation-inline": 1, "title": 4}, "url": "https://math.stackexchange.com/questions/4082284/solving-for-vector-contained-in-a-diagonal-matrix", "content": "Consider the following system of equations:\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{x}$ , $\\mathbf{y}$ , $\\mathbf{z}$ , and $\\mathbf{u}$ are $1\\times n$ vectors and $D_{\\mathbf{x}}$ , $D_{\\mathbf{y}}$ , and $D_{\\mathbf{z}}$ are diagonal $n\\times n$ matrices with $\\mathbf{x}$ , $\\mathbf{y}$ , and $\\mathbf{z}$ , respectively, along their diagonals (i.e., $D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})$ ).\n\nMy question is whether it is possible to solve for $\\mathbf{y}$ here \u2013 both the $\\mathbf{y}$ as a vector and the $\\mathbf{y}$ along the diagonal of $D_{\\mathbf{y}}$ . The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$ . The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write\n\n$$\n(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}\n$$\n\nwhere $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$ . So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$ , not $D_{\\mathbf{y}}$ :\n\n$$\n\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}\n$$\n\nEdit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nbecause rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$ . Then we can write\n\n$$\n\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}\n$$\n\nBut then how would solve for $\\mathbf{y}$ in the following?\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nI think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:\n\n$$\n\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ \u2013 that is, $\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}$ .\n\nNext, what if we had an equation as follows?\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$ , we have some linear combination.\n\nI think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ( $n$ , I believe). Then, we would have a set of vectors ( $n-1$ , I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$ ) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$ .\n\nSo, I have two questions here:\n\n1. Is my general intuition about what this equation would look like correct?\n2. Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?\n\n## 1 Answer\n\n1\n\nI am not sure what the notation $D_\\mathbf{x}$ , $D_\\mathbf{y}$ , and $D_\\mathbf{z}$ represent and if the importance is significant, but assuming $D_{\\mathbf{x}}$ is invertible, you may solve for $\\mathbf{y}$ as follows given the original equation:\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}}\n$$\n\n$$\n\\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1}\n$$\n\nNote: I can't verify that your edit is correct because I don't understand what $D_{\\mathbf{x} + \\mathbf{z}}$ represents.\n\nEdit 1:\n\nThank you for the response as I didn't realize the meaning of the matrices $D_{\\mathbf{x}}$ , $D_{\\mathbf{y}}$ , and $D_{\\mathbf{z}}$ . Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.\n\nSince $D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})$ and similarly for the other matrices, we have\n\n$$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nand multiplying through we have\n\n$$\n\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}\n$$\n\nTherefore, this can be written in matrix form as\n\n$$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nor more concisely as\n\n$$\n\\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}}\n$$\n\nwhich is exactly what is given in your edit:\n\n$$\n\\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}\n$$\n\nEdit 2:\n\nAs for the case where you have $\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}$ where $\\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}$ , you would not be able to solve for $\\mathbf{y}$ as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):\n\n$$\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere we cannot solve for the matrix containing the $y$ variables because we cannot eliminate $\\mathbf{x}$ from the left side of the equation.\n\n- My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$. Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$. Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$. My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$. Commented Mar 31, 2021 at 5:45\n- 1 @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck! \u2013 Ralff Commented Mar 31, 2021 at 6:15\n- Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on. Commented Mar 31, 2021 at 17:38\n- @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result. \u2013 Ralff Commented Mar 31, 2021 at 18:47\n\n## You must log in to answer this question.\n\n## Not the answer you're looking for? Browse other questions tagged\n\n## .\n", "html": "\n\n\n\n\n \n\n \n\n linear algebra - Solving for vector contained in a diagonal matrix - Mathematics Stack Exchange\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n\n\n\n \n \n\n\n\n\n \n\n\n \n\n\n \n \n\n\n \n \n\n\n\n \n \n \n \n\n\n
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                                  Consider the following system of equations:

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                                  $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

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                                  where $\\mathbf{x}$, $\\mathbf{y}$, $\\mathbf{z}$, and $\\mathbf{u}$ are $1\\times n$ vectors and $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$ are diagonal $n\\times n$ matrices with $\\mathbf{x}$, $\\mathbf{y}$, and $\\mathbf{z}$, respectively, along their diagonals (i.e., $D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})$).

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                                  My question is whether it is possible to solve for $\\mathbf{y}$ here \u2013 both the $\\mathbf{y}$ as a vector and the $\\mathbf{y}$ along the diagonal of $D_{\\mathbf{y}}$. The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

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                                  $$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$

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                                  where $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$. So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$, not $D_{\\mathbf{y}}$:

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                                  $$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$

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                                  Edit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as

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                                  $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$

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                                  because rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$. Then we can write

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                                  $$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$

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                                  But then how would solve for $\\mathbf{y}$ in the following?

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                                  $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

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                                  I think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:

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                                  $$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$

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                                  where $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ \u2013 that is, $\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}$.

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                                  Next, what if we had an equation as follows?

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                                  $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$

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                                  where $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$, we have some linear combination.

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                                  I think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ($n$, I believe). Then, we would have a set of vectors ($n-1$, I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$.

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                                  So, I have two questions here:

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                                  1. Is my general intuition about what this equation would look like correct?
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                                  2. Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?
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                                  \n edited Mar 31, 2021 at 17:37\n
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                                  \n Ryan da Silva\n
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                                  \n asked Mar 29, 2021 at 23:19\n
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                                  I am not sure what the notation $D_\\mathbf{x}$, $D_\\mathbf{y}$, and $D_\\mathbf{z}$ represent and if the importance is significant, but assuming $D_{\\mathbf{x}}$ is invertible, you may solve for $\\mathbf{y}$ as follows given the original equation:

                                  \n

                                  $$ \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}} $$\n$$ \\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} $$\n$$ \\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1} $$

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                                  Note: I can't verify that your edit is correct because I don't understand what $D_{\\mathbf{x} + \\mathbf{z}}$ represents.

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                                  Edit 1:

                                  \n

                                  Thank you for the response as I didn't realize the meaning of the matrices $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$. Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.

                                  \n

                                  Since $D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})$ and similarly for the other matrices, we have

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                                  $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

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                                  and multiplying through we have

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                                  $$\n\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}\n$$

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                                  Therefore, this can be written in matrix form as

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                                  $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

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                                  or more concisely as

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                                  $$ \\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}} $$

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                                  which is exactly what is given in your edit:

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                                  $$ \\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}$$

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                                  Edit 2:

                                  \n

                                  As for the case where you have $\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}$ where $\\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}$, you would not be able to solve for $\\mathbf{y}$ as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):

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                                  $$\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

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                                  where we cannot solve for the matrix containing the $y$ variables because we cannot eliminate $\\mathbf{x}$ from the left side of the equation.

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                                  \n edited Mar 31, 2021 at 18:44\n
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                                  \n answered Mar 30, 2021 at 8:19\n
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                                  \"Ralff's
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                                  \n RalffRalff\n
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                                    \n $\\begingroup$\n My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$. Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$. Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$. My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$.\n $\\endgroup$\n
                                    \n– Ryan da Silva\n
                                    \n \n Commented\n Mar 31, 2021 at 5:45\n \n \n \n \n
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                                    \n $\\begingroup$\n @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck!\n $\\endgroup$\n
                                    \n– Ralff\n
                                    \n \n Commented\n Mar 31, 2021 at 6:15\n \n \n \n \n
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                                    \n $\\begingroup$\n Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on.\n $\\endgroup$\n
                                    \n– Ryan da Silva\n
                                    \n \n Commented\n Mar 31, 2021 at 17:38\n \n
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                                    \n $\\begingroup$\n @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result.\n $\\endgroup$\n
                                    \n– Ralff\n
                                    \n \n Commented\n Mar 31, 2021 at 18:47\n \n
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                                  If $f(x,y)$ is a joint pdf,I understand that,

                                  ", "content": [{"c": "If $f(x,y)$ is a joint pdf,I understand that,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                  $$\n \\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty f(x,y) \\ dx \\ dy = 1\n$$

                                  ", "content": {"math_content": "\\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty f(x,y) \\ dx \\ dy = 1", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                  but does this hold for the conditional pdf?

                                  ", "content": [{"c": "but does this hold for the conditional pdf?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                  $$\n \\int_{-\\infty}^\\infty f({y|x}) \\ dy = 1.\n$$

                                  ", "content": {"math_content": "\\int_{-\\infty}^\\infty f({y|x}) \\ dy = 1.", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                  I would argue not. For a given $X=x$ the sum of of the pdf over the support of $Y$ should less than or equal to 1. But according to Bierens it is.

                                  ", "content": [{"c": "I would argue not. For a given $X=x$ the sum of of the pdf over the support of $Y$ should less than or equal to 1. But according to Bierens it is.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                  In particular, in the case (3.4) we have\n \\begin{align}\n\\begin{split}\n E[(Y &- E[Y \\mid X]) I(X \\in B)] \\\\\n &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (y - g(x)) I(x \\in B) f(y, x) \\ dy \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} y f(y \\mid x) \\ dy \\right) I(x \\in B) f_x(x) \\ dx \\\\\n&\\quad - \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} f(y \\mid x) \\ dy \\right) g(x) I (x \\in B) f_x(x) \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} g(x) I(x \\in B) f_x(x) \\ dx - \\int_{-\\infty}^{\\infty} g(x) I (x \\in B) f_x(x) \\ d x = 0.\n\\end{split} \\tag{3.7}\n\\end{align}

                                  ", "content": [{"c": "In particular, in the case (3.4) we have\n \\begin{align}\n\\begin{split}\n E[(Y &- E[Y \\mid X]) I(X \\in B)] \\\\\n &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (y - g(x)) I(x \\in B) f(y, x) \\ dy \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} y f(y \\mid x) \\ dy \\right) I(x \\in B) f_x(x) \\ dx \\\\\n&\\quad - \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} f(y \\mid x) \\ dy \\right) g(x) I (x \\in B) f_x(x) \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} g(x) I(x \\in B) f_x(x) \\ dx - \\int_{-\\infty}^{\\infty} g(x) I (x \\in B) f_x(x) \\ d x = 0.\n\\end{split} \\tag{3.7}\n\\end{align}", "t": "text"}]}]], "main_html": "

                                  If $f(x,y)$ is a joint pdf,I understand that,

                                  $$\n \\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty f(x,y) \\ dx \\ dy = 1\n$$

                                  but does this hold for the conditional pdf?

                                  $$\n \\int_{-\\infty}^\\infty f({y|x}) \\ dy = 1.\n$$

                                  I would argue not. For a given $X=x$ the sum of of the pdf over the support of $Y$ should less than or equal to 1. But according to Bierens it is.

                                  In particular, in the case (3.4) we have\n \\begin{align}\n\\begin{split}\n E[(Y &- E[Y \\mid X]) I(X \\in B)] \\\\\n &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (y - g(x)) I(x \\in B) f(y, x) \\ dy \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} y f(y \\mid x) \\ dy \\right) I(x \\in B) f_x(x) \\ dx \\\\\n&\\quad - \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} f(y \\mid x) \\ dy \\right) g(x) I (x \\in B) f_x(x) \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} g(x) I(x \\in B) f_x(x) \\ dx - \\int_{-\\infty}^{\\infty} g(x) I (x \\in B) f_x(x) \\ d x = 0.\n\\end{split} \\tag{3.7}\n\\end{align}

                                  ", "statics": {"paragraph": 4, "paragraph.text": 4, "equation-interline": 2}, "url": "https://math.stackexchange.com/questions/1561951/integral-over-conditional-pdf", "content": "If $f(x,y)$ is a joint pdf,I understand that,\n\n$$\n\\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty f(x,y) \\ dx \\ dy = 1\n$$\n\nbut does this hold for the conditional pdf?\n\n$$\n\\int_{-\\infty}^\\infty f({y|x}) \\ dy = 1.\n$$\n\nI would argue not. For a given $X=x$ the sum of of the pdf over the support of $Y$ should less than or equal to 1. But according to Bierens it is.\n\nIn particular, in the case (3.4) we have\n \\begin{align}\n\\begin{split}\n E[(Y &- E[Y \\mid X]) I(X \\in B)] \\\\\n &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (y - g(x)) I(x \\in B) f(y, x) \\ dy \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} y f(y \\mid x) \\ dy \\right) I(x \\in B) f_x(x) \\ dx \\\\\n&\\quad - \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} f(y \\mid x) \\ dy \\right) g(x) I (x \\in B) f_x(x) \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} g(x) I(x \\in B) f_x(x) \\ dx - \\int_{-\\infty}^{\\infty} g(x) I (x \\in B) f_x(x) \\ d x = 0.\n\\end{split} \\tag{3.7}\n\\end{align}\n", "html": "\n\n\n\n\n \n\n \n\n probability - Integral Over Conditional PDF - Mathematics Stack Exchange\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n\n\n\n \n \n\n\n\n\n \n\n\n \n\n\n \n \n\n\n \n \n\n\n\n \n \n \n \n\n\n
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                                  If $f(x,y)$ is a joint pdf,I understand that,

                                  \n\n

                                  $$\n \\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty f(x,y) \\ dx \\ dy = 1\n$$

                                  \n\n

                                  but does this hold for the conditional pdf?

                                  \n\n

                                  $$\n \\int_{-\\infty}^\\infty f({y|x}) \\ dy = 1.\n$$

                                  \n\n

                                  I would argue not. For a given $X=x$ the sum of of the pdf over the support of $Y$ should less than or equal to 1. But according to Bierens it is.

                                  \n\n
                                  \n

                                  In particular, in the case (3.4) we have\n \\begin{align}\n\\begin{split}\n E[(Y &- E[Y \\mid X]) I(X \\in B)] \\\\\n &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (y - g(x)) I(x \\in B) f(y, x) \\ dy \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} y f(y \\mid x) \\ dy \\right) I(x \\in B) f_x(x) \\ dx \\\\\n&\\quad - \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} f(y \\mid x) \\ dy \\right) g(x) I (x \\in B) f_x(x) \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} g(x) I(x \\in B) f_x(x) \\ dx - \\int_{-\\infty}^{\\infty} g(x) I (x \\in B) f_x(x) \\ d x = 0.\n\\end{split} \\tag{3.7}\n\\end{align}

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                                  \n edited Dec 6, 2015 at 4:51\n
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                                  \"David
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                                  \n asked Dec 6, 2015 at 3:43\n
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                                  \"jessica's
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                                    \n $\\begingroup$\n Not clear your question and your last expression, that is not the expression for the conditional expectation, is just the integral over a conditional PDF.\n $\\endgroup$\n
                                    \n– Carlos H. Mendoza-Cardenas\n
                                    \n \n Commented\n Dec 6, 2015 at 3:46\n \n
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                                    \n $\\begingroup$\n Sorry it was a typo. I meant over the conditional pdf. Does it sum to 1 over the support of y given a fixed x?\n $\\endgroup$\n
                                    \n– jessica\n
                                    \n \n Commented\n Dec 6, 2015 at 3:48\n \n \n \n \n
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                                    \n $\\begingroup$\n It does. Take a look at the answers and tell us what you think after that.\n $\\endgroup$\n
                                    \n– Carlos H. Mendoza-Cardenas\n
                                    \n \n Commented\n Dec 6, 2015 at 4:22\n \n
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                                    \n $\\begingroup$\n $f_{Y|X}(y|x)$ for any fixed $x$ should be a valid pdf - so it should integrate to $1$.\n $\\endgroup$\n
                                    \n– A.S.\n
                                    \n \n Commented\n Dec 6, 2015 at 4:36\n \n
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                                    \n $\\begingroup$\n I added some graphs and extra explanation to make this more clear, hopefully ;)\n $\\endgroup$\n
                                    \n– Carlos H. Mendoza-Cardenas\n
                                    \n \n Commented\n Dec 6, 2015 at 6:24\n \n
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                                  \n 2 Answers\n 2\n

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                                  By definition,

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                                  $$f_{Y \\mid X}(y \\mid x) = \\frac{f_{X,Y}(x,y)}{f_X(x)}\\qquad \\forall x,y$$

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                                  Here, given $x$, $f_X(x)$ is a constant (we are evaluating at some $x$). Then,

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                                  \\begin{align}\n\\int_{-\\infty}^{\\infty} f_{Y \\mid X}(y \\mid x) dy &= \\frac{\\int_{-\\infty}^\\infty f_{X,Y}(x,y)dy}{f_X(x)}\\\\\n&= \\frac{f_X(x)}{f_X(x)}\\\\\n&= 1\\qquad \\forall x\n\\end{align}

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                                  where in the second equality we have computed the marginal PDF of $X$ by integrating the joint PDF over $y$.

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                                  Let's see now a illustrated example to hopefully gain a better understanding about this result. Suppose that $f_{X,Y}(x,y)$ is as shown below (in red a \"view\" of the joint PDF at $Y=2.5$)

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                                  $\\hspace{2.5cm}$ \"enter

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                                  Suppose we are interested in $f_{X \\mid Y}(x \\mid 2.5)$, which is

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                                  $$f_{X \\mid Y}(x \\mid 2.5) = \\frac{f_{X,Y}(x,2.5)}{f_Y(2.5)} = \\frac{1/4}{2(1/4)} = \\frac{1}{2}\\qquad 1\\leq x \\leq 3$$

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                                  Note that $f_{X \\mid Y}(x \\mid 2.5)$ has the same shape than $f_{X,Y}(x,2.5)$ (a constant equal to $0.25$), but it's divided by $f_Y(2.5)$ to normalize it. That is, to satisfy the normalization axiom! So, although the area under $f_{X,Y}(x, 2.5)$ (the red area above) is clearly smaller than $1$, the area under $f_{X \\mid Y}(x \\mid 2.5)$ is $1$ because of the mentioned normalization. Below the graph for this conditional PDF.

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                                  $\\hspace{2cm}$ \"enter

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                                  EDIT: In response to your comment, Jessica. You are right about the intuition. I just want to add something to it using the example above. We are working with two random variables, but both of them are defined over the same sample space $\\Omega$, which would be represented by that T-shape region in the $x-y$ plane. The probability law defined over $\\Omega$ is $f_{X,Y}$, and the volume under it is $1$. Now, what is $f_{X \\mid Y}$? It is a new probability law defined over a new sample space, the one that results of imposing the restriction $Y=2.5$ to $\\Omega$, that is $\\{1 \\leq X \\leq 3, Y=2.5$}. A key idea here is that when we condition the original model the shape of the original probability law, $f_{X,Y}$, over that new sample space does not change, is just scaled, and that scaling is what help us to hold the normalization axiom. We have reduced the sample space but at the same time we have increased the height of the distribution. It is a beautiful intuitive idea that is right there in the definition of a conditional probability: we start we a given sample space $\\Omega$ and a probability law $P(\\cdot)$ that enable us to compute something like $P(A)$, $P(B)$ or even $P(A \\cap B)$. But what if we know something, for example that $A$ has occurred? Then we think intuitively that our new sample space is $A$. How we compute now $P(B)$? We represent that probability now as $P(B \\mid A)$ to incorporate in the notation that partial knowledge about the result and realize that $B$ occurs only if $A \\cap B$ occurs to computed it as

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                                  $$P(B \\mid A) = \\frac{P(A \\cap B)}{P(B)}$$

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                                  why we divide by $P(B)$? To normalize the non-conditional probabilities so that the conditional ones add up to $1$! Now that our sample space got reduced, is reasonable to increase proportionally the probabilities in the new sample space. This is one of my favorites concepts in probability, the conditional probability.

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                                  \n edited Dec 6, 2015 at 17:58\n
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                                  \n answered Dec 6, 2015 at 3:55\n
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                                  \"Carlos
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                                  \n Carlos H. Mendoza-CardenasCarlos H. Mendoza-Cardenas\n
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                                  Yes, it does\n$$\\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy\n= \\int_{-\\infty}^\\infty\\frac{f_{X,Y}(x,y)}{f_X(x)}dy\n=\\frac{1}{f_X(x)}\\cdot f_X(x) = 1.$$

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                                  As an example, if $Y|X\\sim \\text{Exp}(X)$, then\n$$\\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy = \\int_0^\\infty xe^{-xy}dy = 1.$$

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                                  \n answered Dec 6, 2015 at 4:00\n
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                                  \"Em.'s
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                                  \n\n\n\n\n \n\n\n \n\n\n\n\n \n \n \n \n \n\n \n\n\n\n \n \n"} +{"content_list": [[{"type": "paragraph", "raw_content": "

                                  If f(x,y) is a joint pdf,I understand that,

                                  ", "content": [{"c": "If", "t": "text"}, {"c": "f(x,y)", "t": "equation-inline"}, {"c": "is a joint pdf,I understand that,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\n \\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty f(x,y) \\ dx \\ dy = 1\n", "content": {"math_content": "\\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty f(x,y) \\ dx \\ dy = 1", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                  but does this hold for the conditional pdf?

                                  ", "content": [{"c": "but does this hold for the conditional pdf?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\n \\int_{-\\infty}^\\infty f({y|x}) \\ dy = 1.\n", "content": {"math_content": "\\int_{-\\infty}^\\infty f({y|x}) \\ dy = 1.", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                  I would argue not. For a given X=x the sum of of the pdf over the support of Y should less than or equal to 1. But according to Bierens it is.

                                  ", "content": [{"c": "I would argue not. For a given", "t": "text"}, {"c": "X=x", "t": "equation-inline"}, {"c": "the sum of of the pdf over the support of", "t": "text"}, {"c": "Y", "t": "equation-inline"}, {"c": "should less than or equal to 1. But according to Bierens it is.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                  In particular, in the case (3.4) we have\n

                                  ", "content": [{"c": "In particular, in the case (3.4) we have", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{align}\n\\begin{split}\n E[(Y &- E[Y \\mid X]) I(X \\in B)] \\\\\n &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (y - g(x)) I(x \\in B) f(y, x) \\ dy \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} y f(y \\mid x) \\ dy \\right) I(x \\in B) f_x(x) \\ dx \\\\\n&\\quad - \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} f(y \\mid x) \\ dy \\right) g(x) I (x \\in B) f_x(x) \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} g(x) I(x \\in B) f_x(x) \\ dx - \\int_{-\\infty}^{\\infty} g(x) I (x \\in B) f_x(x) \\ d x = 0.\n\\end{split} \\tag{3.7}\n\\end{align}", "content": {"math_content": "\\begin{align}\n\\begin{split}\n E[(Y &- E[Y \\mid X]) I(X \\in B)] \\\\\n &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (y - g(x)) I(x \\in B) f(y, x) \\ dy \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} y f(y \\mid x) \\ dy \\right) I(x \\in B) f_x(x) \\ dx \\\\\n&\\quad - \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} f(y \\mid x) \\ dy \\right) g(x) I (x \\in B) f_x(x) \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} g(x) I(x \\in B) f_x(x) \\ dx - \\int_{-\\infty}^{\\infty} g(x) I (x \\in B) f_x(x) \\ d x = 0.\n\\end{split} \\tag{3.7}\n\\end{align}", "math_type": "latex", "by": "mathjax"}}, {"type": "list", "raw_content": "
                                    ", "content": {"items": [], "ordered": false}}, {"type": "list", "raw_content": "
                                    • Not clear your question and your last expression, that is not the expression for the conditional expectation, is just the integral over a conditional PDF.CommentedDec 6, 2015 at 3:46
                                    • Sorry it was a typo. I meant over the conditional pdf. Does it sum to 1 over the support of y given a fixed x?
                                      \n\u2013\u00a0jessica
                                      CommentedDec 6, 2015 at 3:48
                                    • It does. Take a look at the answers and tell us what you think after that.CommentedDec 6, 2015 at 4:22
                                    • $f_{Y|X}(y|x)$ for any fixed $x$ should be a valid pdf - so it should integrate to $1$.
                                      \n\u2013\u00a0A.S.
                                      CommentedDec 6, 2015 at 4:36
                                    • I added some graphs and extra explanation to make this more clear, hopefully ;)CommentedDec 6, 2015 at 6:24
                                    ", "content": {"items": [[[{"c": "Not clear your question and your last expression, that is not the expression for the conditional expectation, is just the integral over a conditional PDF.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Dec 6, 2015 at 3:46", "t": "text"}]], [[{"c": "Sorry it was a typo. I meant over the conditional pdf. Does it sum to 1 over the support of y given a fixed x?", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "jessica", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Dec 6, 2015 at 3:48", "t": "text"}]], [[{"c": "It does. Take a look at the answers and tell us what you think after that.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Dec 6, 2015 at 4:22", "t": "text"}]], [[{"c": "$f_{Y|X}(y|x)$ for any fixed $x$ should be a valid pdf - so it should integrate to $1$.", "t": "text"}, {"c": "\n\u2013\u00a0", "t": "text"}, {"c": "A.S.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Dec 6, 2015 at 4:36", "t": "text"}]], [[{"c": "I added some graphs and extra explanation to make this more clear, hopefully ;)", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Dec 6, 2015 at 6:24", "t": "text"}]]], "ordered": false}}, {"type": "title", "raw_content": "

                                    \n 2 Answers\n \n

                                    ", "content": {"title_content": "2 Answers", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                    \n9
                                    ", "content": [{"c": "9", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                    By definition,

                                    ", "content": [{"c": "By definition,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "f_{Y \\mid X}(y \\mid x) = \\frac{f_{X,Y}(x,y)}{f_X(x)}\\qquad \\forall x,y", "content": {"math_content": "f_{Y \\mid X}(y \\mid x) = \\frac{f_{X,Y}(x,y)}{f_X(x)}\\qquad \\forall x,y", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                    Here, given $x$, $f_X(x)$ is a constant (we are evaluating at some $x$). Then,

                                    ", "content": [{"c": "Here, given$x$, $f_X(x)$ is a constant (we are evaluating at some $x$). Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{align}\n\\int_{-\\infty}^{\\infty} f_{Y \\mid X}(y \\mid x) dy &= \\frac{\\int_{-\\infty}^\\infty f_{X,Y}(x,y)dy}{f_X(x)}\\\\\n&= \\frac{f_X(x)}{f_X(x)}\\\\\n&= 1\\qquad \\forall x\n\\end{align}", "content": {"math_content": "\\begin{align}\n\\int_{-\\infty}^{\\infty} f_{Y \\mid X}(y \\mid x) dy &= \\frac{\\int_{-\\infty}^\\infty f_{X,Y}(x,y)dy}{f_X(x)}\\\\\n&= \\frac{f_X(x)}{f_X(x)}\\\\\n&= 1\\qquad \\forall x\n\\end{align}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                    where in the second equality we have computed the marginal PDF of X by integrating the joint PDF over y.

                                    ", "content": [{"c": "where in the second equality we have computed the marginal PDF of", "t": "text"}, {"c": "X", "t": "equation-inline"}, {"c": "by integrating the joint PDF over", "t": "text"}, {"c": "y", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                    Let's see now a illustrated example to hopefully gain a better understanding about this result. Suppose that f_{X,Y}(x,y) is as shown below (in red a \"view\" of the joint PDF at Y=2.5)

                                    ", "content": [{"c": "Let's see now a illustrated example to hopefully gain a better understanding about this result. Suppose that", "t": "text"}, {"c": "f_{X,Y}(x,y)", "t": "equation-inline"}, {"c": "is as shown below (in red a \"view\" of the joint PDF at", "t": "text"}, {"c": "Y=2.5", "t": "equation-inline"}, {"c": ")", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                    $\\hspace{2.5cm}$

                                    ", "content": [{"c": "$\\hspace{2.5cm}$", "t": "text"}]}, {"type": "image", "raw_content": "\"enter", "content": {"url": "https://i.sstatic.net/bmaQx.png", "data": null, "alt": "enter image description here", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

                                    Suppose we are interested in f_{X \\mid Y}(x \\mid 2.5), which is

                                    ", "content": [{"c": "Suppose we are interested in", "t": "text"}, {"c": "f_{X \\mid Y}(x \\mid 2.5)", "t": "equation-inline"}, {"c": ", which is", "t": "text"}]}, {"type": "equation-interline", "raw_content": "f_{X \\mid Y}(x \\mid 2.5) = \\frac{f_{X,Y}(x,2.5)}{f_Y(2.5)} = \\frac{1/4}{2(1/4)} = \\frac{1}{2}\\qquad 1\\leq x \\leq 3", "content": {"math_content": "f_{X \\mid Y}(x \\mid 2.5) = \\frac{f_{X,Y}(x,2.5)}{f_Y(2.5)} = \\frac{1/4}{2(1/4)} = \\frac{1}{2}\\qquad 1\\leq x \\leq 3", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                    Note that f_{X \\mid Y}(x \\mid 2.5) has the same shape than f_{X,Y}(x,2.5) (a constant equal to 0.25), but it's divided by f_Y(2.5) to normalize it. That is, to satisfy the normalization axiom! So, although the area under f_{X,Y}(x, 2.5) (the red area above) is clearly smaller than 1, the area under f_{X \\mid Y}(x \\mid 2.5) is 1 because of the mentioned normalization. Below the graph for this conditional PDF.

                                    ", "content": [{"c": "Note that", "t": "text"}, {"c": "f_{X \\mid Y}(x \\mid 2.5)", "t": "equation-inline"}, {"c": "has the same shape than", "t": "text"}, {"c": "f_{X,Y}(x,2.5)", "t": "equation-inline"}, {"c": "(a constant equal to", "t": "text"}, {"c": "0.25", "t": "equation-inline"}, {"c": "), but it's divided by", "t": "text"}, {"c": "f_Y(2.5)", "t": "equation-inline"}, {"c": "to normalize it. That is, to satisfy the normalization axiom! So, although the area under", "t": "text"}, {"c": "f_{X,Y}(x, 2.5)", "t": "equation-inline"}, {"c": "(the red area above) is clearly smaller than", "t": "text"}, {"c": "1", "t": "equation-inline"}, {"c": ", the area under", "t": "text"}, {"c": "f_{X \\mid Y}(x \\mid 2.5)", "t": "equation-inline"}, {"c": "is", "t": "text"}, {"c": "1", "t": "equation-inline"}, {"c": "because of the mentioned normalization. Below the graph for this conditional PDF.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                    $\\hspace{2cm}$

                                    ", "content": [{"c": "$\\hspace{2cm}$", "t": "text"}]}, {"type": "image", "raw_content": "\"enter", "content": {"url": "https://i.sstatic.net/nmV9v.png", "data": null, "alt": "enter image description here", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

                                    EDIT: In response to your comment, Jessica. You are right about the intuition. I just want to add something to it using the example above. We are working with two random variables, but both of them are defined over the same sample space \\Omega, which would be represented by that T-shape region in the x-y plane. The probability law defined over \\Omega is f_{X,Y}, and the volume under it is 1. Now, what is f_{X \\mid Y}? It is a new probability law defined over a new sample space, the one that results of imposing the restriction Y=2.5 to \\Omega, that is \\{1 \\leq X \\leq 3, Y=2.5}. A key idea here is that when we condition the original model the shape of the original probability law, f_{X,Y}, over that new sample space does not change, is just scaled, and that scaling is what help us to hold the normalization axiom. We have reduced the sample space but at the same time we have increased the height of the distribution. It is a beautiful intuitive idea that is right there in the definition of a conditional probability: we start we a given sample space \\Omega and a probability law P(\\cdot) that enable us to compute something like P(A), P(B) or even P(A \\cap B). But what if we know something, for example that A has occurred? Then we think intuitively that our new sample space is A. How we compute now P(B)? We represent that probability now as P(B \\mid A) to incorporate in the notation that partial knowledge about the result and realize that B occurs only if A \\cap B occurs to computed it as

                                    ", "content": [{"c": "EDIT: In response to your comment, Jessica. You are right about the intuition. I just want to add something to it using the example above. We are working with two random variables, but both of them are defined over the same sample space", "t": "text"}, {"c": "\\Omega", "t": "equation-inline"}, {"c": ", which would be represented by that T-shape region in the", "t": "text"}, {"c": "x-y", "t": "equation-inline"}, {"c": "plane. The probability law defined over", "t": "text"}, {"c": "\\Omega", "t": "equation-inline"}, {"c": "is", "t": "text"}, {"c": "f_{X,Y}", "t": "equation-inline"}, {"c": ", and the volume under it is", "t": "text"}, {"c": "1", "t": "equation-inline"}, {"c": ". Now, what is", "t": "text"}, {"c": "f_{X \\mid Y}", "t": "equation-inline"}, {"c": "? It is a new probability law defined over a new sample space, the one that results of imposing the restriction", "t": "text"}, {"c": "Y=2.5", "t": "equation-inline"}, {"c": "to", "t": "text"}, {"c": "\\Omega", "t": "equation-inline"}, {"c": ", that is", "t": "text"}, {"c": "\\{1 \\leq X \\leq 3, Y=2.5", "t": "equation-inline"}, {"c": "}. A key idea here is that when we condition the original model the shape of the original probability law,", "t": "text"}, {"c": "f_{X,Y}", "t": "equation-inline"}, {"c": ", over that new sample space does not change, is just scaled, and that scaling is what help us to hold the normalization axiom. We have reduced the sample space but at the same time we have increased the height of the distribution. It is a beautiful intuitive idea that is right there in the definition of a conditional probability: we start we a given sample space", "t": "text"}, {"c": "\\Omega", "t": "equation-inline"}, {"c": "and a probability law", "t": "text"}, {"c": "P(\\cdot)", "t": "equation-inline"}, {"c": "that enable us to compute something like", "t": "text"}, {"c": "P(A)", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "P(B)", "t": "equation-inline"}, {"c": "or even", "t": "text"}, {"c": "P(A \\cap B)", "t": "equation-inline"}, {"c": ". But what if we know something, for example that", "t": "text"}, {"c": "A", "t": "equation-inline"}, {"c": "has occurred? Then we think intuitively that our new sample space is", "t": "text"}, {"c": "A", "t": "equation-inline"}, {"c": ". How we compute now", "t": "text"}, {"c": "P(B)", "t": "equation-inline"}, {"c": "? We represent that probability now as", "t": "text"}, {"c": "P(B \\mid A)", "t": "equation-inline"}, {"c": "to incorporate in the notation that partial knowledge about the result and realize that", "t": "text"}, {"c": "B", "t": "equation-inline"}, {"c": "occurs only if", "t": "text"}, {"c": "A \\cap B", "t": "equation-inline"}, {"c": "occurs to computed it as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "P(B \\mid A) = \\frac{P(A \\cap B)}{P(B)}", "content": {"math_content": "P(B \\mid A) = \\frac{P(A \\cap B)}{P(B)}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                    why we divide by $P(B)$? To normalize the non-conditional probabilities so that the conditional ones add up to $1$! Now that our sample space got reduced, is reasonable to increase proportionally the probabilities in the new sample space. This is one of my favorites concepts in probability, the conditional probability.

                                    ", "content": [{"c": "why we divide by $P(B)$? To normalize the non-conditional probabilities so that the conditional ones add up to $1$! Now that our sample space got reduced, is reasonable to increase proportionally the probabilities in the new sample space. This is one of my favorites concepts in probability, the conditional probability.", "t": "text"}]}, {"type": "list", "raw_content": "
                                      ", "content": {"items": [], "ordered": false}}, {"type": "paragraph", "raw_content": "

                                      Yes, it does\n

                                      ", "content": [{"c": "Yes, it does", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy\n= \\int_{-\\infty}^\\infty\\frac{f_{X,Y}(x,y)}{f_X(x)}dy\n=\\frac{1}{f_X(x)}\\cdot f_X(x) = 1.", "content": {"math_content": "\\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy\n= \\int_{-\\infty}^\\infty\\frac{f_{X,Y}(x,y)}{f_X(x)}dy\n=\\frac{1}{f_X(x)}\\cdot f_X(x) = 1.", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                      As an example, if Y|X\\sim \\text{Exp}(X), then\n

                                      ", "content": [{"c": "As an example, if", "t": "text"}, {"c": "Y|X\\sim \\text{Exp}(X)", "t": "equation-inline"}, {"c": ", then", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy = \\int_0^\\infty xe^{-xy}dy = 1.", "content": {"math_content": "\\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy = \\int_0^\\infty xe^{-xy}dy = 1.", "math_type": "latex", "by": "mathjax"}}, {"type": "list", "raw_content": "
                                        ", "content": {"items": [], "ordered": false}}, {"type": "title", "raw_content": "

                                        \n You must log in to answer this question.\n

                                        ", "content": {"title_content": "You must log in to answer this question.", "level": "2"}}, {"type": "title", "raw_content": "

                                        \nNot the answer you're looking for? Browse other questions tagged

                                        ", "content": {"title_content": "Not the answer you're looking for? Browse other questions tagged", "level": "2"}}, {"type": "list", "raw_content": "
                                          . ", "content": {"items": [], "ordered": false}}, {"type": "title", "raw_content": "

                                          .

                                          ", "content": {"title_content": ".", "level": "2"}}]], "main_html": "

                                          If f(x,y) is a joint pdf,I understand that,

                                          \n \\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty f(x,y) \\ dx \\ dy = 1\n

                                          but does this hold for the conditional pdf?

                                          \n \\int_{-\\infty}^\\infty f({y|x}) \\ dy = 1.\n

                                          I would argue not. For a given X=x the sum of of the pdf over the support of Y should less than or equal to 1. But according to Bierens it is.

                                          In particular, in the case (3.4) we have\n

                                          \\begin{align}\n\\begin{split}\n E[(Y &- E[Y \\mid X]) I(X \\in B)] \\\\\n &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (y - g(x)) I(x \\in B) f(y, x) \\ dy \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} y f(y \\mid x) \\ dy \\right) I(x \\in B) f_x(x) \\ dx \\\\\n&\\quad - \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} f(y \\mid x) \\ dy \\right) g(x) I (x \\in B) f_x(x) \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} g(x) I(x \\in B) f_x(x) \\ dx - \\int_{-\\infty}^{\\infty} g(x) I (x \\in B) f_x(x) \\ d x = 0.\n\\end{split} \\tag{3.7}\n\\end{align}
                                            • Not clear your question and your last expression, that is not the expression for the conditional expectation, is just the integral over a conditional PDF.CommentedDec 6, 2015 at 3:46
                                            • Sorry it was a typo. I meant over the conditional pdf. Does it sum to 1 over the support of y given a fixed x?
                                              \n\u2013\u00a0jessica
                                              CommentedDec 6, 2015 at 3:48
                                            • It does. Take a look at the answers and tell us what you think after that.CommentedDec 6, 2015 at 4:22
                                            • $f_{Y|X}(y|x)$ for any fixed $x$ should be a valid pdf - so it should integrate to $1$.
                                              \n\u2013\u00a0A.S.
                                              CommentedDec 6, 2015 at 4:36
                                            • I added some graphs and extra explanation to make this more clear, hopefully ;)CommentedDec 6, 2015 at 6:24

                                            \n 2 Answers\n \n

                                            \n9

                                            By definition,

                                            f_{Y \\mid X}(y \\mid x) = \\frac{f_{X,Y}(x,y)}{f_X(x)}\\qquad \\forall x,y

                                            Here, given $x$, $f_X(x)$ is a constant (we are evaluating at some $x$). Then,

                                            \\begin{align}\n\\int_{-\\infty}^{\\infty} f_{Y \\mid X}(y \\mid x) dy &= \\frac{\\int_{-\\infty}^\\infty f_{X,Y}(x,y)dy}{f_X(x)}\\\\\n&= \\frac{f_X(x)}{f_X(x)}\\\\\n&= 1\\qquad \\forall x\n\\end{align}

                                            where in the second equality we have computed the marginal PDF of X by integrating the joint PDF over y.

                                            Let's see now a illustrated example to hopefully gain a better understanding about this result. Suppose that f_{X,Y}(x,y) is as shown below (in red a \"view\" of the joint PDF at Y=2.5)

                                            $\\hspace{2.5cm}$

                                            \"enter

                                            Suppose we are interested in f_{X \\mid Y}(x \\mid 2.5), which is

                                            f_{X \\mid Y}(x \\mid 2.5) = \\frac{f_{X,Y}(x,2.5)}{f_Y(2.5)} = \\frac{1/4}{2(1/4)} = \\frac{1}{2}\\qquad 1\\leq x \\leq 3

                                            Note that f_{X \\mid Y}(x \\mid 2.5) has the same shape than f_{X,Y}(x,2.5) (a constant equal to 0.25), but it's divided by f_Y(2.5) to normalize it. That is, to satisfy the normalization axiom! So, although the area under f_{X,Y}(x, 2.5) (the red area above) is clearly smaller than 1, the area under f_{X \\mid Y}(x \\mid 2.5) is 1 because of the mentioned normalization. Below the graph for this conditional PDF.

                                            $\\hspace{2cm}$

                                            \"enter

                                            EDIT: In response to your comment, Jessica. You are right about the intuition. I just want to add something to it using the example above. We are working with two random variables, but both of them are defined over the same sample space \\Omega, which would be represented by that T-shape region in the x-y plane. The probability law defined over \\Omega is f_{X,Y}, and the volume under it is 1. Now, what is f_{X \\mid Y}? It is a new probability law defined over a new sample space, the one that results of imposing the restriction Y=2.5 to \\Omega, that is \\{1 \\leq X \\leq 3, Y=2.5}. A key idea here is that when we condition the original model the shape of the original probability law, f_{X,Y}, over that new sample space does not change, is just scaled, and that scaling is what help us to hold the normalization axiom. We have reduced the sample space but at the same time we have increased the height of the distribution. It is a beautiful intuitive idea that is right there in the definition of a conditional probability: we start we a given sample space \\Omega and a probability law P(\\cdot) that enable us to compute something like P(A), P(B) or even P(A \\cap B). But what if we know something, for example that A has occurred? Then we think intuitively that our new sample space is A. How we compute now P(B)? We represent that probability now as P(B \\mid A) to incorporate in the notation that partial knowledge about the result and realize that B occurs only if A \\cap B occurs to computed it as

                                            P(B \\mid A) = \\frac{P(A \\cap B)}{P(B)}

                                            why we divide by $P(B)$? To normalize the non-conditional probabilities so that the conditional ones add up to $1$! Now that our sample space got reduced, is reasonable to increase proportionally the probabilities in the new sample space. This is one of my favorites concepts in probability, the conditional probability.

                                              Yes, it does\n

                                              \\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy\n= \\int_{-\\infty}^\\infty\\frac{f_{X,Y}(x,y)}{f_X(x)}dy\n=\\frac{1}{f_X(x)}\\cdot f_X(x) = 1.

                                              As an example, if Y|X\\sim \\text{Exp}(X), then\n

                                              \\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy = \\int_0^\\infty xe^{-xy}dy = 1.

                                                \n You must log in to answer this question.\n

                                                \nNot the answer you're looking for? Browse other questions tagged

                                                  .

                                                  .

                                                  ", "statics": {"paragraph": 17, "paragraph.text": 55, "paragraph.equation-inline": 38, "equation-interline": 9, "list": 5, "list.text": 19, "title": 4, "image": 2}, "url": "https://math.stackexchange.com/questions/1561951/integral-over-conditional-pdf", "content": "If $f(x,y)$ is a joint pdf,I understand that,\n\n$$\n\\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty f(x,y) \\ dx \\ dy = 1\n$$\n\nbut does this hold for the conditional pdf?\n\n$$\n\\int_{-\\infty}^\\infty f({y|x}) \\ dy = 1.\n$$\n\nI would argue not. For a given $X=x$ the sum of of the pdf over the support of $Y$ should less than or equal to 1. But according to Bierens it is.\n\nIn particular, in the case (3.4) we have\n\n$$\n\\begin{align}\n\\begin{split}\n E[(Y &- E[Y \\mid X]) I(X \\in B)] \\\\\n &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (y - g(x)) I(x \\in B) f(y, x) \\ dy \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} y f(y \\mid x) \\ dy \\right) I(x \\in B) f_x(x) \\ dx \\\\\n&\\quad - \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} f(y \\mid x) \\ dy \\right) g(x) I (x \\in B) f_x(x) \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} g(x) I(x \\in B) f_x(x) \\ dx - \\int_{-\\infty}^{\\infty} g(x) I (x \\in B) f_x(x) \\ d x = 0.\n\\end{split} \\tag{3.7}\n\\end{align}\n$$\n\n- Not clear your question and your last expression, that is not the expression for the conditional expectation, is just the integral over a conditional PDF. Commented Dec 6, 2015 at 3:46\n- Sorry it was a typo. I meant over the conditional pdf. Does it sum to 1 over the support of y given a fixed x? \u2013 jessica Commented Dec 6, 2015 at 3:48\n- It does. Take a look at the answers and tell us what you think after that. Commented Dec 6, 2015 at 4:22\n- $f_{Y|X}(y|x)$ for any fixed $x$ should be a valid pdf - so it should integrate to $1$. \u2013 A.S. Commented Dec 6, 2015 at 4:36\n- I added some graphs and extra explanation to make this more clear, hopefully ;) Commented Dec 6, 2015 at 6:24\n\n## 2 Answers\n\n9\n\nBy definition,\n\n$$\nf_{Y \\mid X}(y \\mid x) = \\frac{f_{X,Y}(x,y)}{f_X(x)}\\qquad \\forall x,y\n$$\n\nHere, given$x$, $f_X(x)$ is a constant (we are evaluating at some $x$). Then,\n\n$$\n\\begin{align}\n\\int_{-\\infty}^{\\infty} f_{Y \\mid X}(y \\mid x) dy &= \\frac{\\int_{-\\infty}^\\infty f_{X,Y}(x,y)dy}{f_X(x)}\\\\\n&= \\frac{f_X(x)}{f_X(x)}\\\\\n&= 1\\qquad \\forall x\n\\end{align}\n$$\n\nwhere in the second equality we have computed the marginal PDF of $X$ by integrating the joint PDF over $y$ .\n\nLet's see now a illustrated example to hopefully gain a better understanding about this result. Suppose that $f_{X,Y}(x,y)$ is as shown below (in red a \"view\" of the joint PDF at $Y=2.5$ )\n\n$\\hspace{2.5cm}$\n\nSuppose we are interested in $f_{X \\mid Y}(x \\mid 2.5)$ , which is\n\n$$\nf_{X \\mid Y}(x \\mid 2.5) = \\frac{f_{X,Y}(x,2.5)}{f_Y(2.5)} = \\frac{1/4}{2(1/4)} = \\frac{1}{2}\\qquad 1\\leq x \\leq 3\n$$\n\nNote that $f_{X \\mid Y}(x \\mid 2.5)$ has the same shape than $f_{X,Y}(x,2.5)$ (a constant equal to $0.25$ ), but it's divided by $f_Y(2.5)$ to normalize it. That is, to satisfy the normalization axiom! So, although the area under $f_{X,Y}(x, 2.5)$ (the red area above) is clearly smaller than $1$ , the area under $f_{X \\mid Y}(x \\mid 2.5)$ is $1$ because of the mentioned normalization. Below the graph for this conditional PDF.\n\n$\\hspace{2cm}$\n\nEDIT: In response to your comment, Jessica. You are right about the intuition. I just want to add something to it using the example above. We are working with two random variables, but both of them are defined over the same sample space $\\Omega$ , which would be represented by that T-shape region in the $x-y$ plane. The probability law defined over $\\Omega$ is $f_{X,Y}$ , and the volume under it is $1$ . Now, what is $f_{X \\mid Y}$ ? It is a new probability law defined over a new sample space, the one that results of imposing the restriction $Y=2.5$ to $\\Omega$ , that is $\\{1 \\leq X \\leq 3, Y=2.5$ }. A key idea here is that when we condition the original model the shape of the original probability law, $f_{X,Y}$ , over that new sample space does not change, is just scaled, and that scaling is what help us to hold the normalization axiom. We have reduced the sample space but at the same time we have increased the height of the distribution. It is a beautiful intuitive idea that is right there in the definition of a conditional probability: we start we a given sample space $\\Omega$ and a probability law $P(\\cdot)$ that enable us to compute something like $P(A)$ , $P(B)$ or even $P(A \\cap B)$ . But what if we know something, for example that $A$ has occurred? Then we think intuitively that our new sample space is $A$ . How we compute now $P(B)$ ? We represent that probability now as $P(B \\mid A)$ to incorporate in the notation that partial knowledge about the result and realize that $B$ occurs only if $A \\cap B$ occurs to computed it as\n\n$$\nP(B \\mid A) = \\frac{P(A \\cap B)}{P(B)}\n$$\n\nwhy we divide by $P(B)$? To normalize the non-conditional probabilities so that the conditional ones add up to $1$! Now that our sample space got reduced, is reasonable to increase proportionally the probabilities in the new sample space. This is one of my favorites concepts in probability, the conditional probability.\n\nYes, it does\n\n$$\n\\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy\n= \\int_{-\\infty}^\\infty\\frac{f_{X,Y}(x,y)}{f_X(x)}dy\n=\\frac{1}{f_X(x)}\\cdot f_X(x) = 1.\n$$\n\nAs an example, if $Y|X\\sim \\text{Exp}(X)$ , then\n\n$$\n\\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy = \\int_0^\\infty xe^{-xy}dy = 1.\n$$\n\n## You must log in to answer this question.\n\n## Not the answer you're looking for? Browse other questions tagged\n\n## .\n", "html": "\n\n\n\n\n \n\n \n\n probability - Integral Over Conditional PDF - Mathematics Stack Exchange\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n\n\n\n \n \n\n\n\n\n \n\n\n \n\n\n \n \n\n\n \n \n\n\n\n \n \n \n \n\n\n
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                                                  If $f(x,y)$ is a joint pdf,I understand that,

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                                                  $$\n \\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty f(x,y) \\ dx \\ dy = 1\n$$

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                                                  but does this hold for the conditional pdf?

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                                                  $$\n \\int_{-\\infty}^\\infty f({y|x}) \\ dy = 1.\n$$

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                                                  I would argue not. For a given $X=x$ the sum of of the pdf over the support of $Y$ should less than or equal to 1. But according to Bierens it is.

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                                                  In particular, in the case (3.4) we have\n \\begin{align}\n\\begin{split}\n E[(Y &- E[Y \\mid X]) I(X \\in B)] \\\\\n &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (y - g(x)) I(x \\in B) f(y, x) \\ dy \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} y f(y \\mid x) \\ dy \\right) I(x \\in B) f_x(x) \\ dx \\\\\n&\\quad - \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} f(y \\mid x) \\ dy \\right) g(x) I (x \\in B) f_x(x) \\ dx \\\\\n&= \\int_{-\\infty}^{\\infty} g(x) I(x \\in B) f_x(x) \\ dx - \\int_{-\\infty}^{\\infty} g(x) I (x \\in B) f_x(x) \\ d x = 0.\n\\end{split} \\tag{3.7}\n\\end{align}

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                                                  \n edited Dec 6, 2015 at 4:51\n
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                                                  \"David
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                                                  \n asked Dec 6, 2015 at 3:43\n
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                                                  \"jessica's
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                                                    \n $\\begingroup$\n Not clear your question and your last expression, that is not the expression for the conditional expectation, is just the integral over a conditional PDF.\n $\\endgroup$\n
                                                    \n– Carlos H. Mendoza-Cardenas\n
                                                    \n \n Commented\n Dec 6, 2015 at 3:46\n \n
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                                                    \n $\\begingroup$\n Sorry it was a typo. I meant over the conditional pdf. Does it sum to 1 over the support of y given a fixed x?\n $\\endgroup$\n
                                                    \n– jessica\n
                                                    \n \n Commented\n Dec 6, 2015 at 3:48\n \n \n \n \n
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                                                    \n $\\begingroup$\n It does. Take a look at the answers and tell us what you think after that.\n $\\endgroup$\n
                                                    \n– Carlos H. Mendoza-Cardenas\n
                                                    \n \n Commented\n Dec 6, 2015 at 4:22\n \n
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                                                    \n $\\begingroup$\n $f_{Y|X}(y|x)$ for any fixed $x$ should be a valid pdf - so it should integrate to $1$.\n $\\endgroup$\n
                                                    \n– A.S.\n
                                                    \n \n Commented\n Dec 6, 2015 at 4:36\n \n
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                                                    \n $\\begingroup$\n I added some graphs and extra explanation to make this more clear, hopefully ;)\n $\\endgroup$\n
                                                    \n– Carlos H. Mendoza-Cardenas\n
                                                    \n \n Commented\n Dec 6, 2015 at 6:24\n \n
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                                                  \n 2 Answers\n 2\n

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                                                  By definition,

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                                                  $$f_{Y \\mid X}(y \\mid x) = \\frac{f_{X,Y}(x,y)}{f_X(x)}\\qquad \\forall x,y$$

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                                                  Here, given $x$, $f_X(x)$ is a constant (we are evaluating at some $x$). Then,

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                                                  \\begin{align}\n\\int_{-\\infty}^{\\infty} f_{Y \\mid X}(y \\mid x) dy &= \\frac{\\int_{-\\infty}^\\infty f_{X,Y}(x,y)dy}{f_X(x)}\\\\\n&= \\frac{f_X(x)}{f_X(x)}\\\\\n&= 1\\qquad \\forall x\n\\end{align}

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                                                  where in the second equality we have computed the marginal PDF of $X$ by integrating the joint PDF over $y$.

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                                                  Let's see now a illustrated example to hopefully gain a better understanding about this result. Suppose that $f_{X,Y}(x,y)$ is as shown below (in red a \"view\" of the joint PDF at $Y=2.5$)

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                                                  $\\hspace{2.5cm}$ \"enter

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                                                  Suppose we are interested in $f_{X \\mid Y}(x \\mid 2.5)$, which is

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                                                  $$f_{X \\mid Y}(x \\mid 2.5) = \\frac{f_{X,Y}(x,2.5)}{f_Y(2.5)} = \\frac{1/4}{2(1/4)} = \\frac{1}{2}\\qquad 1\\leq x \\leq 3$$

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                                                  Note that $f_{X \\mid Y}(x \\mid 2.5)$ has the same shape than $f_{X,Y}(x,2.5)$ (a constant equal to $0.25$), but it's divided by $f_Y(2.5)$ to normalize it. That is, to satisfy the normalization axiom! So, although the area under $f_{X,Y}(x, 2.5)$ (the red area above) is clearly smaller than $1$, the area under $f_{X \\mid Y}(x \\mid 2.5)$ is $1$ because of the mentioned normalization. Below the graph for this conditional PDF.

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                                                  $\\hspace{2cm}$ \"enter

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                                                  EDIT: In response to your comment, Jessica. You are right about the intuition. I just want to add something to it using the example above. We are working with two random variables, but both of them are defined over the same sample space $\\Omega$, which would be represented by that T-shape region in the $x-y$ plane. The probability law defined over $\\Omega$ is $f_{X,Y}$, and the volume under it is $1$. Now, what is $f_{X \\mid Y}$? It is a new probability law defined over a new sample space, the one that results of imposing the restriction $Y=2.5$ to $\\Omega$, that is $\\{1 \\leq X \\leq 3, Y=2.5$}. A key idea here is that when we condition the original model the shape of the original probability law, $f_{X,Y}$, over that new sample space does not change, is just scaled, and that scaling is what help us to hold the normalization axiom. We have reduced the sample space but at the same time we have increased the height of the distribution. It is a beautiful intuitive idea that is right there in the definition of a conditional probability: we start we a given sample space $\\Omega$ and a probability law $P(\\cdot)$ that enable us to compute something like $P(A)$, $P(B)$ or even $P(A \\cap B)$. But what if we know something, for example that $A$ has occurred? Then we think intuitively that our new sample space is $A$. How we compute now $P(B)$? We represent that probability now as $P(B \\mid A)$ to incorporate in the notation that partial knowledge about the result and realize that $B$ occurs only if $A \\cap B$ occurs to computed it as

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                                                  $$P(B \\mid A) = \\frac{P(A \\cap B)}{P(B)}$$

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                                                  why we divide by $P(B)$? To normalize the non-conditional probabilities so that the conditional ones add up to $1$! Now that our sample space got reduced, is reasonable to increase proportionally the probabilities in the new sample space. This is one of my favorites concepts in probability, the conditional probability.

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                                                  \n edited Dec 6, 2015 at 17:58\n
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                                                  \n answered Dec 6, 2015 at 3:55\n
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                                                  \"Carlos
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                                                  \n Carlos H. Mendoza-CardenasCarlos H. Mendoza-Cardenas\n
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                                                  Yes, it does\n$$\\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy\n= \\int_{-\\infty}^\\infty\\frac{f_{X,Y}(x,y)}{f_X(x)}dy\n=\\frac{1}{f_X(x)}\\cdot f_X(x) = 1.$$

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                                                  As an example, if $Y|X\\sim \\text{Exp}(X)$, then\n$$\\int_{-\\infty}^\\infty f_{Y|X}(y|x)\\,dy = \\int_0^\\infty xe^{-xy}dy = 1.$$

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                                                  \n answered Dec 6, 2015 at 4:00\n
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                                                  \"Em.'s
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                                                  \n Em.Em.\n
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                                                  \n                                                   \n Attempt at computing various values given a joint PDF\n
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                                                  \n                                                   \n From Joint pdf to Conditional pdf and from Conditional pdf to Joint pdf\n
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                                                  \n                                                   \n What is the PDF of the sum of two dependent variables?\n
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                                                  \n \n Hot Network Questions\n \n

                                                  \n \n\n \n more hot questions\n \n
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                                                  \n\n\n\n\n \n\n\n \n\n\n\n\n \n \n \n \n \n\n \n\n\n\n \n \n"} diff --git a/bench/data/groundtruth/math_mathjax_latex_6.jsonl b/bench/data/groundtruth/math_mathjax_latex_6.jsonl index 91121ed7..7544059c 100644 --- a/bench/data/groundtruth/math_mathjax_latex_6.jsonl +++ b/bench/data/groundtruth/math_mathjax_latex_6.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "title", "raw_content": "

                                                  The short version

                                                  ", "content": {"title_content": "The short version", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                  Here is an extremely natural hyperplane arrangement in \\mathbb{R}^n, which I will callR_nforresonance arrangement.

                                                  ", "content": [{"c": "Here is an extremely natural hyperplane arrangement in", "t": "text"}, {"c": "\\mathbb{R}^n", "t": "equation-inline"}, {"c": ", which I will call", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": "for resonance arrangement.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  Let x_ibe the standard coordinates on\\mathbb{R}^n. For each nonemptyI\\subseteq [n]=\\{1,\\dots,n\\}, define the hyperplaneH_Ito be the hyperplane given by

                                                  ", "content": [{"c": "Let", "t": "text"}, {"c": "x_i", "t": "equation-inline"}, {"c": "be the standard coordinates on", "t": "text"}, {"c": "\\mathbb{R}^n", "t": "equation-inline"}, {"c": ". For each nonempty", "t": "text"}, {"c": "I\\subseteq [n]=\\{1,\\dots,n\\}", "t": "equation-inline"}, {"c": ", define the hyperplane", "t": "text"}, {"c": "H_I", "t": "equation-inline"}, {"c": "to be the hyperplane given by", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\sum_{i\\in I} x_i=0.$$\nThe resonance arrangement is given by all ", "content": {"math_content": "\\sum_{i\\in I} x_i=0.", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                  The resonance arrangement is given by all2^n-1hyperplanesH_I. The arrangementR_nis natural enough that it arises in many contexts -- to first order, my question is simply: have you come across it yourself?

                                                  ", "content": [{"c": "The resonance arrangement is given by all", "t": "text"}, {"c": "2^n-1", "t": "equation-inline"}, {"c": "hyperplanes", "t": "text"}, {"c": "H_I", "t": "equation-inline"}, {"c": ". The arrangement", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": "is natural enough that it arises in many contexts -- to first order, my question is simply: have you come across it yourself?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  This feels rather vague to be a good question, and after giving some background on where I've seen this I'll try to be a bit more specific about what I'm looking for, but my point is this arrangement has a rather simple and natural definition, and so crops up in multiple places, and I'd be curious to hear about more of them even if you can't specifically connect it to what follows.

                                                  ", "content": [{"c": "This feels rather vague to be a good question, and after giving some background on where I've seen this I'll try to be a bit more specific about what I'm looking for, but my point is this arrangement has a rather simple and natural definition, and so crops up in multiple places, and I'd be curious to hear about more of them even if you can't specifically connect it to what follows.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                  What I knew until this week

                                                  ", "content": {"title_content": "What I knew until this week", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                  I came across this arrangement in my study of double Hurwitz numbers -- they are piecewise polynomial, and the chambers of the resonance arrangement are the chambers of polynomiality. I don't want to go into this much more, as it's unimportant to most of what follows Though I will say that conjecturally double Hurwitz numbers could be related to compactified Picard varieties in a way which would connect this arrangement up with birational transformations of those. Also, the name \"resonance arrangement\" was essentially introduce in this context, by Shadrin, Shapiro and Vainshtein.

                                                  ", "content": [{"c": "I came across this arrangement in my study of double Hurwitz numbers -- they are piecewise polynomial, and the chambers of the resonance arrangement are the chambers of polynomiality. I don't want to go into this much more, as it's unimportant to most of what follows Though I will say that conjecturally double Hurwitz numbers could be related to compactified Picard varieties in a way which would connect this arrangement up with birational transformations of those. Also, the name \"resonance arrangement\" was essentially introduce in this context, by Shadrin, Shapiro and Vainshtein.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  It apparently comes up in physics -- I only know this because the number of regions of R_n, starting atn=2, is 2, 6, 32, 370, 11292, 1066044, 347326352 ... Sloane sequenceA034997, which you will see was entered as \"Number of Generalized Retarded Functions in Quantum Field Theory\" by a physicist.

                                                  ", "content": [{"c": "It apparently comes up in physics -- I only know this because the number of regions of", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": ", starting at", "t": "text"}, {"c": "n=2", "t": "equation-inline"}, {"c": ", is 2, 6, 32, 370, 11292, 1066044, 347326352 ... Sloane sequence A034997, which you will see was entered as \"Number of Generalized Retarded Functions in Quantum Field Theory\" by a physicist.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  You might expect by that rate of growth that this hyperplane arrangement is completely intractable, and more specifically, what I would love from an answer is some kind qualitative statement about how ugly the R_nget. Which brings us to:

                                                  ", "content": [{"c": "You might expect by that rate of growth that this hyperplane arrangement is completely intractable, and more specifically, what I would love from an answer is some kind qualitative statement about how ugly the", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": "get. Which brings us to:", "t": "text"}]}, {"type": "title", "raw_content": "

                                                  Connection to the GGMS decomposition

                                                  ", "content": {"title_content": "Connection to the GGMS decomposition", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                  I got to thinking about this again now and decided to post on MO of it because of Noah's question about the vertices of a variation of GIT problem, where this arrangement is lurking around -- see there for more detail. Allen's brief comment there prompted me to skim some of his and related papers to that general area, and I found the introduction to Positroid varieties I: juggling and geometry most enlightening, together with the discussion at Noah's question could give another suggestion why R_nis perhaps intractable. Briefly:

                                                  ", "content": [{"c": "I got to thinking about this again now and decided to post on MO of it because of Noah's question about the vertices of a variation of GIT problem, where this arrangement is lurking around -- see there for more detail. Allen's brief comment there prompted me to skim some of his and related papers to that general area, and I found the introduction to Positroid varieties I: juggling and geometry most enlightening, together with the discussion at Noah's question could give another suggestion why", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": "is perhaps intractable. Briefly:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  The arrangement R_nis a natural extension of theA_narrangement. One common description of theA_narrangement is as the\\binom{n+1}{2}hyperplanesy_i-y_j=0, i,j\\in [n]andy_i=0, i\\in [n]. However, one can consider the triangular change of variables

                                                  ", "content": [{"c": "The arrangement", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": "is a natural extension of the", "t": "text"}, {"c": "A_n", "t": "equation-inline"}, {"c": "arrangement. One common description of the", "t": "text"}, {"c": "A_n", "t": "equation-inline"}, {"c": "arrangement is as the", "t": "text"}, {"c": "\\binom{n+1}{2}", "t": "equation-inline"}, {"c": "hyperplanes", "t": "text"}, {"c": "y_i-y_j=0, i,j\\in [n]", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "y_i=0, i\\in [n]", "t": "equation-inline"}, {"c": ". However, one can consider the triangular change of variables", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$y_k\\mapsto \\sum_{j\\leq k} y_j$$.", "content": {"math_content": "y_k\\mapsto \\sum_{j\\leq k} y_j", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                  .

                                                  ", "content": [{"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  This changes the hyperplanes to\n

                                                  ", "content": [{"c": "This changes the hyperplanes to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "$$\\sum_{i\\leq k \\leq j} y_k=0.$$", "content": {"math_content": "\\sum_{i\\leq k \\leq j} y_k=0.", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                  These hyperplanes are no longer invariant under permutation of the coordinates, and if we proceed to add the entire S_norbit of them, we get the resonance arrangementR_n.

                                                  ", "content": [{"c": "These hyperplanes are no longer invariant under permutation of the coordinates, and if we proceed to add the entire", "t": "text"}, {"c": "S_n", "t": "equation-inline"}, {"c": "orbit of them, we get the resonance arrangement", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  From the discussion on Noah's question and the introduction to the Positroid paper, we see that this manipulation is a shadow of the GGMS decomposition of the Grassmannian, and describing this decomposition in general seems to be intractable in that it requires identifying whether matroids are representable or not. So, what I'd really like to know how is much of the \"GGMS abyss\", as they refer to it, is reflected in the resonance arrangement? Is it hopeless to describe and count its chambers?

                                                  ", "content": [{"c": "From the discussion on Noah's question and the introduction to the Positroid paper, we see that this manipulation is a shadow of the GGMS decomposition of the Grassmannian, and describing this decomposition in general seems to be intractable in that it requires identifying whether matroids are representable or not. So, what I'd really like to know how is much of the \"GGMS abyss\", as they refer to it, is reflected in the resonance arrangement? Is it hopeless to describe and count its chambers?", "t": "text"}]}]], "main_html": "

                                                  The short version

                                                  Here is an extremely natural hyperplane arrangement in \\mathbb{R}^n, which I will callR_nforresonance arrangement.

                                                  Let x_ibe the standard coordinates on\\mathbb{R}^n. For each nonemptyI\\subseteq [n]=\\{1,\\dots,n\\}, define the hyperplaneH_Ito be the hyperplane given by

                                                  $$\\sum_{i\\in I} x_i=0.$$\nThe resonance arrangement is given by all

                                                  The resonance arrangement is given by all2^n-1hyperplanesH_I. The arrangementR_nis natural enough that it arises in many contexts -- to first order, my question is simply: have you come across it yourself?

                                                  This feels rather vague to be a good question, and after giving some background on where I've seen this I'll try to be a bit more specific about what I'm looking for, but my point is this arrangement has a rather simple and natural definition, and so crops up in multiple places, and I'd be curious to hear about more of them even if you can't specifically connect it to what follows.

                                                  What I knew until this week

                                                  I came across this arrangement in my study of double Hurwitz numbers -- they are piecewise polynomial, and the chambers of the resonance arrangement are the chambers of polynomiality. I don't want to go into this much more, as it's unimportant to most of what follows Though I will say that conjecturally double Hurwitz numbers could be related to compactified Picard varieties in a way which would connect this arrangement up with birational transformations of those. Also, the name \"resonance arrangement\" was essentially introduce in this context, by Shadrin, Shapiro and Vainshtein.

                                                  It apparently comes up in physics -- I only know this because the number of regions of R_n, starting atn=2, is 2, 6, 32, 370, 11292, 1066044, 347326352 ... Sloane sequenceA034997, which you will see was entered as \"Number of Generalized Retarded Functions in Quantum Field Theory\" by a physicist.

                                                  You might expect by that rate of growth that this hyperplane arrangement is completely intractable, and more specifically, what I would love from an answer is some kind qualitative statement about how ugly the R_nget. Which brings us to:

                                                  Connection to the GGMS decomposition

                                                  I got to thinking about this again now and decided to post on MO of it because of Noah's question about the vertices of a variation of GIT problem, where this arrangement is lurking around -- see there for more detail. Allen's brief comment there prompted me to skim some of his and related papers to that general area, and I found the introduction to Positroid varieties I: juggling and geometry most enlightening, together with the discussion at Noah's question could give another suggestion why R_nis perhaps intractable. Briefly:

                                                  The arrangement R_nis a natural extension of theA_narrangement. One common description of theA_narrangement is as the\\binom{n+1}{2}hyperplanesy_i-y_j=0, i,j\\in [n]andy_i=0, i\\in [n]. However, one can consider the triangular change of variables

                                                  $$y_k\\mapsto \\sum_{j\\leq k} y_j$$.

                                                  .

                                                  This changes the hyperplanes to\n

                                                  $$\\sum_{i\\leq k \\leq j} y_k=0.$$

                                                  These hyperplanes are no longer invariant under permutation of the coordinates, and if we proceed to add the entire S_norbit of them, we get the resonance arrangementR_n.

                                                  From the discussion on Noah's question and the introduction to the Positroid paper, we see that this manipulation is a shadow of the GGMS decomposition of the Grassmannian, and describing this decomposition in general seems to be intractable in that it requires identifying whether matroids are representable or not. So, what I'd really like to know how is much of the \"GGMS abyss\", as they refer to it, is reflected in the resonance arrangement? Is it hopeless to describe and count its chambers?

                                                  ", "statics": {"title": 3, "paragraph": 13, "paragraph.text": 34, "paragraph.equation-inline": 21, "equation-interline": 3}, "url": "https://mathoverflow.net/questions/62764/a-natural-refinement-of-the-a-n-arrangement-is-to-consider-all-2n-1-hyperpl?rq=1", "content": "## The short version\n\nHere is an extremely natural hyperplane arrangement in $\\mathbb{R}^n$ , which I will call $R_n$ for resonance arrangement.\n\nLet $x_i$ be the standard coordinates on $\\mathbb{R}^n$ . For each nonempty $I\\subseteq [n]=\\{1,\\dots,n\\}$ , define the hyperplane $H_I$ to be the hyperplane given by\n\n$$\n\\sum_{i\\in I} x_i=0.\n$$\n\nThe resonance arrangement is given by all $2^n-1$ hyperplanes $H_I$ . The arrangement $R_n$ is natural enough that it arises in many contexts -- to first order, my question is simply: have you come across it yourself?\n\nThis feels rather vague to be a good question, and after giving some background on where I've seen this I'll try to be a bit more specific about what I'm looking for, but my point is this arrangement has a rather simple and natural definition, and so crops up in multiple places, and I'd be curious to hear about more of them even if you can't specifically connect it to what follows.\n\n## What I knew until this week\n\nI came across this arrangement in my study of double Hurwitz numbers -- they are piecewise polynomial, and the chambers of the resonance arrangement are the chambers of polynomiality. I don't want to go into this much more, as it's unimportant to most of what follows Though I will say that conjecturally double Hurwitz numbers could be related to compactified Picard varieties in a way which would connect this arrangement up with birational transformations of those. Also, the name \"resonance arrangement\" was essentially introduce in this context, by Shadrin, Shapiro and Vainshtein.\n\nIt apparently comes up in physics -- I only know this because the number of regions of $R_n$ , starting at $n=2$ , is 2, 6, 32, 370, 11292, 1066044, 347326352 ... Sloane sequence A034997, which you will see was entered as \"Number of Generalized Retarded Functions in Quantum Field Theory\" by a physicist.\n\nYou might expect by that rate of growth that this hyperplane arrangement is completely intractable, and more specifically, what I would love from an answer is some kind qualitative statement about how ugly the $R_n$ get. Which brings us to:\n\n## Connection to the GGMS decomposition\n\nI got to thinking about this again now and decided to post on MO of it because of Noah's question about the vertices of a variation of GIT problem, where this arrangement is lurking around -- see there for more detail. Allen's brief comment there prompted me to skim some of his and related papers to that general area, and I found the introduction to Positroid varieties I: juggling and geometry most enlightening, together with the discussion at Noah's question could give another suggestion why $R_n$ is perhaps intractable. Briefly:\n\nThe arrangement $R_n$ is a natural extension of the $A_n$ arrangement. One common description of the $A_n$ arrangement is as the $\\binom{n+1}{2}$ hyperplanes $y_i-y_j=0, i,j\\in [n]$ and $y_i=0, i\\in [n]$ . However, one can consider the triangular change of variables\n\n$$\ny_k\\mapsto \\sum_{j\\leq k} y_j\n$$\n\n.\n\nThis changes the hyperplanes to\n\n$$\n\\sum_{i\\leq k \\leq j} y_k=0.\n$$\n\nThese hyperplanes are no longer invariant under permutation of the coordinates, and if we proceed to add the entire $S_n$ orbit of them, we get the resonance arrangement $R_n$ .\n\nFrom the discussion on Noah's question and the introduction to the Positroid paper, we see that this manipulation is a shadow of the GGMS decomposition of the Grassmannian, and describing this decomposition in general seems to be intractable in that it requires identifying whether matroids are representable or not. So, what I'd really like to know how is much of the \"GGMS abyss\", as they refer to it, is reflected in the resonance arrangement? Is it hopeless to describe and count its chambers?\n", "html": "\n\n\n\n\n \n\n \n\n co.combinatorics - A natural refinement of the $A_n$ arrangement is to consider all $2^n-1$ hyperplanes given by the sums of the coordinate functions. Have you seen this arrangement? Is it completely intractable? - MathOverflow\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n\n\n\n \n \n\n\n\n\n \n\n\n \n\n\n \n \n\n\n \n \n\n\n\n \n \n \n \n\n\n
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                                                  The short version

                                                  \n

                                                  Here is an extremely natural hyperplane arrangement in $\\mathbb{R}^n$, which I will call $R_n$ for resonance arrangement.

                                                  \n

                                                  Let $x_i$ be the standard coordinates on $\\mathbb{R}^n$. For each nonempty $I\\subseteq [n]=\\{1,\\dots,n\\}$, define the hyperplane $H_I$ to be the hyperplane given by\n$$\\sum_{i\\in I} x_i=0.$$\nThe resonance arrangement is given by all $2^n-1$ hyperplanes $H_I$. The arrangement $R_n$ is natural enough that it arises in many contexts -- to first order, my question is simply: have you come across it yourself?

                                                  \n

                                                  This feels rather vague to be a good question, and after giving some background on where I've seen this I'll try to be a bit more specific about what I'm looking for, but my point is this arrangement has a rather simple and natural definition, and so crops up in multiple places, and I'd be curious to hear about more of them even if you can't specifically connect it to what follows.

                                                  \n

                                                  What I knew until this week

                                                  \n

                                                  I came across this arrangement in my study of double Hurwitz numbers -- they are piecewise polynomial, and the chambers of the resonance arrangement are the chambers of polynomiality. I don't want to go into this much more, as it's unimportant to most of what follows Though I will say that conjecturally double Hurwitz numbers could be related to compactified Picard varieties in a way which would connect this arrangement up with birational transformations of those. Also, the name "resonance arrangement" was essentially introduce in this context, by Shadrin, Shapiro and Vainshtein.

                                                  \n

                                                  It apparently comes up in physics -- I only know this because the number of regions of $R_n$, starting at $n=2$, is 2, 6, 32, 370, 11292, 1066044, 347326352 ... Sloane sequence A034997, which you will see was entered as "Number of Generalized Retarded Functions in Quantum Field Theory" by a physicist.

                                                  \n

                                                  You might expect by that rate of growth that this hyperplane arrangement is completely intractable, and more specifically, what I would love from an answer is some kind qualitative statement about how ugly the $R_n$ get. Which brings us to:

                                                  \n

                                                  Connection to the GGMS decomposition

                                                  \n

                                                  I got to thinking about this again now and decided to post on MO of it because of Noah's question about the vertices of a variation of GIT problem, where this arrangement is lurking around -- see there for more detail. Allen's brief comment there prompted me to skim some of his and related papers to that general area, and I found the introduction to Positroid varieties I: juggling and geometry most enlightening, together with the discussion at Noah's question could give another suggestion why $R_n$ is perhaps intractable. Briefly:

                                                  \n

                                                  The arrangement $R_n$ is a natural extension of the $A_n$ arrangement. One common description of the $A_n$ arrangement is as the $\\binom{n+1}{2}$ hyperplanes $y_i-y_j=0, i,j\\in [n]$ and $y_i=0, i\\in [n]$. However, one can consider the triangular change of variables\n$$y_k\\mapsto \\sum_{j\\leq k} y_j$$.

                                                  \n

                                                  This changes the hyperplanes to\n$$\\sum_{i\\leq k \\leq j} y_k=0.$$

                                                  \n

                                                  These hyperplanes are no longer invariant under permutation of the coordinates, and if we proceed to add the entire $S_n$ orbit of them, we get the resonance arrangement $R_n$.

                                                  \n

                                                  From the discussion on Noah's question and the introduction to the Positroid paper, we see that this manipulation is a shadow of the GGMS decomposition of the Grassmannian, and describing this decomposition in general seems to be intractable in that it requires identifying whether matroids are representable or not. So, what I'd really like to know how is much of the "GGMS abyss", as they refer to it, is reflected in the resonance arrangement? Is it hopeless to describe and count its chambers?

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                                                  \n edited Jun 15, 2020 at 7:27\n
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                                                  \"Community's
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                                                  \n CommunityBot\n
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                                                  \n asked Apr 23, 2011 at 18:36\n
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                                                  \n Paul JohnsonPaul Johnson\n
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                                                    \n $\\begingroup$\n Several of my students and I have thought about the characteristic polynomial and number of regions of this arrangement over the years, but we never came up with anything interesting. \n $\\endgroup$\n
                                                    \n– Richard Stanley\n
                                                    \n \n Commented\n Apr 23, 2011 at 19:09\n \n
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                                                    \n $\\begingroup$\n -1 for not having a more specific question, but +2 for lots of details and interesting discussion and good title. Also, I think there's something funny with the TeX between "However, one can consider the triangular ..." and "These hyperplanes are no longer ...".\n $\\endgroup$\n
                                                    \n– Theo Johnson-Freyd\n
                                                    \n \n Commented\n Apr 23, 2011 at 21:04\n \n
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                                                    \n $\\begingroup$\n Think I fixed the TeX. And I was a bit torn about it not being specific enough myself, but it was never going to get asked if I kept worrying about it. Next time will be better.\n $\\endgroup$\n
                                                    \n– Paul Johnson\n
                                                    \n \n Commented\n Apr 23, 2011 at 21:48\n \n
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                                                    \n $\\begingroup$\n @Theo: Also, you are half responsible for the long title: somewhere you mentioned that a title can be about a "tweet and a half" long, and that was an odd enough description that it stuck in my head. Good work.\n $\\endgroup$\n
                                                    \n– Paul Johnson\n
                                                    \n \n Commented\n Apr 23, 2011 at 21:56\n \n
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                                                    \n $\\begingroup$\n Just commenting to thank Richard -- this is one kind of thing I wanted to hear: that people had put some real thought into it and not gotten anything interesting. I wasn't up-voting that you didn't get anywhere.\n $\\endgroup$\n
                                                    \n– Paul Johnson\n
                                                    \n \n Commented\n Apr 27, 2011 at 19:41\n \n
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                                                  Here's what I know about this arrangement.

                                                  \n\n

                                                  Regarding the number of chambers in this arrangment, Zuev obtained the lower bound $2^{(1-o(1))n^2}$. The proof uses Zaslavsky's theorem and a difficult estimate due to Odlyzko.

                                                  \n\n

                                                  http://www.doiserbia.nb.rs/img/doi/0350-1302/2007/0350-13020796129K.pdf describes an improvement to that lower bound (including references to Zuev's and Odlyzko's articles).

                                                  \n\n

                                                  http://arxiv.org/pdf/1209.2309v1.pdf studies a closely related arrangement and gives a related (but weaker) lower bound using a very elegant method. It turns out the arrangment becomes much easier mod $2$ (even than one might expect).

                                                  \n\n

                                                  Whenever one has a central hyperplane arrangement one has a zonotope dual to it. Klivans and Reiner [ http://arxiv.org/pdf/math/0610787v2.pdf ] fix $k$ and look at the zonotope (Minkowski sum of the line segments) generated by all 0/1-vectors of length $n$ with exactly $k$ ones. In particular they are interested in the zonotope considered as symmetric polynomial (the sum $\\sum x^m$ over all lattice points $m$ in the zonotope).\nThus the Minkowski sum of their zonotopes over $1\\leq k \\leq n$ is the zonotope dual to your arrangement. Degree sequences of hypergraphs correspond to integer points in this latter zonotope. Surprisingly the converse is false! This was showed by Liu [http://arxiv.org/abs/1201.5989 ].

                                                  \n\n

                                                  The vertices in the zonotope, and thus the regions in the arrangement, correspond almost to linear threshold hypergraphs (better known as linear threshold (Boolean) functions). A linear threshold hypergraph is determined by $n+1$ real potentially negative numbers $q,w_1,\\dots,w_n$; any set $S\\subseteq [n]$ with $\\sum_{i\\in S} x_i \\leq q$ is declared an edge. \"Almost\" above means that the vertices in this zonotope correspond exactly to the linear threshold hypergraphs that can be given with $q = 0$.

                                                  \n\n

                                                  What if $q \\neq 0$? Given a hypergraph $\\mathcal{H}$ with $s$ simplices and degree sequence $v_1,\\dots,v_n$, one can look at the values $s-2v_i$, $1\\leq i\\leq n$. If $\\mathcal{H}$ happens to be down-closed, this is what game theorists call the Banzhaf value of the game represented by $\\mathcal{H}$ and what computer scientists call the Boolean influence of the Boolean function represented by $\\mathcal{H}$. Anyway it is more natural to look at the vector $(s,v_1,\\dots,v_n)$ rather than just $(v_1,\\dots,v_n)$. This corresponds to looking instead at the zonotope generated by $(1,0,0),(1,0,1),(1,1,0),(1,1,1)$ et.c. (all binary vectors of length $n$, prepended by $1$). The vertices (obviously more than when we didn't have the leading 1's in the generators) in this zonotope correspond exactly to linear threshold hypergraphs. Since each $v_i$ is an integer $0$ and $2^{n-1}$, and $s$ is between $2^n$ there are at most $2^{n + n(n-1)} = 2^{n^2}$ of them, giving an almost matching upper bound to the lower bound above.

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                                                  The latter zonotope is briefly mentioned in the end of the article http://arxiv.org/pdf/0908.4425.pdf .

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                                                  I remember having seen a reference to a paper of Terao computing the characteristic polynomial of the arrangemnt for some small values, but cannot locate that reference right now.

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                                                  \n edited Jun 19, 2013 at 22:13\n
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                                                  \n answered Jun 19, 2013 at 22:06\n
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                                                  \"Erik
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                                                    \n $\\begingroup$\n Thanks! Haven't digested this at all, but was hoping that there'd be more about this in the literature, and you've supplied that well.\n $\\endgroup$\n
                                                    \n– Paul Johnson\n
                                                    \n \n Commented\n Jun 21, 2013 at 12:20\n \n
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                                                  \n\n\n\n\n \n\n\n\n\n \n \n \n \n \n\n \n\n\n\n \n \n"} +{"content_list": [[{"type": "title", "raw_content": "

                                                  The short version

                                                  ", "content": {"title_content": "The short version", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                  Here is an extremely natural hyperplane arrangement in \\mathbb{R}^n, which I will call R_n for resonance arrangement.

                                                  ", "content": [{"c": "Here is an extremely natural hyperplane arrangement in", "t": "text"}, {"c": "\\mathbb{R}^n", "t": "equation-inline"}, {"c": ", which I will call", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": "for resonance arrangement.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  Let x_i be the standard coordinates on \\mathbb{R}^n. For each nonempty I\\subseteq [n]=\\{1,\\dots,n\\}, define the hyperplane H_I to be the hyperplane given by\n

                                                  ", "content": [{"c": "Let", "t": "text"}, {"c": "x_i", "t": "equation-inline"}, {"c": "be the standard coordinates on", "t": "text"}, {"c": "\\mathbb{R}^n", "t": "equation-inline"}, {"c": ". For each nonempty", "t": "text"}, {"c": "I\\subseteq [n]=\\{1,\\dots,n\\}", "t": "equation-inline"}, {"c": ", define the hyperplane", "t": "text"}, {"c": "H_I", "t": "equation-inline"}, {"c": "to be the hyperplane given by", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\sum_{i\\in I} x_i=0.", "content": {"math_content": "\\sum_{i\\in I} x_i=0.", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                  \nThe resonance arrangement is given by all 2^n-1 hyperplanes H_I. The arrangement R_n is natural enough that it arises in many contexts -- to first order, my question is simply: have you come across it yourself?

                                                  ", "content": [{"c": "The resonance arrangement is given by all", "t": "text"}, {"c": "2^n-1", "t": "equation-inline"}, {"c": "hyperplanes", "t": "text"}, {"c": "H_I", "t": "equation-inline"}, {"c": ". The arrangement", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": "is natural enough that it arises in many contexts -- to first order, my question is simply: have you come across it yourself?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  This feels rather vague to be a good question, and after giving some background on where I've seen this I'll try to be a bit more specific about what I'm looking for, but my point is this arrangement has a rather simple and natural definition, and so crops up in multiple places, and I'd be curious to hear about more of them even if you can't specifically connect it to what follows.

                                                  ", "content": [{"c": "This feels rather vague to be a good question, and after giving some background on where I've seen this I'll try to be a bit more specific about what I'm looking for, but my point is this arrangement has a rather simple and natural definition, and so crops up in multiple places, and I'd be curious to hear about more of them even if you can't specifically connect it to what follows.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                  What I knew until this week

                                                  ", "content": {"title_content": "What I knew until this week", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                  I came across this arrangement in my study of double Hurwitz numbers -- they are piecewise polynomial, and the chambers of the resonance arrangement are the chambers of polynomiality. I don't want to go into this much more, as it's unimportant to most of what follows Though I will say that conjecturally double Hurwitz numbers could be related to compactified Picard varieties in a way which would connect this arrangement up with birational transformations of those. Also, the name \"resonance arrangement\" was essentially introduce in this context, by Shadrin, Shapiro and Vainshtein.

                                                  ", "content": [{"c": "I came across this arrangement in my study of double Hurwitz numbers -- they are piecewise polynomial, and the chambers of the resonance arrangement are the chambers of polynomiality. I don't want to go into this much more, as it's unimportant to most of what follows Though I will say that conjecturally double Hurwitz numbers could be related to compactified Picard varieties in a way which would connect this arrangement up with birational transformations of those. Also, the name \"resonance arrangement\" was essentially introduce in this context, by Shadrin, Shapiro and Vainshtein.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  It apparently comes up in physics -- I only know this because the number of regions of R_n, starting at n=2, is 2, 6, 32, 370, 11292, 1066044, 347326352 ... Sloane sequence A034997, which you will see was entered as \"Number of Generalized Retarded Functions in Quantum Field Theory\" by a physicist.

                                                  ", "content": [{"c": "It apparently comes up in physics -- I only know this because the number of regions of", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": ", starting at", "t": "text"}, {"c": "n=2", "t": "equation-inline"}, {"c": ", is 2, 6, 32, 370, 11292, 1066044, 347326352 ... Sloane sequence A034997, which you will see was entered as \"Number of Generalized Retarded Functions in Quantum Field Theory\" by a physicist.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  You might expect by that rate of growth that this hyperplane arrangement is completely intractable, and more specifically, what I would love from an answer is some kind qualitative statement about how ugly the R_n get. Which brings us to:

                                                  ", "content": [{"c": "You might expect by that rate of growth that this hyperplane arrangement is completely intractable, and more specifically, what I would love from an answer is some kind qualitative statement about how ugly the", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": "get. Which brings us to:", "t": "text"}]}, {"type": "title", "raw_content": "

                                                  Connection to the GGMS decomposition

                                                  ", "content": {"title_content": "Connection to the GGMS decomposition", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                  I got to thinking about this again now and decided to post on MO of it because of Noah's question about the vertices of a variation of GIT problem, where this arrangement is lurking around -- see there for more detail. Allen's brief comment there prompted me to skim some of his and related papers to that general area, and I found the introduction to Positroid varieties I: juggling and geometry most enlightening, together with the discussion at Noah's question could give another suggestion why R_n is perhaps intractable. Briefly:

                                                  ", "content": [{"c": "I got to thinking about this again now and decided to post on MO of it because of Noah's question about the vertices of a variation of GIT problem, where this arrangement is lurking around -- see there for more detail. Allen's brief comment there prompted me to skim some of his and related papers to that general area, and I found the introduction to Positroid varieties I: juggling and geometry most enlightening, together with the discussion at Noah's question could give another suggestion why", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": "is perhaps intractable. Briefly:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  The arrangement R_n is a natural extension of the A_n arrangement. One common description of the A_n arrangement is as the \\binom{n+1}{2} hyperplanes y_i-y_j=0, i,j\\in [n] and y_i=0, i\\in [n]. However, one can consider the triangular change of variables\n

                                                  ", "content": [{"c": "The arrangement", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": "is a natural extension of the", "t": "text"}, {"c": "A_n", "t": "equation-inline"}, {"c": "arrangement. One common description of the", "t": "text"}, {"c": "A_n", "t": "equation-inline"}, {"c": "arrangement is as the", "t": "text"}, {"c": "\\binom{n+1}{2}", "t": "equation-inline"}, {"c": "hyperplanes", "t": "text"}, {"c": "y_i-y_j=0, i,j\\in [n]", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "y_i=0, i\\in [n]", "t": "equation-inline"}, {"c": ". However, one can consider the triangular change of variables", "t": "text"}]}, {"type": "equation-interline", "raw_content": "y_k\\mapsto \\sum_{j\\leq k} y_j", "content": {"math_content": "y_k\\mapsto \\sum_{j\\leq k} y_j", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                  .

                                                  ", "content": [{"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  This changes the hyperplanes to\n

                                                  ", "content": [{"c": "This changes the hyperplanes to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\sum_{i\\leq k \\leq j} y_k=0.", "content": {"math_content": "\\sum_{i\\leq k \\leq j} y_k=0.", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                  These hyperplanes are no longer invariant under permutation of the coordinates, and if we proceed to add the entire S_n orbit of them, we get the resonance arrangement R_n.

                                                  ", "content": [{"c": "These hyperplanes are no longer invariant under permutation of the coordinates, and if we proceed to add the entire", "t": "text"}, {"c": "S_n", "t": "equation-inline"}, {"c": "orbit of them, we get the resonance arrangement", "t": "text"}, {"c": "R_n", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                  From the discussion on Noah's question and the introduction to the Positroid paper, we see that this manipulation is a shadow of the GGMS decomposition of the Grassmannian, and describing this decomposition in general seems to be intractable in that it requires identifying whether matroids are representable or not. So, what I'd really like to know how is much of the \"GGMS abyss\", as they refer to it, is reflected in the resonance arrangement? Is it hopeless to describe and count its chambers?

                                                  ", "content": [{"c": "From the discussion on Noah's question and the introduction to the Positroid paper, we see that this manipulation is a shadow of the GGMS decomposition of the Grassmannian, and describing this decomposition in general seems to be intractable in that it requires identifying whether matroids are representable or not. So, what I'd really like to know how is much of the \"GGMS abyss\", as they refer to it, is reflected in the resonance arrangement? Is it hopeless to describe and count its chambers?", "t": "text"}]}, {"type": "list", "raw_content": "
                                                    ", "content": {"items": [], "ordered": false}}, {"type": "list", "raw_content": "
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                                                      Several of my students and I have thought about the characteristic polynomial and number of regions of this arrangement over the years, but we never came up with anything interesting. CommentedApr 23, 2011 at 19:09
                                                    • -1 for not having a more specific question, but +2 for lots of details and interesting discussion and good title. Also, I think there's something funny with the TeX between \"However, one can consider the triangular ...\" and \"These hyperplanes are no longer ...\".CommentedApr 23, 2011 at 21:04
                                                    • Think I fixed the TeX. And I was a bit torn about it not being specific enough myself, but it was never going to get asked if I kept worrying about it. Next time will be better.CommentedApr 23, 2011 at 21:48
                                                    • @Theo: Also, you are half responsible for the long title: somewhere you mentioned that a title can be about a \"tweet and a half\" long, and that was an odd enough description that it stuck in my head. Good work.CommentedApr 23, 2011 at 21:56
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                                                      Just commenting to thank Richard -- this is one kind of thing I wanted to hear: that people had put some real thought into it and not gotten anything interesting. I wasn't up-voting that you didn't get anywhere.CommentedApr 27, 2011 at 19:41
                                                    ", "content": {"items": [[[{"c": "8", "t": "text"}, {"c": "Several of my students and I have thought about the characteristic polynomial and number of regions of this arrangement over the years, but we never came up with anything interesting. ", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Apr 23, 2011 at 19:09", "t": "text"}]], [[{"c": "-1 for not having a more specific question, but +2 for lots of details and interesting discussion and good title. Also, I think there's something funny with the TeX between \"However, one can consider the triangular ...\" and \"These hyperplanes are no longer ...\".", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Apr 23, 2011 at 21:04", "t": "text"}]], [[{"c": "Think I fixed the TeX. And I was a bit torn about it not being specific enough myself, but it was never going to get asked if I kept worrying about it. Next time will be better.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Apr 23, 2011 at 21:48", "t": "text"}]], [[{"c": "@Theo: Also, you are half responsible for the long title: somewhere you mentioned that a title can be about a \"tweet and a half\" long, and that was an odd enough description that it stuck in my head. Good work.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Apr 23, 2011 at 21:56", "t": "text"}]], [[{"c": "1", "t": "text"}, {"c": "Just commenting to thank Richard -- this is one kind of thing I wanted to hear: that people had put some real thought into it and not gotten anything interesting. I wasn't up-voting that you didn't get anywhere.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Apr 27, 2011 at 19:41", "t": "text"}]]], "ordered": false}}, {"type": "title", "raw_content": "

                                                    \n 1 Answer\n \n

                                                    ", "content": {"title_content": "1 Answer", "level": "2"}}, {"type": "paragraph", "raw_content": "
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                                                    ", "content": [{"c": "8", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                    Here's what I know about this arrangement.

                                                    ", "content": [{"c": "Here's what I know about this arrangement.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                    Regarding the number of chambers in this arrangment, Zuev obtained the lower bound 2^{(1-o(1))n^2}. The proof uses Zaslavsky's theorem and a difficult estimate due to Odlyzko.

                                                    ", "content": [{"c": "Regarding the number of chambers in this arrangment, Zuev obtained the lower bound", "t": "text"}, {"c": "2^{(1-o(1))n^2}", "t": "equation-inline"}, {"c": ". The proof uses Zaslavsky's theorem and a difficult estimate due to Odlyzko.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                    http://www.doiserbia.nb.rs/img/doi/0350-1302/2007/0350-13020796129K.pdf describes an improvement to that lower bound (including references to Zuev's and Odlyzko's articles).

                                                    ", "content": [{"c": "http://www.doiserbia.nb.rs/img/doi/0350-1302/2007/0350-13020796129K.pdf describes an improvement to that lower bound (including references to Zuev's and Odlyzko's articles).", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                    http://arxiv.org/pdf/1209.2309v1.pdf studies a closely related arrangement and gives a related (but weaker) lower bound using a very elegant method. It turns out the arrangment becomes much easier mod $2$ (even than one might expect).

                                                    ", "content": [{"c": "http://arxiv.org/pdf/1209.2309v1.pdf studies a closely related arrangement and gives a related (but weaker) lower bound using a very elegant method. It turns out the arrangment becomes much easier mod $2$ (even than one might expect).", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                    Whenever one has a central hyperplane arrangement one has a zonotope dual to it. Klivans and Reiner [ http://arxiv.org/pdf/math/0610787v2.pdf ] fix $k$ and look at the zonotope (Minkowski sum of the line segments) generated by all 0/1-vectors of length $n$ with exactly $k$ ones. In particular they are interested in the zonotope considered as symmetric polynomial (the sum $\\sum x^m$ over all lattice points $m$ in the zonotope).\nThus the Minkowski sum of their zonotopes over $1\\leq k \\leq n$ is the zonotope dual to your arrangement. Degree sequences of hypergraphs correspond to integer points in this latter zonotope. Surprisingly the converse is false! This was showed by Liu [http://arxiv.org/abs/1201.5989 ].

                                                    ", "content": [{"c": "Whenever one has a central hyperplane arrangement one has a zonotope dual to it. Klivans and Reiner [ http://arxiv.org/pdf/math/0610787v2.pdf] fix $k$ and look at the zonotope (Minkowski sum of the line segments) generated by all 0/1-vectors of length $n$ with exactly $k$ ones. In particular they are interested in the zonotope considered as symmetric polynomial (the sum $\\sum x^m$ over all lattice points $m$ in the zonotope).\nThus the Minkowski sum of their zonotopes over $1\\leq k \\leq n$ is the zonotope dual to your arrangement. Degree sequences of hypergraphs correspond to integer points in this latter zonotope. Surprisingly the converse is false! This was showed by Liu [ http://arxiv.org/abs/1201.5989].", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                    The vertices in the zonotope, and thus the regions in the arrangement, correspond almost to linear threshold hypergraphs (better known as linear threshold (Boolean) functions). A linear threshold hypergraph is determined by n+1 real potentially negative numbers q,w_1,\\dots,w_n; any set S\\subseteq [n] with \\sum_{i\\in S} x_i \\leq q is declared an edge. \"Almost\" above means that the vertices in this zonotope correspond exactly to the linear threshold hypergraphs that can be given with q = 0.

                                                    ", "content": [{"c": "The vertices in the zonotope, and thus the regions in the arrangement, correspond almost to linear threshold hypergraphs (better known as linear threshold (Boolean) functions). A linear threshold hypergraph is determined by", "t": "text"}, {"c": "n+1", "t": "equation-inline"}, {"c": "real potentially negative numbers", "t": "text"}, {"c": "q,w_1,\\dots,w_n", "t": "equation-inline"}, {"c": "; any set", "t": "text"}, {"c": "S\\subseteq [n]", "t": "equation-inline"}, {"c": "with", "t": "text"}, {"c": "\\sum_{i\\in S} x_i \\leq q", "t": "equation-inline"}, {"c": "is declared an edge. \"Almost\" above means that the vertices in this zonotope correspond exactly to the linear threshold hypergraphs that can be given with", "t": "text"}, {"c": "q = 0", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                    What if q \\neq 0? Given a hypergraph \\mathcal{H} with s simplices and degree sequence v_1,\\dots,v_n, one can look at the values s-2v_i, 1\\leq i\\leq n. If \\mathcal{H} happens to be down-closed, this is what game theorists call the Banzhaf value of the game represented by \\mathcal{H} and what computer scientists call the Boolean influence of the Boolean function represented by \\mathcal{H}. Anyway it is more natural to look at the vector (s,v_1,\\dots,v_n) rather than just (v_1,\\dots,v_n). This corresponds to looking instead at the zonotope generated by (1,0,0),(1,0,1),(1,1,0),(1,1,1) et.c. (all binary vectors of length n, prepended by 1). The vertices (obviously more than when we didn't have the leading 1's in the generators) in this zonotope correspond exactly to linear threshold hypergraphs. Since each v_i is an integer 0 and 2^{n-1}, and s is between 2^n there are at most 2^{n + n(n-1)} = 2^{n^2} of them, giving an almost matching upper bound to the lower bound above.

                                                    ", "content": [{"c": "What if", "t": "text"}, {"c": "q \\neq 0", "t": "equation-inline"}, {"c": "? Given a hypergraph", "t": "text"}, {"c": "\\mathcal{H}", "t": "equation-inline"}, {"c": "with", "t": "text"}, {"c": "s", "t": "equation-inline"}, {"c": "simplices and degree sequence", "t": "text"}, {"c": "v_1,\\dots,v_n", "t": "equation-inline"}, {"c": ", one can look at the values", "t": "text"}, {"c": "s-2v_i", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "1\\leq i\\leq n", "t": "equation-inline"}, {"c": ". If", "t": "text"}, {"c": "\\mathcal{H}", "t": "equation-inline"}, {"c": "happens to be down-closed, this is what game theorists call the Banzhaf value of the game represented by", "t": "text"}, {"c": "\\mathcal{H}", "t": "equation-inline"}, {"c": "and what computer scientists call the Boolean influence of the Boolean function represented by", "t": "text"}, {"c": "\\mathcal{H}", "t": "equation-inline"}, {"c": ". Anyway it is more natural to look at the vector", "t": "text"}, {"c": "(s,v_1,\\dots,v_n)", "t": "equation-inline"}, {"c": "rather than just", "t": "text"}, {"c": "(v_1,\\dots,v_n)", "t": "equation-inline"}, {"c": ". This corresponds to looking instead at the zonotope generated by", "t": "text"}, {"c": "(1,0,0),(1,0,1),(1,1,0),(1,1,1)", "t": "equation-inline"}, {"c": "et.c. (all binary vectors of length", "t": "text"}, {"c": "n", "t": "equation-inline"}, {"c": ", prepended by", "t": "text"}, {"c": "1", "t": "equation-inline"}, {"c": "). The vertices (obviously more than when we didn't have the leading 1's in the generators) in this zonotope correspond exactly to linear threshold hypergraphs. Since each", "t": "text"}, {"c": "v_i", "t": "equation-inline"}, {"c": "is an integer", "t": "text"}, {"c": "0", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "2^{n-1}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "s", "t": "equation-inline"}, {"c": "is between", "t": "text"}, {"c": "2^n", "t": "equation-inline"}, {"c": "there are at most", "t": "text"}, {"c": "2^{n + n(n-1)} = 2^{n^2}", "t": "equation-inline"}, {"c": "of them, giving an almost matching upper bound to the lower bound above.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                    The latter zonotope is briefly mentioned in the end of the article http://arxiv.org/pdf/0908.4425.pdf .

                                                    ", "content": [{"c": "The latter zonotope is briefly mentioned in the end of the article http://arxiv.org/pdf/0908.4425.pdf.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                    I remember having seen a reference to a paper of Terao computing the characteristic polynomial of the arrangemnt for some small values, but cannot locate that reference right now.

                                                    ", "content": [{"c": "I remember having seen a reference to a paper of Terao computing the characteristic polynomial of the arrangemnt for some small values, but cannot locate that reference right now.", "t": "text"}]}, {"type": "list", "raw_content": "
                                                    • Thanks! Haven't digested this at all, but was hoping that there'd be more about this in the literature, and you've supplied that well.CommentedJun 21, 2013 at 12:20
                                                    ", "content": {"items": [[[{"c": "Thanks! Haven't digested this at all, but was hoping that there'd be more about this in the literature, and you've supplied that well.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Jun 21, 2013 at 12:20", "t": "text"}]]], "ordered": false}}, {"type": "title", "raw_content": "

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                                                      The short version

                                                      Here is an extremely natural hyperplane arrangement in \\mathbb{R}^n, which I will call R_n for resonance arrangement.

                                                      Let x_i be the standard coordinates on \\mathbb{R}^n. For each nonempty I\\subseteq [n]=\\{1,\\dots,n\\}, define the hyperplane H_I to be the hyperplane given by\n

                                                      \\sum_{i\\in I} x_i=0.

                                                      \nThe resonance arrangement is given by all 2^n-1 hyperplanes H_I. The arrangement R_n is natural enough that it arises in many contexts -- to first order, my question is simply: have you come across it yourself?

                                                      This feels rather vague to be a good question, and after giving some background on where I've seen this I'll try to be a bit more specific about what I'm looking for, but my point is this arrangement has a rather simple and natural definition, and so crops up in multiple places, and I'd be curious to hear about more of them even if you can't specifically connect it to what follows.

                                                      What I knew until this week

                                                      I came across this arrangement in my study of double Hurwitz numbers -- they are piecewise polynomial, and the chambers of the resonance arrangement are the chambers of polynomiality. I don't want to go into this much more, as it's unimportant to most of what follows Though I will say that conjecturally double Hurwitz numbers could be related to compactified Picard varieties in a way which would connect this arrangement up with birational transformations of those. Also, the name \"resonance arrangement\" was essentially introduce in this context, by Shadrin, Shapiro and Vainshtein.

                                                      It apparently comes up in physics -- I only know this because the number of regions of R_n, starting at n=2, is 2, 6, 32, 370, 11292, 1066044, 347326352 ... Sloane sequence A034997, which you will see was entered as \"Number of Generalized Retarded Functions in Quantum Field Theory\" by a physicist.

                                                      You might expect by that rate of growth that this hyperplane arrangement is completely intractable, and more specifically, what I would love from an answer is some kind qualitative statement about how ugly the R_n get. Which brings us to:

                                                      Connection to the GGMS decomposition

                                                      I got to thinking about this again now and decided to post on MO of it because of Noah's question about the vertices of a variation of GIT problem, where this arrangement is lurking around -- see there for more detail. Allen's brief comment there prompted me to skim some of his and related papers to that general area, and I found the introduction to Positroid varieties I: juggling and geometry most enlightening, together with the discussion at Noah's question could give another suggestion why R_n is perhaps intractable. Briefly:

                                                      The arrangement R_n is a natural extension of the A_n arrangement. One common description of the A_n arrangement is as the \\binom{n+1}{2} hyperplanes y_i-y_j=0, i,j\\in [n] and y_i=0, i\\in [n]. However, one can consider the triangular change of variables\n

                                                      y_k\\mapsto \\sum_{j\\leq k} y_j

                                                      .

                                                      This changes the hyperplanes to\n

                                                      \\sum_{i\\leq k \\leq j} y_k=0.

                                                      These hyperplanes are no longer invariant under permutation of the coordinates, and if we proceed to add the entire S_n orbit of them, we get the resonance arrangement R_n.

                                                      From the discussion on Noah's question and the introduction to the Positroid paper, we see that this manipulation is a shadow of the GGMS decomposition of the Grassmannian, and describing this decomposition in general seems to be intractable in that it requires identifying whether matroids are representable or not. So, what I'd really like to know how is much of the \"GGMS abyss\", as they refer to it, is reflected in the resonance arrangement? Is it hopeless to describe and count its chambers?

                                                        • 8
                                                          Several of my students and I have thought about the characteristic polynomial and number of regions of this arrangement over the years, but we never came up with anything interesting. CommentedApr 23, 2011 at 19:09
                                                        • -1 for not having a more specific question, but +2 for lots of details and interesting discussion and good title. Also, I think there's something funny with the TeX between \"However, one can consider the triangular ...\" and \"These hyperplanes are no longer ...\".CommentedApr 23, 2011 at 21:04
                                                        • Think I fixed the TeX. And I was a bit torn about it not being specific enough myself, but it was never going to get asked if I kept worrying about it. Next time will be better.CommentedApr 23, 2011 at 21:48
                                                        • @Theo: Also, you are half responsible for the long title: somewhere you mentioned that a title can be about a \"tweet and a half\" long, and that was an odd enough description that it stuck in my head. Good work.CommentedApr 23, 2011 at 21:56
                                                        • 1
                                                          Just commenting to thank Richard -- this is one kind of thing I wanted to hear: that people had put some real thought into it and not gotten anything interesting. I wasn't up-voting that you didn't get anywhere.CommentedApr 27, 2011 at 19:41

                                                        \n 1 Answer\n \n

                                                        \n8

                                                        Here's what I know about this arrangement.

                                                        Regarding the number of chambers in this arrangment, Zuev obtained the lower bound 2^{(1-o(1))n^2}. The proof uses Zaslavsky's theorem and a difficult estimate due to Odlyzko.

                                                        http://www.doiserbia.nb.rs/img/doi/0350-1302/2007/0350-13020796129K.pdf describes an improvement to that lower bound (including references to Zuev's and Odlyzko's articles).

                                                        http://arxiv.org/pdf/1209.2309v1.pdf studies a closely related arrangement and gives a related (but weaker) lower bound using a very elegant method. It turns out the arrangment becomes much easier mod $2$ (even than one might expect).

                                                        Whenever one has a central hyperplane arrangement one has a zonotope dual to it. Klivans and Reiner [ http://arxiv.org/pdf/math/0610787v2.pdf ] fix $k$ and look at the zonotope (Minkowski sum of the line segments) generated by all 0/1-vectors of length $n$ with exactly $k$ ones. In particular they are interested in the zonotope considered as symmetric polynomial (the sum $\\sum x^m$ over all lattice points $m$ in the zonotope).\nThus the Minkowski sum of their zonotopes over $1\\leq k \\leq n$ is the zonotope dual to your arrangement. Degree sequences of hypergraphs correspond to integer points in this latter zonotope. Surprisingly the converse is false! This was showed by Liu [http://arxiv.org/abs/1201.5989 ].

                                                        The vertices in the zonotope, and thus the regions in the arrangement, correspond almost to linear threshold hypergraphs (better known as linear threshold (Boolean) functions). A linear threshold hypergraph is determined by n+1 real potentially negative numbers q,w_1,\\dots,w_n; any set S\\subseteq [n] with \\sum_{i\\in S} x_i \\leq q is declared an edge. \"Almost\" above means that the vertices in this zonotope correspond exactly to the linear threshold hypergraphs that can be given with q = 0.

                                                        What if q \\neq 0? Given a hypergraph \\mathcal{H} with s simplices and degree sequence v_1,\\dots,v_n, one can look at the values s-2v_i, 1\\leq i\\leq n. If \\mathcal{H} happens to be down-closed, this is what game theorists call the Banzhaf value of the game represented by \\mathcal{H} and what computer scientists call the Boolean influence of the Boolean function represented by \\mathcal{H}. Anyway it is more natural to look at the vector (s,v_1,\\dots,v_n) rather than just (v_1,\\dots,v_n). This corresponds to looking instead at the zonotope generated by (1,0,0),(1,0,1),(1,1,0),(1,1,1) et.c. (all binary vectors of length n, prepended by 1). The vertices (obviously more than when we didn't have the leading 1's in the generators) in this zonotope correspond exactly to linear threshold hypergraphs. Since each v_i is an integer 0 and 2^{n-1}, and s is between 2^n there are at most 2^{n + n(n-1)} = 2^{n^2} of them, giving an almost matching upper bound to the lower bound above.

                                                        The latter zonotope is briefly mentioned in the end of the article http://arxiv.org/pdf/0908.4425.pdf .

                                                        I remember having seen a reference to a paper of Terao computing the characteristic polynomial of the arrangemnt for some small values, but cannot locate that reference right now.

                                                        • Thanks! Haven't digested this at all, but was hoping that there'd be more about this in the literature, and you've supplied that well.CommentedJun 21, 2013 at 12:20

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                                                          .

                                                          .

                                                          ", "statics": {"title": 7, "paragraph": 23, "paragraph.text": 70, "paragraph.equation-inline": 47, "equation-interline": 3, "list": 4, "list.text": 20}, "url": "https://mathoverflow.net/questions/62764/a-natural-refinement-of-the-a-n-arrangement-is-to-consider-all-2n-1-hyperpl?rq=1", "content": "## The short version\n\nHere is an extremely natural hyperplane arrangement in $\\mathbb{R}^n$ , which I will call $R_n$ for resonance arrangement.\n\nLet $x_i$ be the standard coordinates on $\\mathbb{R}^n$ . For each nonempty $I\\subseteq [n]=\\{1,\\dots,n\\}$ , define the hyperplane $H_I$ to be the hyperplane given by\n\n$$\n\\sum_{i\\in I} x_i=0.\n$$\n\nThe resonance arrangement is given by all $2^n-1$ hyperplanes $H_I$ . The arrangement $R_n$ is natural enough that it arises in many contexts -- to first order, my question is simply: have you come across it yourself?\n\nThis feels rather vague to be a good question, and after giving some background on where I've seen this I'll try to be a bit more specific about what I'm looking for, but my point is this arrangement has a rather simple and natural definition, and so crops up in multiple places, and I'd be curious to hear about more of them even if you can't specifically connect it to what follows.\n\n## What I knew until this week\n\nI came across this arrangement in my study of double Hurwitz numbers -- they are piecewise polynomial, and the chambers of the resonance arrangement are the chambers of polynomiality. I don't want to go into this much more, as it's unimportant to most of what follows Though I will say that conjecturally double Hurwitz numbers could be related to compactified Picard varieties in a way which would connect this arrangement up with birational transformations of those. Also, the name \"resonance arrangement\" was essentially introduce in this context, by Shadrin, Shapiro and Vainshtein.\n\nIt apparently comes up in physics -- I only know this because the number of regions of $R_n$ , starting at $n=2$ , is 2, 6, 32, 370, 11292, 1066044, 347326352 ... Sloane sequence A034997, which you will see was entered as \"Number of Generalized Retarded Functions in Quantum Field Theory\" by a physicist.\n\nYou might expect by that rate of growth that this hyperplane arrangement is completely intractable, and more specifically, what I would love from an answer is some kind qualitative statement about how ugly the $R_n$ get. Which brings us to:\n\n## Connection to the GGMS decomposition\n\nI got to thinking about this again now and decided to post on MO of it because of Noah's question about the vertices of a variation of GIT problem, where this arrangement is lurking around -- see there for more detail. Allen's brief comment there prompted me to skim some of his and related papers to that general area, and I found the introduction to Positroid varieties I: juggling and geometry most enlightening, together with the discussion at Noah's question could give another suggestion why $R_n$ is perhaps intractable. Briefly:\n\nThe arrangement $R_n$ is a natural extension of the $A_n$ arrangement. One common description of the $A_n$ arrangement is as the $\\binom{n+1}{2}$ hyperplanes $y_i-y_j=0, i,j\\in [n]$ and $y_i=0, i\\in [n]$ . However, one can consider the triangular change of variables\n\n$$\ny_k\\mapsto \\sum_{j\\leq k} y_j\n$$\n\n.\n\nThis changes the hyperplanes to\n\n$$\n\\sum_{i\\leq k \\leq j} y_k=0.\n$$\n\nThese hyperplanes are no longer invariant under permutation of the coordinates, and if we proceed to add the entire $S_n$ orbit of them, we get the resonance arrangement $R_n$ .\n\nFrom the discussion on Noah's question and the introduction to the Positroid paper, we see that this manipulation is a shadow of the GGMS decomposition of the Grassmannian, and describing this decomposition in general seems to be intractable in that it requires identifying whether matroids are representable or not. So, what I'd really like to know how is much of the \"GGMS abyss\", as they refer to it, is reflected in the resonance arrangement? Is it hopeless to describe and count its chambers?\n\n- 8 Several of my students and I have thought about the characteristic polynomial and number of regions of this arrangement over the years, but we never came up with anything interesting. Commented Apr 23, 2011 at 19:09\n- -1 for not having a more specific question, but +2 for lots of details and interesting discussion and good title. Also, I think there's something funny with the TeX between \"However, one can consider the triangular ...\" and \"These hyperplanes are no longer ...\". Commented Apr 23, 2011 at 21:04\n- Think I fixed the TeX. And I was a bit torn about it not being specific enough myself, but it was never going to get asked if I kept worrying about it. Next time will be better. Commented Apr 23, 2011 at 21:48\n- @Theo: Also, you are half responsible for the long title: somewhere you mentioned that a title can be about a \"tweet and a half\" long, and that was an odd enough description that it stuck in my head. Good work. Commented Apr 23, 2011 at 21:56\n- 1 Just commenting to thank Richard -- this is one kind of thing I wanted to hear: that people had put some real thought into it and not gotten anything interesting. I wasn't up-voting that you didn't get anywhere. Commented Apr 27, 2011 at 19:41\n\n## 1 Answer\n\n8\n\nHere's what I know about this arrangement.\n\nRegarding the number of chambers in this arrangment, Zuev obtained the lower bound $2^{(1-o(1))n^2}$ . The proof uses Zaslavsky's theorem and a difficult estimate due to Odlyzko.\n\nhttp://www.doiserbia.nb.rs/img/doi/0350-1302/2007/0350-13020796129K.pdf describes an improvement to that lower bound (including references to Zuev's and Odlyzko's articles).\n\nhttp://arxiv.org/pdf/1209.2309v1.pdf studies a closely related arrangement and gives a related (but weaker) lower bound using a very elegant method. It turns out the arrangment becomes much easier mod $2$ (even than one might expect).\n\nWhenever one has a central hyperplane arrangement one has a zonotope dual to it. Klivans and Reiner [ http://arxiv.org/pdf/math/0610787v2.pdf] fix $k$ and look at the zonotope (Minkowski sum of the line segments) generated by all 0/1-vectors of length $n$ with exactly $k$ ones. In particular they are interested in the zonotope considered as symmetric polynomial (the sum $\\sum x^m$ over all lattice points $m$ in the zonotope).\nThus the Minkowski sum of their zonotopes over $1\\leq k \\leq n$ is the zonotope dual to your arrangement. Degree sequences of hypergraphs correspond to integer points in this latter zonotope. Surprisingly the converse is false! This was showed by Liu [ http://arxiv.org/abs/1201.5989].\n\nThe vertices in the zonotope, and thus the regions in the arrangement, correspond almost to linear threshold hypergraphs (better known as linear threshold (Boolean) functions). A linear threshold hypergraph is determined by $n+1$ real potentially negative numbers $q,w_1,\\dots,w_n$ ; any set $S\\subseteq [n]$ with $\\sum_{i\\in S} x_i \\leq q$ is declared an edge. \"Almost\" above means that the vertices in this zonotope correspond exactly to the linear threshold hypergraphs that can be given with $q = 0$ .\n\nWhat if $q \\neq 0$ ? Given a hypergraph $\\mathcal{H}$ with $s$ simplices and degree sequence $v_1,\\dots,v_n$ , one can look at the values $s-2v_i$ , $1\\leq i\\leq n$ . If $\\mathcal{H}$ happens to be down-closed, this is what game theorists call the Banzhaf value of the game represented by $\\mathcal{H}$ and what computer scientists call the Boolean influence of the Boolean function represented by $\\mathcal{H}$ . Anyway it is more natural to look at the vector $(s,v_1,\\dots,v_n)$ rather than just $(v_1,\\dots,v_n)$ . This corresponds to looking instead at the zonotope generated by $(1,0,0),(1,0,1),(1,1,0),(1,1,1)$ et.c. (all binary vectors of length $n$ , prepended by $1$ ). The vertices (obviously more than when we didn't have the leading 1's in the generators) in this zonotope correspond exactly to linear threshold hypergraphs. Since each $v_i$ is an integer $0$ and $2^{n-1}$ , and $s$ is between $2^n$ there are at most $2^{n + n(n-1)} = 2^{n^2}$ of them, giving an almost matching upper bound to the lower bound above.\n\nThe latter zonotope is briefly mentioned in the end of the article http://arxiv.org/pdf/0908.4425.pdf.\n\nI remember having seen a reference to a paper of Terao computing the characteristic polynomial of the arrangemnt for some small values, but cannot locate that reference right now.\n\n- Thanks! Haven't digested this at all, but was hoping that there'd be more about this in the literature, and you've supplied that well. Commented Jun 21, 2013 at 12:20\n\n## You must log in to answer this question.\n\n## Not the answer you're looking for? Browse other questions tagged\n\n## .\n", "html": "\n\n\n\n\n \n\n \n\n co.combinatorics - A natural refinement of the $A_n$ arrangement is to consider all $2^n-1$ hyperplanes given by the sums of the coordinate functions. Have you seen this arrangement? Is it completely intractable? - MathOverflow\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n\n\n\n \n \n\n\n\n\n \n\n\n \n\n\n \n \n\n\n \n \n\n\n\n \n \n \n \n\n\n
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                                                          The short version

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                                                          Here is an extremely natural hyperplane arrangement in $\\mathbb{R}^n$, which I will call $R_n$ for resonance arrangement.

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                                                          Let $x_i$ be the standard coordinates on $\\mathbb{R}^n$. For each nonempty $I\\subseteq [n]=\\{1,\\dots,n\\}$, define the hyperplane $H_I$ to be the hyperplane given by\n$$\\sum_{i\\in I} x_i=0.$$\nThe resonance arrangement is given by all $2^n-1$ hyperplanes $H_I$. The arrangement $R_n$ is natural enough that it arises in many contexts -- to first order, my question is simply: have you come across it yourself?

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                                                          This feels rather vague to be a good question, and after giving some background on where I've seen this I'll try to be a bit more specific about what I'm looking for, but my point is this arrangement has a rather simple and natural definition, and so crops up in multiple places, and I'd be curious to hear about more of them even if you can't specifically connect it to what follows.

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                                                          What I knew until this week

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                                                          I came across this arrangement in my study of double Hurwitz numbers -- they are piecewise polynomial, and the chambers of the resonance arrangement are the chambers of polynomiality. I don't want to go into this much more, as it's unimportant to most of what follows Though I will say that conjecturally double Hurwitz numbers could be related to compactified Picard varieties in a way which would connect this arrangement up with birational transformations of those. Also, the name "resonance arrangement" was essentially introduce in this context, by Shadrin, Shapiro and Vainshtein.

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                                                          It apparently comes up in physics -- I only know this because the number of regions of $R_n$, starting at $n=2$, is 2, 6, 32, 370, 11292, 1066044, 347326352 ... Sloane sequence A034997, which you will see was entered as "Number of Generalized Retarded Functions in Quantum Field Theory" by a physicist.

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                                                          You might expect by that rate of growth that this hyperplane arrangement is completely intractable, and more specifically, what I would love from an answer is some kind qualitative statement about how ugly the $R_n$ get. Which brings us to:

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                                                          Connection to the GGMS decomposition

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                                                          I got to thinking about this again now and decided to post on MO of it because of Noah's question about the vertices of a variation of GIT problem, where this arrangement is lurking around -- see there for more detail. Allen's brief comment there prompted me to skim some of his and related papers to that general area, and I found the introduction to Positroid varieties I: juggling and geometry most enlightening, together with the discussion at Noah's question could give another suggestion why $R_n$ is perhaps intractable. Briefly:

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                                                          The arrangement $R_n$ is a natural extension of the $A_n$ arrangement. One common description of the $A_n$ arrangement is as the $\\binom{n+1}{2}$ hyperplanes $y_i-y_j=0, i,j\\in [n]$ and $y_i=0, i\\in [n]$. However, one can consider the triangular change of variables\n$$y_k\\mapsto \\sum_{j\\leq k} y_j$$.

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                                                          This changes the hyperplanes to\n$$\\sum_{i\\leq k \\leq j} y_k=0.$$

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                                                          These hyperplanes are no longer invariant under permutation of the coordinates, and if we proceed to add the entire $S_n$ orbit of them, we get the resonance arrangement $R_n$.

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                                                          From the discussion on Noah's question and the introduction to the Positroid paper, we see that this manipulation is a shadow of the GGMS decomposition of the Grassmannian, and describing this decomposition in general seems to be intractable in that it requires identifying whether matroids are representable or not. So, what I'd really like to know how is much of the "GGMS abyss", as they refer to it, is reflected in the resonance arrangement? Is it hopeless to describe and count its chambers?

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                                                          \n edited Jun 15, 2020 at 7:27\n
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                                                          \n asked Apr 23, 2011 at 18:36\n
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                                                            \n $\\begingroup$\n Several of my students and I have thought about the characteristic polynomial and number of regions of this arrangement over the years, but we never came up with anything interesting. \n $\\endgroup$\n
                                                            \n– Richard Stanley\n
                                                            \n \n Commented\n Apr 23, 2011 at 19:09\n \n
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                                                            \n $\\begingroup$\n -1 for not having a more specific question, but +2 for lots of details and interesting discussion and good title. Also, I think there's something funny with the TeX between "However, one can consider the triangular ..." and "These hyperplanes are no longer ...".\n $\\endgroup$\n
                                                            \n– Theo Johnson-Freyd\n
                                                            \n \n Commented\n Apr 23, 2011 at 21:04\n \n
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                                                            \n $\\begingroup$\n Think I fixed the TeX. And I was a bit torn about it not being specific enough myself, but it was never going to get asked if I kept worrying about it. Next time will be better.\n $\\endgroup$\n
                                                            \n– Paul Johnson\n
                                                            \n \n Commented\n Apr 23, 2011 at 21:48\n \n
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                                                            \n $\\begingroup$\n @Theo: Also, you are half responsible for the long title: somewhere you mentioned that a title can be about a "tweet and a half" long, and that was an odd enough description that it stuck in my head. Good work.\n $\\endgroup$\n
                                                            \n– Paul Johnson\n
                                                            \n \n Commented\n Apr 23, 2011 at 21:56\n \n
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                                                            \n $\\begingroup$\n Just commenting to thank Richard -- this is one kind of thing I wanted to hear: that people had put some real thought into it and not gotten anything interesting. I wasn't up-voting that you didn't get anywhere.\n $\\endgroup$\n
                                                            \n– Paul Johnson\n
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                                                          Here's what I know about this arrangement.

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                                                          Regarding the number of chambers in this arrangment, Zuev obtained the lower bound $2^{(1-o(1))n^2}$. The proof uses Zaslavsky's theorem and a difficult estimate due to Odlyzko.

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                                                          http://www.doiserbia.nb.rs/img/doi/0350-1302/2007/0350-13020796129K.pdf describes an improvement to that lower bound (including references to Zuev's and Odlyzko's articles).

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                                                          http://arxiv.org/pdf/1209.2309v1.pdf studies a closely related arrangement and gives a related (but weaker) lower bound using a very elegant method. It turns out the arrangment becomes much easier mod $2$ (even than one might expect).

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                                                          Whenever one has a central hyperplane arrangement one has a zonotope dual to it. Klivans and Reiner [ http://arxiv.org/pdf/math/0610787v2.pdf ] fix $k$ and look at the zonotope (Minkowski sum of the line segments) generated by all 0/1-vectors of length $n$ with exactly $k$ ones. In particular they are interested in the zonotope considered as symmetric polynomial (the sum $\\sum x^m$ over all lattice points $m$ in the zonotope).\nThus the Minkowski sum of their zonotopes over $1\\leq k \\leq n$ is the zonotope dual to your arrangement. Degree sequences of hypergraphs correspond to integer points in this latter zonotope. Surprisingly the converse is false! This was showed by Liu [http://arxiv.org/abs/1201.5989 ].

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                                                          The vertices in the zonotope, and thus the regions in the arrangement, correspond almost to linear threshold hypergraphs (better known as linear threshold (Boolean) functions). A linear threshold hypergraph is determined by $n+1$ real potentially negative numbers $q,w_1,\\dots,w_n$; any set $S\\subseteq [n]$ with $\\sum_{i\\in S} x_i \\leq q$ is declared an edge. \"Almost\" above means that the vertices in this zonotope correspond exactly to the linear threshold hypergraphs that can be given with $q = 0$.

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                                                          What if $q \\neq 0$? Given a hypergraph $\\mathcal{H}$ with $s$ simplices and degree sequence $v_1,\\dots,v_n$, one can look at the values $s-2v_i$, $1\\leq i\\leq n$. If $\\mathcal{H}$ happens to be down-closed, this is what game theorists call the Banzhaf value of the game represented by $\\mathcal{H}$ and what computer scientists call the Boolean influence of the Boolean function represented by $\\mathcal{H}$. Anyway it is more natural to look at the vector $(s,v_1,\\dots,v_n)$ rather than just $(v_1,\\dots,v_n)$. This corresponds to looking instead at the zonotope generated by $(1,0,0),(1,0,1),(1,1,0),(1,1,1)$ et.c. (all binary vectors of length $n$, prepended by $1$). The vertices (obviously more than when we didn't have the leading 1's in the generators) in this zonotope correspond exactly to linear threshold hypergraphs. Since each $v_i$ is an integer $0$ and $2^{n-1}$, and $s$ is between $2^n$ there are at most $2^{n + n(n-1)} = 2^{n^2}$ of them, giving an almost matching upper bound to the lower bound above.

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                                                          The latter zonotope is briefly mentioned in the end of the article http://arxiv.org/pdf/0908.4425.pdf .

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                                                          I remember having seen a reference to a paper of Terao computing the characteristic polynomial of the arrangemnt for some small values, but cannot locate that reference right now.

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                                                          \n edited Jun 19, 2013 at 22:13\n
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                                                          \n answered Jun 19, 2013 at 22:06\n
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                                                          \n Erik AasErik Aas\n
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                                                            \n $\\begingroup$\n Thanks! Haven't digested this at all, but was hoping that there'd be more about this in the literature, and you've supplied that well.\n $\\endgroup$\n
                                                            \n– Paul Johnson\n
                                                            \n \n Commented\n Jun 21, 2013 at 12:20\n \n
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                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          By the end of this section, you will be able to:

                                                          ", "content": [{"c": "By the end of this section, you will be able to:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          • Multiply integers
                                                          • Divide integers
                                                          • Simplify expressions with integers
                                                          • Evaluate variable expressions with integers
                                                          • Translate English phrases to algebraic expressions
                                                          • Use integers in applications
                                                          ", "content": {"items": [[[{"c": "Multiply integers", "t": "text"}]], [[{"c": "Divide integers", "t": "text"}]], [[{"c": "Simplify expressions with integers", "t": "text"}]], [[{"c": "Evaluate variable expressions with integers", "t": "text"}]], [[{"c": "Translate English phrases to algebraic expressions", "t": "text"}]], [[{"c": "Use integers in applications", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

                                                          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

                                                          ", "content": [{"c": "A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Multiply Integers

                                                          ", "content": {"title_content": "Multiply Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

                                                          ", "content": [{"c": "Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          We remember that \\(a\\cdot b\\) means add \\(a,\\, b\\) times. Here, we are using the model just to help us discover the pattern.

                                                          ", "content": [{"c": "We remember that \\(a\\cdot b\\) means add \\(a,\\, b\\) times. Here, we are using the model just to help us discover the pattern.", "t": "text"}]}, {"type": "image", "raw_content": "
                                                          \"Two
                                                          Figure \\(\\PageIndex{1}\\)
                                                          ", "content": {"url": "https://math.libretexts.org/@api/deki/files/17395/CNX_ElemAlg_Figure_01_04_001_img_new.jpg?revision=1", "data": null, "alt": "Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads \u201cadd 5, 3 times.\u201d Below this depicts three rows of blue counters, with five counters in each row. Under this, it says \u201c15 positives.\u201d Under thisis the equation\u201c5 times 3 equals 15.\u201d The image on the right reads \u201cnegative 5 times three. The three is in parentheses. Below this it reads, \u201cadd negative five, three times.\u201d Under this are fifteen red counters in three rows of five. Below this it reads\u201d \u201c15 negatives\u201d. Below this is the equation negative five times 3 equals negative 15.\u201d", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

                                                          The next two examples are more interesting.

                                                          ", "content": [{"c": "The next two examples are more interesting.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          What does it mean to multiply \\(5\\) by \\(\u22123\\)? It means subtract \\(5, 3\\) times. Looking at subtraction as \u201ctaking away,\u201d it means to take away \\(5, 3\\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \\(5\\) three times.

                                                          ", "content": [{"c": "What does it mean to multiply \\(5\\) by \\(\u22123\\)? It means subtract \\(5, 3\\) times. Looking at subtraction as \u201ctaking away,\u201d it means to take away \\(5, 3\\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \\(5\\) three times.", "t": "text"}]}, {"type": "image", "raw_content": "
                                                          \"This
                                                          Figure \\(\\PageIndex{2}\\)
                                                          ", "content": {"url": "https://math.libretexts.org/@api/deki/files/17306/CNX_ElemAlg_Figure_01_04_002_img_new.jpg?revision=1", "data": null, "alt": "This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

                                                          In summary:

                                                          ", "content": [{"c": "In summary:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\\]

                                                          ", "content": {"math_content": "\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Notice that for multiplication of two signed numbers, when the:

                                                          ", "content": [{"c": "Notice that for multiplication of two signed numbers, when the:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          • signs are the same, the product is positive.
                                                          • signs are different, the product is negative.
                                                          ", "content": {"items": [[[{"c": "signs are the ", "t": "text"}, {"c": "same", "t": "text"}, {"c": ", the product is ", "t": "text"}, {"c": "positive", "t": "text"}, {"c": ".", "t": "text"}]], [[{"c": "signs are ", "t": "text"}, {"c": "different", "t": "text"}, {"c": ", the product is ", "t": "text"}, {"c": "negative", "t": "text"}, {"c": ".", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

                                                          We\u2019ll put this all together in the chart below.

                                                          ", "content": [{"c": "We\u2019ll put this all together in the chart below.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          For multiplication of two signed numbers:

                                                          ", "content": [{"c": "For multiplication of two signed numbers:", "t": "text"}]}, {"type": "table", "raw_content": "
                                                          Same signsProductExample
                                                          Two positivesPositive\\(7\\cdot 4 = 28\\)
                                                          Two negativesPositive\\(-8(-6) = 48\\)
                                                          Table \\(\\PageIndex{1}\\)
                                                          ", "content": {"html": "
                                                          Same signsProductExample
                                                          Two positivesPositive\\(7\\cdot 4 = 28\\)
                                                          Two negativesPositive\\(-8(-6) = 48\\)
                                                          Table \\(\\PageIndex{1}\\)
                                                          ", "is_complex": false}}, {"type": "table", "raw_content": "
                                                          Different signsProductExample
                                                          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
                                                          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
                                                          Table \\(\\PageIndex{2}\\)
                                                          ", "content": {"html": "
                                                          Different signsProductExample
                                                          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
                                                          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
                                                          Table \\(\\PageIndex{2}\\)
                                                          ", "is_complex": false}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-9\\cdot 3\\)
                                                          2. \\(-2(-5)\\)
                                                          3. \\(4(-8)\\)
                                                          4. \\(7\\cdot 6\\)
                                                          ", "content": {"items": [[[{"c": "\\(-9\\cdot 3\\)", "t": "text"}]], [[{"c": "\\(-2(-5)\\)", "t": "text"}]], [[{"c": "\\(4(-8)\\)", "t": "text"}]], [[{"c": "\\(7\\cdot 6\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]
                                                          3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]
                                                          4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-6\\cdot 8\\)
                                                          2. \\(-4(-7)\\)
                                                          3. \\(9(-7)\\)
                                                          4. \\(5\\cdot 12\\)
                                                          ", "content": {"items": [[[{"c": "\\(-6\\cdot 8\\)", "t": "text"}]], [[{"c": "\\(-4(-7)\\)", "t": "text"}]], [[{"c": "\\(9(-7)\\)", "t": "text"}]], [[{"c": "\\(5\\cdot 12\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-48\\)
                                                          2. \\(28\\)
                                                          3. \\(-63\\)
                                                          4. \\(60\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-48\\)", "t": "text"}, {"c": "\\(28\\)", "t": "text"}, {"c": "\\(-63\\)", "t": "text"}, {"c": "\\(60\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-8\\cdot 7\\)
                                                          2. \\(-6(-9)\\)
                                                          3. \\(7(-4)\\)
                                                          4. \\(3\\cdot 13\\)
                                                          ", "content": {"items": [[[{"c": "\\(-8\\cdot 7\\)", "t": "text"}]], [[{"c": "\\(-6(-9)\\)", "t": "text"}]], [[{"c": "\\(7(-4)\\)", "t": "text"}]], [[{"c": "\\(3\\cdot 13\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-56\\)
                                                          2. \\(54\\)
                                                          3. \\(-28\\)
                                                          4. \\(39\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-56\\)", "t": "text"}, {"c": "\\(54\\)", "t": "text"}, {"c": "\\(-28\\)", "t": "text"}, {"c": "\\(39\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          When we multiply a number by \\(1\\), the result is the same number. What happens when we multiply a number by \\(\u22121\\)? Let\u2019s multiply a positive number and then a negative number by \\(\u22121\\) to see what we get.

                                                          ", "content": [{"c": "When we multiply a number by \\(1\\), the result is the same number. What happens when we multiply a number by \\(\u22121\\)? Let\u2019s multiply a positive number and then a negative number by \\(\u22121\\) to see what we get.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]
                                                          \nEach time we multiply a number by \\(\u22121\\), we get its opposite!

                                                          ", "content": [{"c": "\\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\] Each time we multiply a number by \\(\u22121\\), we get its opposite!", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          MULTIPLICATION BY \u22121

                                                          ", "content": [{"c": "MULTIPLICATION BY \u22121", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\u22121a=\u2212a\\]

                                                          ", "content": {"math_content": "\u22121a=\u2212a", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Multiplying a number by \\(\u22121\\) gives its opposite.

                                                          ", "content": [{"c": "Multiplying a number by \\(\u22121\\) gives its opposite.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-1 \\cdot 7\\)
                                                          2. \\(-1(-11)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-1 \\cdot 7\\)", "t": "text"}]], [[{"c": "\\(-1(-11)\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-1\\cdot 9\\)
                                                          2. \\(-1\\cdot(-17)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-1\\cdot 9\\)", "t": "text"}]], [[{"c": "\\(-1\\cdot(-17)\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-9\\)
                                                          2. \\(17\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-9\\)", "t": "text"}, {"c": "\\(17\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-1\\cdot 8\\)
                                                          2. \\(-1\\cdot(-16)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-1\\cdot 8\\)", "t": "text"}]], [[{"c": "\\(-1\\cdot(-16)\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-8\\)
                                                          2. \\(16\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-8\\)", "t": "text"}, {"c": "\\(16\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Divide Integers

                                                          ", "content": {"title_content": "Divide Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.

                                                          ", "content": [{"c": "What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\\]

                                                          ", "content": {"math_content": "\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Division follows the same rules as multiplication!

                                                          ", "content": [{"c": "Division follows the same rules as multiplication!", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          For division of two signed numbers, when the:

                                                          ", "content": [{"c": "For division of two signed numbers, when the:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          • signs are the same, the quotient is positive.
                                                          • signs are different, the quotient is negative.
                                                          ", "content": {"items": [[[{"c": "signs are the ", "t": "text"}, {"c": "same", "t": "text"}, {"c": ", the quotient is ", "t": "text"}, {"c": "positive", "t": "text"}, {"c": ".", "t": "text"}]], [[{"c": "signs are ", "t": "text"}, {"c": "different", "t": "text"}, {"c": ", the quotient is ", "t": "text"}, {"c": "negative", "t": "text"}, {"c": ".", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

                                                          And remember that we can always check the answer of a division problem by multiplying.

                                                          ", "content": [{"c": "And remember that we can always check the answer of a division problem by multiplying.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          For multiplication and division of two signed numbers:

                                                          ", "content": [{"c": "For multiplication and division of two signed numbers:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          • If the signs are the same, the result is positive.
                                                          • If the signs are different, the result is negative.
                                                          ", "content": {"items": [[[{"c": "If the signs are the same, the result is positive.", "t": "text"}]], [[{"c": "If the signs are different, the result is negative.", "t": "text"}]]], "ordered": false}}, {"type": "table", "raw_content": "
                                                          Same signsResult
                                                          Two positivesPositive
                                                          Two negativesPositive
                                                          If the signs are the same, the result is positive.
                                                          Table \\(\\PageIndex{3}\\)
                                                          ", "content": {"html": "
                                                          Same signsResult
                                                          Two positivesPositive
                                                          Two negativesPositive
                                                          If the signs are the same, the result is positive.
                                                          Table \\(\\PageIndex{3}\\)
                                                          ", "is_complex": false}}, {"type": "table", "raw_content": "
                                                          Different signsResult
                                                          Positive and negativeNegative
                                                          Negative and positiveNegative
                                                          If the signs are different, the result is negative.
                                                          Table \\(\\PageIndex{4}\\)
                                                          ", "content": {"html": "
                                                          Different signsResult
                                                          Positive and negativeNegative
                                                          Negative and positiveNegative
                                                          If the signs are different, the result is negative.
                                                          Table \\(\\PageIndex{4}\\)
                                                          ", "is_complex": false}}, {"type": "list", "raw_content": "
                                                          1. \\(-27\\div 3\\)
                                                          2. \\(-100\\div (-4)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-27\\div 3\\)", "t": "text"}]], [[{"c": "\\(-100\\div (-4)\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Divide:

                                                          ", "content": [{"c": "Divide:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-42\\div 6\\)
                                                          2. \\(-117\\div (-3)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-42\\div 6\\)", "t": "text"}]], [[{"c": "\\(-117\\div (-3)\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-7\\)
                                                          2. \\(39\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-7\\)", "t": "text"}, {"c": "\\(39\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Divide:

                                                          ", "content": [{"c": "Divide:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-63\\div 7\\)
                                                          2. \\(-115\\div (-5)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-63\\div 7\\)", "t": "text"}]], [[{"c": "\\(-115\\div (-5)\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-9\\)
                                                          2. \\(23\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-9\\)", "t": "text"}, {"c": "\\(23\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Simplify Expressions with Integers

                                                          ", "content": {"title_content": "Simplify Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

                                                          ", "content": [{"c": "What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Let\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.

                                                          ", "content": [{"c": "Let\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          7(-2)+4(-7)-6
                                                          ", "content": [{"c": "7(-2)+4(-7)-6", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\\]

                                                          ", "content": {"math_content": "\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          8(-3)+5(-7)-4
                                                          ", "content": [{"c": "8(-3)+5(-7)-4", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          -63
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-63", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          9(-3)+7(-8)-1
                                                          ", "content": [{"c": "9(-3)+7(-8)-1", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          -84
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-84", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\((-2)^{4}\\)
                                                          2. \\(-2^{4}\\)
                                                          ", "content": {"items": [[[{"c": "\\((-2)^{4}\\)", "t": "text"}]], [[{"c": "\\(-2^{4}\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \\((\u22122)\\) to the \\(4^{th}\\) power. In part (2), the exponent means to raise just the \\(2\\) to the \\(4^{th}\\) power and then take the opposite.

                                                          ", "content": [{"c": "Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \\((\u22122)\\) to the \\(4^{th}\\) power. In part (2), the exponent means to raise just the \\(2\\) to the \\(4^{th}\\) power and then take the opposite.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\((-3)^{4}\\)
                                                          2. \\(-3^{4}\\)
                                                          ", "content": {"items": [[[{"c": "\\((-3)^{4}\\)", "t": "text"}]], [[{"c": "\\(-3^{4}\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(81\\)
                                                          2. \\(-81\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(81\\)", "t": "text"}, {"c": "\\(-81\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\((-7)^{2}\\)
                                                          2. \\(-7^{2}\\)
                                                          ", "content": {"items": [[[{"c": "\\((-7)^{2}\\)", "t": "text"}]], [[{"c": "\\(-7^{2}\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(49\\)
                                                          2. \\(-49\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(49\\)", "t": "text"}, {"c": "\\(-49\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          The next example reminds us to simplify inside parentheses first.

                                                          ", "content": [{"c": "The next example reminds us to simplify inside parentheses first.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          12-3(9 - 12)
                                                          ", "content": [{"c": "12-3(9 - 12)", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\\]

                                                          ", "content": {"math_content": "\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          17 - 4(8 - 11)
                                                          ", "content": [{"c": "17 - 4(8 - 11)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          29
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "29", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          16 - 6(7 - 13)
                                                          ", "content": [{"c": "16 - 6(7 - 13)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          52
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "52", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          8(-9)\\div (-2)^{3}
                                                          ", "content": [{"c": "8(-9)\\div (-2)^{3}", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\\]

                                                          ", "content": {"math_content": "\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          12(-9)\\div (-3)^{3}
                                                          ", "content": [{"c": "12(-9)\\div (-3)^{3}", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          4
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "4", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          18(-4)\\div (-2)^{3}
                                                          ", "content": [{"c": "18(-4)\\div (-2)^{3}", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          9
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "9", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          -30\\div 2 + (-3)(-7)
                                                          ", "content": [{"c": "-30\\div 2 + (-3)(-7)", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\\]

                                                          ", "content": {"math_content": "\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          -27\\div 3 + (-5)(-6)
                                                          ", "content": [{"c": "-27\\div 3 + (-5)(-6)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          21
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "21", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                          -32\\div 4 + (-2)(-7)
                                                          ", "content": [{"c": "-32\\div 4 + (-2)(-7)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          6
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "6", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Evaluate Variable Expressions with Integers

                                                          ", "content": {"title_content": "Evaluate Variable Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

                                                          ", "content": [{"c": "Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          When \\(n=\u22125\\), evaluate:

                                                          ", "content": [{"c": "When \\(n=\u22125\\), evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(n+1\\)
                                                          2. \\(\u2212n+1\\).
                                                          ", "content": {"items": [[[{"c": "\\(n+1\\)", "t": "text"}]], [[{"c": "\\(\u2212n+1\\).", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1\u00a0\\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{-\u00a0{\\color{red}{(-5)}}\u00a0+1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1\u00a0\\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{-\u00a0{\\color{red}{(-5)}}\u00a0+1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          When \\(n=\u22128\\), evaluate:

                                                          ", "content": [{"c": "When \\(n=\u22128\\), evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(n+2\\)
                                                          2. \\(\u2212n+2\\).
                                                          ", "content": {"items": [[[{"c": "\\(n+2\\)", "t": "text"}]], [[{"c": "\\(\u2212n+2\\).", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-6\\)
                                                          2. \\(10\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-6\\)", "t": "text"}, {"c": "\\(10\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          When \\(y=\u22129\\), evaluate:

                                                          ", "content": [{"c": "When \\(y=\u22129\\), evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(y+8\\)
                                                          2. \\(\u2212y+8\\).
                                                          ", "content": {"items": [[[{"c": "\\(y+8\\)", "t": "text"}]], [[{"c": "\\(\u2212y+8\\).", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-1\\)
                                                          2. \\(17\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-1\\)", "t": "text"}, {"c": "\\(17\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate \\((x+y)^{2}\\) when \\(x = -18\\) and \\(y = 24\\).

                                                          ", "content": [{"c": "Evaluate \\((x+y)^{2}\\) when \\(x = -18\\) and \\(y = 24\\).", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\\]

                                                          ", "content": {"math_content": "\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate \\((x+y)^{2}\\) when \\(x = -15\\) and \\(y = 29\\).

                                                          ", "content": [{"c": "Evaluate \\((x+y)^{2}\\) when \\(x = -15\\) and \\(y = 29\\).", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          196
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "196", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate \\((x+y)^{3}\\) when \\(x = -8\\) and \\(y = 10\\).

                                                          ", "content": [{"c": "Evaluate \\((x+y)^{3}\\) when \\(x = -8\\) and \\(y = 10\\).", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          8
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "8", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate \\(20 -z \\) when

                                                          ", "content": [{"c": "Evaluate \\(20 -z \\) when", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(z = 12\\)
                                                          2. \\(z = -12\\)
                                                          ", "content": {"items": [[[{"c": "\\(z = 12\\)", "t": "text"}]], [[{"c": "\\(z = -12\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate \\(17 - k\\) when

                                                          ", "content": [{"c": "Evaluate \\(17 - k\\) when", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(k = 19\\)
                                                          2. \\(k = -19\\)
                                                          ", "content": {"items": [[[{"c": "\\(k = 19\\)", "t": "text"}]], [[{"c": "\\(k = -19\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-2\\)
                                                          2. \\(36\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-2\\)", "t": "text"}, {"c": "\\(36\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate \\(-5 - b\\) when

                                                          ", "content": [{"c": "Evaluate \\(-5 - b\\) when", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(b = 14\\)
                                                          2. \\(b = -14\\)
                                                          ", "content": {"items": [[[{"c": "\\(b = 14\\)", "t": "text"}]], [[{"c": "\\(b = -14\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-19\\)
                                                          2. \\(9\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-19\\)", "t": "text"}, {"c": "\\(9\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate:

                                                          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          \\(2x^{2} + 3x + 8\\) when \\(x = 4\\).

                                                          ", "content": [{"c": "\\(2x^{2} + 3x + 8\\) when \\(x = 4\\).", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Substitute \\(4\\) for \\(x\\). Use parentheses to show multiplication.

                                                          ", "content": [{"c": "Substitute \\(4\\) for \\(x\\). Use parentheses to show multiplication.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\\]

                                                          ", "content": {"math_content": "\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate:

                                                          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          \\(3x^{2} - 2x + 6\\) when \\(x =-3\\).

                                                          ", "content": [{"c": "\\(3x^{2} - 2x + 6\\) when \\(x =-3\\).", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          39
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "39", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate:

                                                          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          \\(4x^{2} - x - 5\\) when \\(x = -2\\).

                                                          ", "content": [{"c": "\\(4x^{2} - x - 5\\) when \\(x = -2\\).", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          13
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "13", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Translate Phrases to Expressions with Integers

                                                          ", "content": {"title_content": "Translate Phrases to Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

                                                          ", "content": [{"c": "Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate and simplify: the sum of \\(8\\) and \\(\u221212\\), increased by \\(3\\).

                                                          ", "content": [{"c": "Translate and simplify: the sum of \\(8\\) and \\(\u221212\\), increased by \\(3\\).", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\\]

                                                          ", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate and simplify: the sum of \\(9\\) and \\(\u221216\\), increased by \\(4\\).

                                                          ", "content": [{"c": "Translate and simplify: the sum of \\(9\\) and \\(\u221216\\), increased by \\(4\\).", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          (9 + (-16)) + 4 - 3
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "(9 + (-16)) + 4 - 3", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate and simplify: the sum of \\(-8\\) and \\(\u221212\\), increased by \\(7\\).

                                                          ", "content": [{"c": "Translate and simplify: the sum of \\(-8\\) and \\(\u221212\\), increased by \\(7\\).", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          (-8 + (-12)) + 7 - 13
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "(-8 + (-12)) + 7 - 13", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

                                                          ", "content": [{"c": "When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.", "t": "text"}]}, {"type": "table", "raw_content": "
                                                          \\(a\u2212b\\)
                                                          \\(a\\) minus \\(b\\)
                                                          \n the difference of \\(a\\) and \\(b\\)
                                                          \n \\(b\\) subtracted from \\(a\\)
                                                          \n \\(b\\) less than \\(a\\)
                                                          Table \\(\\PageIndex{5}\\)
                                                          ", "content": {"html": "
                                                          \\(a\u2212b\\)
                                                          \\(a\\) minus \\(b\\)<br>the difference of \\(a\\) and \\(b\\)<br>\\(b\\) subtracted from \\(a\\)<br>\\(b\\) less than \\(a\\)
                                                          Table \\(\\PageIndex{5}\\)
                                                          ", "is_complex": false}}, {"type": "paragraph", "raw_content": "

                                                          Be careful to get a and b in the right order!

                                                          ", "content": [{"c": "Be careful to get a and b in the right order!", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate and then simplify

                                                          ", "content": [{"c": "Translate and then simplify", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. the difference of \\(13\\) and \\(\u221221\\)
                                                          2. subtract \\(24\\) from \\(\u221219\\).
                                                          ", "content": {"items": [[[{"c": "the difference of \\(13\\) and \\(\u221221\\)", "t": "text"}]], [[{"c": "subtract \\(24\\) from \\(\u221219\\).", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate and simplify

                                                          ", "content": [{"c": "Translate and simplify", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. the difference of \\(14\\) and \\(\u221223\\)
                                                          2. subtract \\(21\\) from \\(\u221217\\).
                                                          ", "content": {"items": [[[{"c": "the difference of \\(14\\) and \\(\u221223\\)", "t": "text"}]], [[{"c": "subtract \\(21\\) from \\(\u221217\\).", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(14 - (-23); 37\\)
                                                          2. \\(-17 - 21; -38\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(14 - (-23); 37\\)", "t": "text"}, {"c": "\\(-17 - 21; -38\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate and simplify

                                                          ", "content": [{"c": "Translate and simplify", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. the difference of \\(11\\) and \\(\u221219\\)
                                                          2. subtract \\(18\\) from \\(\u221211\\).
                                                          ", "content": {"items": [[[{"c": "the difference of \\(11\\) and \\(\u221219\\)", "t": "text"}]], [[{"c": "subtract \\(18\\) from \\(\u221211\\).", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(11 - (-19); 30\\)
                                                          2. \\(-11 - 18; -29\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(11 - (-19); 30\\)", "t": "text"}, {"c": "\\(-11 - 18; -29\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is \u201cproduct\u201d and for division is \u201cquotient.\u201d

                                                          ", "content": [{"c": "Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is \u201c product \u201d and for division is \u201c quotient.\u201d", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(\u22122\\) and \\(14\\).

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of \\(\u22122\\) and \\(14\\).", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\\]

                                                          ", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(\u22125\\) and \\(12\\).

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of \\(\u22125\\) and \\(12\\).", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          -5(12); -60
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-5(12); -60", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(8\\) and \\(-13\\).

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of \\(8\\) and \\(-13\\).", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          -8(13); -104
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-8(13); -104", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(\u221256\\) and \\(\u22127\\).

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of \\(\u221256\\) and \\(\u22127\\).", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

                                                          \\[\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\\]

                                                          ", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(\u221263\\) and \\(\u22129\\).

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of \\(\u221263\\) and \\(\u22129\\).", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          -63\\div (-9); 7
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-63\\div (-9); 7", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(\u221272\\) and \\(\u22129\\).

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of \\(\u221272\\) and \\(\u22129\\).", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer
                                                          -72\\div (-9); 8
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-72\\div (-9); 8", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Use Integers in Applications

                                                          ", "content": {"title_content": "Use Integers in Applications", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          We\u2019ll outline a plan to solve applications. It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we\u2019ll write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

                                                          ", "content": [{"c": "We\u2019ll outline a plan to solve applications. It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we\u2019ll write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          How to Apply a Strategy to Solve Applications with Integers

                                                          ", "content": [{"c": "How to Apply a Strategy to Solve Applications with Integers", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          The temperature in Urbana, Illinois one morning was \\(11\\) degrees. By mid-afternoon, the temperature had dropped to \\(\u22129\\) degrees. What was the difference of the morning and afternoon temperatures?

                                                          ", "content": [{"c": "The temperature in Urbana, Illinois one morning was \\(11\\) degrees. By mid-afternoon, the temperature had dropped to \\(\u22129\\) degrees. What was the difference of the morning and afternoon temperatures?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "table", "raw_content": "
                                                          Step 1. Read the problem. Make sure all the words and ideas are understood.
                                                          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
                                                          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
                                                          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
                                                          Step 5. Simplify the expression.\\(20\\)
                                                          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
                                                          ", "content": {"html": "
                                                          Step 1<br>. Read the problem. Make sure all the words and ideas are understood.
                                                          Step 2<br>. Identify what we are asked to find.the difference of the morning and afternoon temperatures
                                                          Step 3<br>. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
                                                          Step 4.<br>Translate the phrase to an expression.\\(11 - (-9)\\)
                                                          Step 5<br>. Simplify the expression.\\(20\\)
                                                          Step 6<br>. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
                                                          ", "is_complex": false}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          The temperature in Anchorage, Alaska one morning was \\(15\\) degrees. By mid-afternoon the temperature had dropped to \\(30\\) degrees below zero. What was the difference in the morning and afternoon temperatures?

                                                          ", "content": [{"c": "The temperature in Anchorage, Alaska one morning was \\(15\\) degrees. By mid-afternoon the temperature had dropped to \\(30\\) degrees below zero. What was the difference in the morning and afternoon temperatures?", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          The difference in temperatures was \\(45\\) degrees.

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "The difference in temperatures was \\(45\\) degrees.", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          The temperature in Denver was \\(\u22126\\) degrees at lunchtime. By sunset the temperature had dropped to \\(\u221215\\) degrees. What was the difference in the lunchtime and sunset temperatures?

                                                          ", "content": [{"c": "The temperature in Denver was \\(\u22126\\) degrees at lunchtime. By sunset the temperature had dropped to \\(\u221215\\) degrees. What was the difference in the lunchtime and sunset temperatures?", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          The difference in temperatures was \\(9\\) degrees.

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "The difference in temperatures was \\(9\\) degrees.", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          1. Read the problem. Make sure all the words and ideas are understood
                                                          2. Identify what we are asked to find.
                                                          3. Write a phrase that gives the information to find it.
                                                          4. Translate the phrase to an expression.
                                                          5. Simplify the expression.
                                                          6. Answer the question with a complete sentence.
                                                          ", "content": {"items": [[[{"c": "Read the problem. Make sure all the words and ideas are understood", "t": "text"}]], [[{"c": "Identify what we are asked to find.", "t": "text"}]], [[{"c": "Write a phrase that gives the information to find it.", "t": "text"}]], [[{"c": "Translate the phrase to an expression.", "t": "text"}]], [[{"c": "Simplify the expression.", "t": "text"}]], [[{"c": "Answer the question with a complete sentence.", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

                                                          ", "content": [{"c": "The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "table", "raw_content": "
                                                          Step 1. Read the problem. Make sure all the words and ideas are understood.
                                                          Step 2. Identify what we are asked to find.the number of yards lost
                                                          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
                                                          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
                                                          Step 5. Simplify the expression.\\(-45\\)
                                                          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.
                                                          ", "content": {"html": "
                                                          Step 1<br>. Read the problem. Make sure all the words and ideas are understood.
                                                          Step 2<br>. Identify what we are asked to find.the number of yards lost
                                                          Step 3<br>. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
                                                          Step 4.<br>Translate the phrase to an expression.\\(3(-15)\\)
                                                          Step 5<br>. Simplify the expression.\\(-45\\)
                                                          Step 6<br>. Write a complete sentence that answers the question.The team lost \\(45\\) yards.
                                                          ", "is_complex": false}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \\(15\\) yards. What is the number of yards lost due to penalties?

                                                          ", "content": [{"c": "The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \\(15\\) yards. What is the number of yards lost due to penalties?", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          The Bears lost \\(105\\) yards.

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "The Bears lost \\(105\\) yards.", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "
                                                          ", "content": {"title_content": null, "level": "5"}}, {"type": "paragraph", "raw_content": "

                                                          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

                                                          ", "content": [{"c": "Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          A $16 fee was deducted from his checking account.

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "A $16 fee was deducted from his checking account.", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Key Concepts

                                                          ", "content": {"title_content": "Key Concepts", "level": "2"}}, {"type": "list", "raw_content": "
                                                          • Multiplication and Division of Two Signed Numbers
                                                            • Same signs\u2014Product is positive
                                                            • Different signs\u2014Product is negative
                                                          • Strategy for Applications
                                                            1. Identify what you are asked to find.
                                                            2. Write a phrase that gives the information to find it.
                                                            3. Translate the phrase to an expression.
                                                            4. Simplify the expression.
                                                            5. Answer the question with a complete sentence.
                                                          ", "content": {"items": [[[{"c": "Multiplication and Division of Two Signed Numbers", "t": "text"}, {"c": "Same signs\u2014Product is positive", "t": "text"}, {"c": "Different signs\u2014Product is negative", "t": "text"}]], [[{"c": "Strategy for Applications", "t": "text"}, {"c": "Identify what you are asked to find.", "t": "text"}, {"c": "Write a phrase that gives the information to find it.", "t": "text"}, {"c": "Translate the phrase to an expression.", "t": "text"}, {"c": "Simplify the expression.", "t": "text"}, {"c": "Answer the question with a complete sentence.", "t": "text"}]]], "ordered": false}}]], "main_html": "

                                                          \n 1.5: Multiply and Divide Integers\n

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                                                          30345
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                                                          • By the end of this section, you will be able to:

                                                          • Multiply integers
                                                          • Divide integers
                                                          • Simplify expressions with integers
                                                          • Evaluate variable expressions with integers
                                                          • Translate English phrases to algebraic expressions
                                                          • Use integers in applications

                                                          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

                                                          Multiply Integers

                                                          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

                                                          We remember that \\(a\\cdot b\\) means add \\(a,\\, b\\) times. Here, we are using the model just to help us discover the pattern.

                                                          \"Two
                                                          Figure \\(\\PageIndex{1}\\)

                                                          The next two examples are more interesting.

                                                          What does it mean to multiply \\(5\\) by \\(\u22123\\)? It means subtract \\(5, 3\\) times. Looking at subtraction as \u201ctaking away,\u201d it means to take away \\(5, 3\\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \\(5\\) three times.

                                                          \"This
                                                          Figure \\(\\PageIndex{2}\\)

                                                          In summary:

                                                          \\[\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\\]

                                                          Notice that for multiplication of two signed numbers, when the:

                                                          • signs are the same, the product is positive.
                                                          • signs are different, the product is negative.

                                                          We\u2019ll put this all together in the chart below.

                                                          For multiplication of two signed numbers:

                                                          Same signsProductExample
                                                          Two positivesPositive\\(7\\cdot 4 = 28\\)
                                                          Two negativesPositive\\(-8(-6) = 48\\)
                                                          Table \\(\\PageIndex{1}\\)
                                                          Different signsProductExample
                                                          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
                                                          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
                                                          Table \\(\\PageIndex{2}\\)

                                                          Multiply:

                                                          1. \\(-9\\cdot 3\\)
                                                          2. \\(-2(-5)\\)
                                                          3. \\(4(-8)\\)
                                                          4. \\(7\\cdot 6\\)

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]
                                                          3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]
                                                          4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]

                                                          Multiply:

                                                          1. \\(-6\\cdot 8\\)
                                                          2. \\(-4(-7)\\)
                                                          3. \\(9(-7)\\)
                                                          4. \\(5\\cdot 12\\)
                                                          Answer
                                                          1. \\(-48\\)
                                                          2. \\(28\\)
                                                          3. \\(-63\\)
                                                          4. \\(60\\)

                                                          Multiply:

                                                          1. \\(-8\\cdot 7\\)
                                                          2. \\(-6(-9)\\)
                                                          3. \\(7(-4)\\)
                                                          4. \\(3\\cdot 13\\)
                                                          Answer
                                                          1. \\(-56\\)
                                                          2. \\(54\\)
                                                          3. \\(-28\\)
                                                          4. \\(39\\)

                                                          When we multiply a number by \\(1\\), the result is the same number. What happens when we multiply a number by \\(\u22121\\)? Let\u2019s multiply a positive number and then a negative number by \\(\u22121\\) to see what we get.

                                                          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]
                                                          \nEach time we multiply a number by \\(\u22121\\), we get its opposite!

                                                          MULTIPLICATION BY \u22121

                                                          \\[\u22121a=\u2212a\\]

                                                          Multiplying a number by \\(\u22121\\) gives its opposite.

                                                          Multiply:

                                                          1. \\(-1 \\cdot 7\\)
                                                          2. \\(-1(-11)\\)

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]

                                                          Multiply:

                                                          1. \\(-1\\cdot 9\\)
                                                          2. \\(-1\\cdot(-17)\\)
                                                          Answer
                                                          1. \\(-9\\)
                                                          2. \\(17\\)

                                                          Multiply:

                                                          1. \\(-1\\cdot 8\\)
                                                          2. \\(-1\\cdot(-16)\\)
                                                          Answer
                                                          1. \\(-8\\)
                                                          2. \\(16\\)

                                                          Divide Integers

                                                          What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.

                                                          \\[\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\\]

                                                          Division follows the same rules as multiplication!

                                                          For division of two signed numbers, when the:

                                                          • signs are the same, the quotient is positive.
                                                          • signs are different, the quotient is negative.

                                                          And remember that we can always check the answer of a division problem by multiplying.

                                                          For multiplication and division of two signed numbers:

                                                          • If the signs are the same, the result is positive.
                                                          • If the signs are different, the result is negative.
                                                          Same signsResult
                                                          Two positivesPositive
                                                          Two negativesPositive
                                                          If the signs are the same, the result is positive.
                                                          Table \\(\\PageIndex{3}\\)
                                                          Different signsResult
                                                          Positive and negativeNegative
                                                          Negative and positiveNegative
                                                          If the signs are different, the result is negative.
                                                          Table \\(\\PageIndex{4}\\)
                                                          1. \\(-27\\div 3\\)
                                                          2. \\(-100\\div (-4)\\)

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]

                                                          Divide:

                                                          1. \\(-42\\div 6\\)
                                                          2. \\(-117\\div (-3)\\)
                                                          Answer
                                                          1. \\(-7\\)
                                                          2. \\(39\\)

                                                          Divide:

                                                          1. \\(-63\\div 7\\)
                                                          2. \\(-115\\div (-5)\\)
                                                          Answer
                                                          1. \\(-9\\)
                                                          2. \\(23\\)

                                                          Simplify Expressions with Integers

                                                          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

                                                          Let\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.

                                                          Simplify:

                                                          7(-2)+4(-7)-6

                                                          Solution

                                                          \\[\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\\]

                                                          Simplify:

                                                          8(-3)+5(-7)-4
                                                          Answer
                                                          -63

                                                          Simplify:

                                                          9(-3)+7(-8)-1
                                                          Answer
                                                          -84

                                                          Simplify:

                                                          1. \\((-2)^{4}\\)
                                                          2. \\(-2^{4}\\)

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]

                                                          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \\((\u22122)\\) to the \\(4^{th}\\) power. In part (2), the exponent means to raise just the \\(2\\) to the \\(4^{th}\\) power and then take the opposite.

                                                          Simplify:

                                                          1. \\((-3)^{4}\\)
                                                          2. \\(-3^{4}\\)
                                                          Answer
                                                          1. \\(81\\)
                                                          2. \\(-81\\)

                                                          Simplify:

                                                          1. \\((-7)^{2}\\)
                                                          2. \\(-7^{2}\\)
                                                          Answer
                                                          1. \\(49\\)
                                                          2. \\(-49\\)

                                                          The next example reminds us to simplify inside parentheses first.

                                                          Simplify:

                                                          12-3(9 - 12)

                                                          Solution

                                                          \\[\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\\]

                                                          Simplify:

                                                          17 - 4(8 - 11)
                                                          Answer
                                                          29

                                                          Simplify:

                                                          16 - 6(7 - 13)
                                                          Answer
                                                          52

                                                          Simplify:

                                                          8(-9)\\div (-2)^{3}

                                                          Solution

                                                          \\[\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\\]

                                                          Simplify:

                                                          12(-9)\\div (-3)^{3}
                                                          Answer
                                                          4

                                                          Simplify:

                                                          18(-4)\\div (-2)^{3}
                                                          Answer
                                                          9

                                                          Simplify:

                                                          -30\\div 2 + (-3)(-7)

                                                          Solution

                                                          \\[\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\\]

                                                          Simplify:

                                                          -27\\div 3 + (-5)(-6)
                                                          Answer
                                                          21

                                                          Simplify:

                                                          -32\\div 4 + (-2)(-7)
                                                          Answer
                                                          6

                                                          Evaluate Variable Expressions with Integers

                                                          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

                                                          When \\(n=\u22125\\), evaluate:

                                                          1. \\(n+1\\)
                                                          2. \\(\u2212n+1\\).

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1\u00a0\\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{-\u00a0{\\color{red}{(-5)}}\u00a0+1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]

                                                          When \\(n=\u22128\\), evaluate:

                                                          1. \\(n+2\\)
                                                          2. \\(\u2212n+2\\).
                                                          Answer
                                                          1. \\(-6\\)
                                                          2. \\(10\\)

                                                          When \\(y=\u22129\\), evaluate:

                                                          1. \\(y+8\\)
                                                          2. \\(\u2212y+8\\).
                                                          Answer
                                                          1. \\(-1\\)
                                                          2. \\(17\\)

                                                          Evaluate \\((x+y)^{2}\\) when \\(x = -18\\) and \\(y = 24\\).

                                                          Solution

                                                          \\[\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\\]

                                                          Evaluate \\((x+y)^{2}\\) when \\(x = -15\\) and \\(y = 29\\).

                                                          Answer
                                                          196

                                                          Evaluate \\((x+y)^{3}\\) when \\(x = -8\\) and \\(y = 10\\).

                                                          Answer
                                                          8

                                                          Evaluate \\(20 -z \\) when

                                                          1. \\(z = 12\\)
                                                          2. \\(z = -12\\)

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]

                                                          Evaluate \\(17 - k\\) when

                                                          1. \\(k = 19\\)
                                                          2. \\(k = -19\\)
                                                          Answer
                                                          1. \\(-2\\)
                                                          2. \\(36\\)

                                                          Evaluate \\(-5 - b\\) when

                                                          1. \\(b = 14\\)
                                                          2. \\(b = -14\\)
                                                          Answer
                                                          1. \\(-19\\)
                                                          2. \\(9\\)

                                                          Evaluate:

                                                          \\(2x^{2} + 3x + 8\\) when \\(x = 4\\).

                                                          Solution

                                                          Substitute \\(4\\) for \\(x\\). Use parentheses to show multiplication.

                                                          \\[\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\\]

                                                          Evaluate:

                                                          \\(3x^{2} - 2x + 6\\) when \\(x =-3\\).

                                                          Answer
                                                          39

                                                          Evaluate:

                                                          \\(4x^{2} - x - 5\\) when \\(x = -2\\).

                                                          Answer
                                                          13

                                                          Translate Phrases to Expressions with Integers

                                                          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

                                                          Translate and simplify: the sum of \\(8\\) and \\(\u221212\\), increased by \\(3\\).

                                                          Solution

                                                          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\\]

                                                          Translate and simplify: the sum of \\(9\\) and \\(\u221216\\), increased by \\(4\\).

                                                          Answer
                                                          (9 + (-16)) + 4 - 3

                                                          Translate and simplify: the sum of \\(-8\\) and \\(\u221212\\), increased by \\(7\\).

                                                          Answer
                                                          (-8 + (-12)) + 7 - 13

                                                          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

                                                          \\(a\u2212b\\)
                                                          \\(a\\) minus \\(b\\)
                                                          \n the difference of \\(a\\) and \\(b\\)
                                                          \n \\(b\\) subtracted from \\(a\\)
                                                          \n \\(b\\) less than \\(a\\)
                                                          Table \\(\\PageIndex{5}\\)

                                                          Be careful to get a and b in the right order!

                                                          Translate and then simplify

                                                          1. the difference of \\(13\\) and \\(\u221221\\)
                                                          2. subtract \\(24\\) from \\(\u221219\\).

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]

                                                          Translate and simplify

                                                          1. the difference of \\(14\\) and \\(\u221223\\)
                                                          2. subtract \\(21\\) from \\(\u221217\\).
                                                          Answer
                                                          1. \\(14 - (-23); 37\\)
                                                          2. \\(-17 - 21; -38\\)

                                                          Translate and simplify

                                                          1. the difference of \\(11\\) and \\(\u221219\\)
                                                          2. subtract \\(18\\) from \\(\u221211\\).
                                                          Answer
                                                          1. \\(11 - (-19); 30\\)
                                                          2. \\(-11 - 18; -29\\)

                                                          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is \u201cproduct\u201d and for division is \u201cquotient.\u201d

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(\u22122\\) and \\(14\\).

                                                          Solution

                                                          \\[\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\\]

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(\u22125\\) and \\(12\\).

                                                          Answer
                                                          -5(12); -60

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(8\\) and \\(-13\\).

                                                          Answer
                                                          -8(13); -104

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(\u221256\\) and \\(\u22127\\).

                                                          Solution

                                                          \\[\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\\]

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(\u221263\\) and \\(\u22129\\).

                                                          Answer
                                                          -63\\div (-9); 7

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(\u221272\\) and \\(\u22129\\).

                                                          Answer
                                                          -72\\div (-9); 8

                                                          Use Integers in Applications

                                                          We\u2019ll outline a plan to solve applications. It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we\u2019ll write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

                                                          How to Apply a Strategy to Solve Applications with Integers

                                                          The temperature in Urbana, Illinois one morning was \\(11\\) degrees. By mid-afternoon, the temperature had dropped to \\(\u22129\\) degrees. What was the difference of the morning and afternoon temperatures?

                                                          Solution

                                                          Step 1. Read the problem. Make sure all the words and ideas are understood.
                                                          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
                                                          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
                                                          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
                                                          Step 5. Simplify the expression.\\(20\\)
                                                          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.

                                                          The temperature in Anchorage, Alaska one morning was \\(15\\) degrees. By mid-afternoon the temperature had dropped to \\(30\\) degrees below zero. What was the difference in the morning and afternoon temperatures?

                                                          Answer

                                                          The difference in temperatures was \\(45\\) degrees.

                                                          The temperature in Denver was \\(\u22126\\) degrees at lunchtime. By sunset the temperature had dropped to \\(\u221215\\) degrees. What was the difference in the lunchtime and sunset temperatures?

                                                          Answer

                                                          The difference in temperatures was \\(9\\) degrees.

                                                          1. Read the problem. Make sure all the words and ideas are understood
                                                          2. Identify what we are asked to find.
                                                          3. Write a phrase that gives the information to find it.
                                                          4. Translate the phrase to an expression.
                                                          5. Simplify the expression.
                                                          6. Answer the question with a complete sentence.

                                                          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

                                                          Solution

                                                          Step 1. Read the problem. Make sure all the words and ideas are understood.
                                                          Step 2. Identify what we are asked to find.the number of yards lost
                                                          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
                                                          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
                                                          Step 5. Simplify the expression.\\(-45\\)
                                                          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.

                                                          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \\(15\\) yards. What is the number of yards lost due to penalties?

                                                          Answer

                                                          The Bears lost \\(105\\) yards.

                                                          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

                                                          Answer

                                                          A $16 fee was deducted from his checking account.

                                                          Key Concepts

                                                          • Multiplication and Division of Two Signed Numbers
                                                            • Same signs\u2014Product is positive
                                                            • Different signs\u2014Product is negative
                                                          • Strategy for Applications
                                                            1. Identify what you are asked to find.
                                                            2. Write a phrase that gives the information to find it.
                                                            3. Translate the phrase to an expression.
                                                            4. Simplify the expression.
                                                            5. Answer the question with a complete sentence.
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\\(\\definecolor{fillinmathshade}{gray}{0.9}\\)\n\nBy the end of this section, you will be able to:\n\n- Multiply integers\n- Divide integers\n- Simplify expressions with integers\n- Evaluate variable expressions with integers\n- Translate English phrases to algebraic expressions\n- Use integers in applications\n\nA more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.\n\n## Multiply Integers\n\nSince multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.\n\nWe remember that \\(a\\cdot b\\) means add \\(a,\\, b\\) times. Here, we are using the model just to help us discover the pattern.\n\nThe next two examples are more interesting.\n\nWhat does it mean to multiply \\(5\\) by \\(\u22123\\)? It means subtract \\(5, 3\\) times. Looking at subtraction as \u201ctaking away,\u201d it means to take away \\(5, 3\\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \\(5\\) three times.\n\nIn summary:\n\n$$\n\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\n$$\n\nNotice that for multiplication of two signed numbers, when the:\n\n- signs are the same , the product is positive .\n- signs are different , the product is negative .\n\nWe\u2019ll put this all together in the chart below.\n\nFor multiplication of two signed numbers:\n\n| Same signs | Product | Example |\n|---|---|---|\n| Two positives | Positive | \\(7\\cdot 4 = 28\\) |\n| Two negatives | Positive | \\(-8(-6) = 48\\) |\n\n| Different signs | Product | Example |\n|---|---|---|\n| Positives \\(\\cdot\\) negative | Negative | \\(7(-9) = -63\\) |\n| Negative \\(\\cdot\\) positives | Negative | \\(-5\\cdot 10= -50\\) |\n\nMultiply:\n\n1. \\(-9\\cdot 3\\)\n2. \\(-2(-5)\\)\n3. \\(4(-8)\\)\n4. \\(7\\cdot 6\\)\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]\n3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]\n4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]\n\nMultiply:\n\n1. \\(-6\\cdot 8\\)\n2. \\(-4(-7)\\)\n3. \\(9(-7)\\)\n4. \\(5\\cdot 12\\)\n\n1. Answer\n2. \\(-48\\) \\(28\\) \\(-63\\) \\(60\\)\n\nMultiply:\n\n1. \\(-8\\cdot 7\\)\n2. \\(-6(-9)\\)\n3. \\(7(-4)\\)\n4. \\(3\\cdot 13\\)\n\n1. Answer\n2. \\(-56\\) \\(54\\) \\(-28\\) \\(39\\)\n\nWhen we multiply a number by \\(1\\), the result is the same number. What happens when we multiply a number by \\(\u22121\\)? Let\u2019s multiply a positive number and then a negative number by \\(\u22121\\) to see what we get.\n\n\\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\] Each time we multiply a number by \\(\u22121\\), we get its opposite!\n\nMULTIPLICATION BY \u22121\n\n$$\n\u22121a=\u2212a\n$$\n\nMultiplying a number by \\(\u22121\\) gives its opposite.\n\nMultiply:\n\n1. \\(-1 \\cdot 7\\)\n2. \\(-1(-11)\\)\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]\n\nMultiply:\n\n1. \\(-1\\cdot 9\\)\n2. \\(-1\\cdot(-17)\\)\n\n1. Answer\n2. \\(-9\\) \\(17\\)\n\nMultiply:\n\n1. \\(-1\\cdot 8\\)\n2. \\(-1\\cdot(-16)\\)\n\n1. Answer\n2. \\(-8\\) \\(16\\)\n\n## Divide Integers\n\nWhat about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.\n\n$$\n\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\n$$\n\nDivision follows the same rules as multiplication!\n\nFor division of two signed numbers, when the:\n\n- signs are the same , the quotient is positive .\n- signs are different , the quotient is negative .\n\nAnd remember that we can always check the answer of a division problem by multiplying.\n\nFor multiplication and division of two signed numbers:\n\n- If the signs are the same, the result is positive.\n- If the signs are different, the result is negative.\n\n| Same signs | Result |\n|---|---|\n| Two positives | Positive |\n| Two negatives | Positive |\n| If the signs are the same, the result is positive. | |\n\n| Different signs | Result |\n|---|---|\n| Positive and negative | Negative |\n| Negative and positive | Negative |\n| If the signs are different, the result is negative. | |\n\n1. \\(-27\\div 3\\)\n2. \\(-100\\div (-4)\\)\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]\n\nDivide:\n\n1. \\(-42\\div 6\\)\n2. \\(-117\\div (-3)\\)\n\n1. Answer\n2. \\(-7\\) \\(39\\)\n\nDivide:\n\n1. \\(-63\\div 7\\)\n2. \\(-115\\div (-5)\\)\n\n1. Answer\n2. \\(-9\\) \\(23\\)\n\n## Simplify Expressions with Integers\n\nWhat happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?\n\nLet\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.\n\nSimplify:\n\n$7(-2)+4(-7)-6$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\n$$\n\nSimplify:\n\n$8(-3)+5(-7)-4$\n\n1. Answer\n2. $-63$\n\nSimplify:\n\n$9(-3)+7(-8)-1$\n\n1. Answer\n2. $-84$\n\nSimplify:\n\n1. \\((-2)^{4}\\)\n2. \\(-2^{4}\\)\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]\n\nNotice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \\((\u22122)\\) to the \\(4^{th}\\) power. In part (2), the exponent means to raise just the \\(2\\) to the \\(4^{th}\\) power and then take the opposite.\n\nSimplify:\n\n1. \\((-3)^{4}\\)\n2. \\(-3^{4}\\)\n\n1. Answer\n2. \\(81\\) \\(-81\\)\n\nSimplify:\n\n1. \\((-7)^{2}\\)\n2. \\(-7^{2}\\)\n\n1. Answer\n2. \\(49\\) \\(-49\\)\n\nThe next example reminds us to simplify inside parentheses first.\n\nSimplify:\n\n$12-3(9 - 12)$\n\nSolution\n\n$$\n\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\n$$\n\nSimplify:\n\n$17 - 4(8 - 11)$\n\n1. Answer\n2. $29$\n\nSimplify:\n\n$16 - 6(7 - 13)$\n\n1. Answer\n2. $52$\n\nSimplify:\n\n$8(-9)\\div (-2)^{3}$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\n$$\n\nSimplify:\n\n$12(-9)\\div (-3)^{3}$\n\n1. Answer\n2. $4$\n\nSimplify:\n\n$18(-4)\\div (-2)^{3}$\n\n1. Answer\n2. $9$\n\nSimplify:\n\n$-30\\div 2 + (-3)(-7)$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\n$$\n\nSimplify:\n\n$-27\\div 3 + (-5)(-6)$\n\n1. Answer\n2. $21$\n\nSimplify:\n\n$-32\\div 4 + (-2)(-7)$\n\n1. Answer\n2. $6$\n\n## Evaluate Variable Expressions with Integers\n\nRemember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.\n\nWhen \\(n=\u22125\\), evaluate:\n\n1. \\(n+1\\)\n2. \\(\u2212n+1\\).\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1\u00a0\\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{-\u00a0{\\color{red}{(-5)}}\u00a0+1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]\n\nWhen \\(n=\u22128\\), evaluate:\n\n1. \\(n+2\\)\n2. \\(\u2212n+2\\).\n\n1. Answer\n2. \\(-6\\) \\(10\\)\n\nWhen \\(y=\u22129\\), evaluate:\n\n1. \\(y+8\\)\n2. \\(\u2212y+8\\).\n\n1. Answer\n2. \\(-1\\) \\(17\\)\n\nEvaluate \\((x+y)^{2}\\) when \\(x = -18\\) and \\(y = 24\\).\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\n$$\n\nEvaluate \\((x+y)^{2}\\) when \\(x = -15\\) and \\(y = 29\\).\n\n1. Answer\n2. $196$\n\nEvaluate \\((x+y)^{3}\\) when \\(x = -8\\) and \\(y = 10\\).\n\n1. Answer\n2. $8$\n\nEvaluate \\(20 -z \\) when\n\n1. \\(z = 12\\)\n2. \\(z = -12\\)\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]\n\nEvaluate \\(17 - k\\) when\n\n1. \\(k = 19\\)\n2. \\(k = -19\\)\n\n1. Answer\n2. \\(-2\\) \\(36\\)\n\nEvaluate \\(-5 - b\\) when\n\n1. \\(b = 14\\)\n2. \\(b = -14\\)\n\n1. Answer\n2. \\(-19\\) \\(9\\)\n\nEvaluate:\n\n\\(2x^{2} + 3x + 8\\) when \\(x = 4\\).\n\nSolution\n\nSubstitute \\(4\\) for \\(x\\). Use parentheses to show multiplication.\n\n$$\n\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\n$$\n\nEvaluate:\n\n\\(3x^{2} - 2x + 6\\) when \\(x =-3\\).\n\n1. Answer\n2. $39$\n\nEvaluate:\n\n\\(4x^{2} - x - 5\\) when \\(x = -2\\).\n\n1. Answer\n2. $13$\n\n## Translate Phrases to Expressions with Integers\n\nOur earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.\n\nTranslate and simplify: the sum of \\(8\\) and \\(\u221212\\), increased by \\(3\\).\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\n$$\n\nTranslate and simplify: the sum of \\(9\\) and \\(\u221216\\), increased by \\(4\\).\n\n1. Answer\n2. $(9 + (-16)) + 4 - 3$\n\nTranslate and simplify: the sum of \\(-8\\) and \\(\u221212\\), increased by \\(7\\).\n\n1. Answer\n2. $(-8 + (-12)) + 7 - 13$\n\nWhen we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.\n\n| \\(a\u2212b\\) |\n|---|\n| \\(a\\) minus \\(b\\)
                                                          the difference of \\(a\\) and \\(b\\)
                                                          \\(b\\) subtracted from \\(a\\)
                                                          \\(b\\) less than \\(a\\) |\n\nBe careful to get a and b in the right order!\n\nTranslate and then simplify\n\n1. the difference of \\(13\\) and \\(\u221221\\)\n2. subtract \\(24\\) from \\(\u221219\\).\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]\n\nTranslate and simplify\n\n1. the difference of \\(14\\) and \\(\u221223\\)\n2. subtract \\(21\\) from \\(\u221217\\).\n\n1. Answer\n2. \\(14 - (-23); 37\\) \\(-17 - 21; -38\\)\n\nTranslate and simplify\n\n1. the difference of \\(11\\) and \\(\u221219\\)\n2. subtract \\(18\\) from \\(\u221211\\).\n\n1. Answer\n2. \\(11 - (-19); 30\\) \\(-11 - 18; -29\\)\n\nOnce again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is \u201c product \u201d and for division is \u201c quotient.\u201d\n\nTranslate to an algebraic expression and simplify if possible: the product of \\(\u22122\\) and \\(14\\).\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\n$$\n\nTranslate to an algebraic expression and simplify if possible: the product of \\(\u22125\\) and \\(12\\).\n\n1. Answer\n2. $-5(12); -60$\n\nTranslate to an algebraic expression and simplify if possible: the product of \\(8\\) and \\(-13\\).\n\n1. Answer\n2. $-8(13); -104$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of \\(\u221256\\) and \\(\u22127\\).\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\n$$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of \\(\u221263\\) and \\(\u22129\\).\n\n1. Answer\n2. $-63\\div (-9); 7$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of \\(\u221272\\) and \\(\u22129\\).\n\n1. Answer\n2. $-72\\div (-9); 8$\n\n## Use Integers in Applications\n\nWe\u2019ll outline a plan to solve applications. It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we\u2019ll write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.\n\nHow to Apply a Strategy to Solve Applications with Integers\n\nThe temperature in Urbana, Illinois one morning was \\(11\\) degrees. By mid-afternoon, the temperature had dropped to \\(\u22129\\) degrees. What was the difference of the morning and afternoon temperatures?\n\nSolution\n\n| Step 1
                                                          . Read the problem. Make sure all the words and ideas are understood. | |\n|---|---|\n| Step 2
                                                          . Identify what we are asked to find. | the difference of the morning and afternoon temperatures |\n| Step 3
                                                          . Write a phrase that gives the information to find it. | the difference of \\(11\\) and \\(-9\\) |\n| Step 4.
                                                          Translate the phrase to an expression. | \\(11 - (-9)\\) |\n| Step 5
                                                          . Simplify the expression. | \\(20\\) |\n| Step 6
                                                          . Write a complete sentence that answers the question. | The difference in temperatures was 20 degrees. |\n\nThe temperature in Anchorage, Alaska one morning was \\(15\\) degrees. By mid-afternoon the temperature had dropped to \\(30\\) degrees below zero. What was the difference in the morning and afternoon temperatures?\n\n1. Answer\n2. The difference in temperatures was \\(45\\) degrees.\n\nThe temperature in Denver was \\(\u22126\\) degrees at lunchtime. By sunset the temperature had dropped to \\(\u221215\\) degrees. What was the difference in the lunchtime and sunset temperatures?\n\n1. Answer\n2. The difference in temperatures was \\(9\\) degrees.\n\n1. Read the problem. Make sure all the words and ideas are understood\n2. Identify what we are asked to find.\n3. Write a phrase that gives the information to find it.\n4. Translate the phrase to an expression.\n5. Simplify the expression.\n6. Answer the question with a complete sentence.\n\nThe Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?\n\nSolution\n\n| Step 1
                                                          . Read the problem. Make sure all the words and ideas are understood. | |\n|---|---|\n| Step 2
                                                          . Identify what we are asked to find. | the number of yards lost |\n| Step 3
                                                          . Write a phrase that gives the information to find it. | three times a \\(15\\)-yard penalty |\n| Step 4.
                                                          Translate the phrase to an expression. | \\(3(-15)\\) |\n| Step 5
                                                          . Simplify the expression. | \\(-45\\) |\n| Step 6
                                                          . Write a complete sentence that answers the question. | The team lost \\(45\\) yards. |\n\nThe Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \\(15\\) yards. What is the number of yards lost due to penalties?\n\n1. Answer\n2. The Bears lost \\(105\\) yards.\n\nBill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?\n\n1. Answer\n2. A $16 fee was deducted from his checking account.\n\n## Key Concepts\n\n- Multiplication and Division of Two Signed Numbers Same signs\u2014Product is positive Different signs\u2014Product is negative\n- Strategy for Applications Identify what you are asked to find. Write a phrase that gives the information to find it. Translate the phrase to an expression. Simplify the expression. Answer the question with a complete sentence.\n", "html": "\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n\n 1.5: Multiply and Divide Integers - Mathematics LibreTexts\n\n\n \n\n \n\n\n \n\n\n \n \n\n\n \n\n\n\n\n\n\nSkip to main content
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                                                          1: Foundations
                                                          MTH 098 Elementary Algebra
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                                                          Mon, 06 Jan 2020 03:19:01 GMT
                                                          1.5: Multiply and Divide Integers
                                                          30345
                                                          30345
                                                          Paul Seeburger
                                                          { }
                                                          Anonymous
                                                          Anonymous
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                                                          [ "article:topic", "license:ccby", "showtoc:no", "transcluded:yes", "source[1]-math-15120" ]
                                                          [ "article:topic", "license:ccby", "showtoc:no", "transcluded:yes", "source[1]-math-15120" ]
                                                          https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FUnder_Construction%2FPurgatory%2FRemixer_University%2FUsername%253A_pseeburger%2FMTH_098_Elementary_Algebra%2F1%253A_Foundations%2F1.5%253A_Multiply_and_Divide_Integers
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                                                          \n 1.5: Multiply and Divide Integers\n

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                                                        • Page ID
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                                                          Learning Objectives
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                                                          By the end of this section, you will be able to:

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                                                          • Multiply integers
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                                                          • Divide integers
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                                                          • Simplify expressions with integers
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                                                          • Evaluate variable expressions with integers
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                                                          • Translate English phrases to algebraic expressions
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                                                          • Use integers in applications
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                                                          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

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                                                          Multiply Integers

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                                                          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

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                                                          We remember that \\(a\\cdot b\\) means add \\(a,\\, b\\) times. Here, we are using the model just to help us discover the pattern.

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                                                          \"Two\n
                                                          Figure \\(\\PageIndex{1}\\)
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                                                          The next two examples are more interesting.

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                                                          What does it mean to multiply \\(5\\) by \\(−3\\)? It means subtract \\(5, 3\\) times. Looking at subtraction as “taking away,” it means to take away \\(5, 3\\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \\(5\\) three times.

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                                                          \"This\n
                                                          Figure \\(\\PageIndex{2}\\)
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                                                          In summary:

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                                                          \\[\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\\]

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                                                          Notice that for multiplication of two signed numbers, when the:

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                                                          • signs are the same, the product is positive.
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                                                          • signs are different, the product is negative.
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                                                          We’ll put this all together in the chart below.

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                                                          MULTIPLICATION OF SIGNED NUMBERS
                                                          \n\n

                                                          For multiplication of two signed numbers:

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                                                          Same signsProductExample
                                                          Two positivesPositive\\(7\\cdot 4 = 28\\)
                                                          Two negativesPositive\\(-8(-6) = 48\\)
                                                          Table \\(\\PageIndex{1}\\)
                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                          Different signsProductExample
                                                          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
                                                          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
                                                          Table \\(\\PageIndex{2}\\)
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                                                          Example \\(\\PageIndex{1}\\)
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                                                          Multiply:

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                                                            \n
                                                          1. \\(-9\\cdot 3\\)
                                                            2. \n
                                                          2. \\(-2(-5)\\)
                                                            3. \n
                                                          3. \\(4(-8)\\)
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                                                          4. \\(7\\cdot 6\\)
                                                            5. \n
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                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]
                                                            3. \n
                                                          3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]
                                                            4. \n
                                                          4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]
                                                            5. \n
                                                          \n
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                                                          \n
                                                          Try It \\(\\PageIndex{2}\\)
                                                          \n\n

                                                          Multiply:

                                                          \n\n
                                                            \n
                                                          1. \\(-6\\cdot 8\\)
                                                            2. \n
                                                          2. \\(-4(-7)\\)
                                                            3. \n
                                                          3. \\(9(-7)\\)
                                                            4. \n
                                                          4. \\(5\\cdot 12\\)
                                                            5. \n
                                                          \n\n
                                                          \n
                                                          Answer
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                                                          \n
                                                            \n
                                                          1. \\(-48\\)
                                                            2. \n
                                                          2. \\(28\\)
                                                            3. \n
                                                          3. \\(-63\\)
                                                            4. \n
                                                          4. \\(60\\)
                                                            5. \n
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                                                          \n
                                                          Try It \\(\\PageIndex{3}\\)
                                                          \n\n

                                                          Multiply:

                                                          \n\n
                                                            \n
                                                          1. \\(-8\\cdot 7\\)
                                                            2. \n
                                                          2. \\(-6(-9)\\)
                                                            3. \n
                                                          3. \\(7(-4)\\)
                                                            4. \n
                                                          4. \\(3\\cdot 13\\)
                                                            5. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-56\\)
                                                            2. \n
                                                          2. \\(54\\)
                                                            3. \n
                                                          3. \\(-28\\)
                                                            4. \n
                                                          4. \\(39\\)
                                                            5. \n
                                                          \n
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                                                          \n\n

                                                          When we multiply a number by \\(1\\), the result is the same number. What happens when we multiply a number by \\(−1\\)? Let’s multiply a positive number and then a negative number by \\(−1\\) to see what we get.

                                                          \n\n

                                                          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]
                                                          \nEach time we multiply a number by \\(−1\\), we get its opposite!

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                                                          \n
                                                           
                                                          \n\n

                                                          MULTIPLICATION BY −1

                                                          \n\n

                                                          \\[−1a=−a\\]

                                                          \n\n

                                                          Multiplying a number by \\(−1\\) gives its opposite.

                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{4}\\)
                                                          \n\n

                                                          Multiply:

                                                          \n\n
                                                            \n
                                                          1. \\(-1 \\cdot 7\\)
                                                            2. \n
                                                          2. \\(-1(-11)\\)
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]
                                                            3. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{5}\\)
                                                          \n\n

                                                          Multiply:

                                                          \n\n
                                                            \n
                                                          1. \\(-1\\cdot 9\\)
                                                            2. \n
                                                          2. \\(-1\\cdot(-17)\\)
                                                            3. \n
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                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-9\\)
                                                            2. \n
                                                          2. \\(17\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{6}\\)
                                                          \n\n

                                                          Multiply:

                                                          \n\n
                                                            \n
                                                          1. \\(-1\\cdot 8\\)
                                                            2. \n
                                                          2. \\(-1\\cdot(-16)\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-8\\)
                                                            2. \n
                                                          2. \\(16\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Divide Integers

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                                                          What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.

                                                          \n\n

                                                          \\[\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\\]

                                                          \n\n

                                                          Division follows the same rules as multiplication!

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                                                          For division of two signed numbers, when the:

                                                          \n\n
                                                            \n
                                                          • signs are the same, the quotient is positive.
                                                            • \n
                                                          • signs are different, the quotient is negative.
                                                            • \n
                                                          \n\n

                                                          And remember that we can always check the answer of a division problem by multiplying.

                                                          \n\n
                                                          \n
                                                          MULTIPLICATION AND DIVISION OF SIGNED NUMBERS
                                                          \n\n

                                                          For multiplication and division of two signed numbers:

                                                          \n\n
                                                            \n
                                                          • If the signs are the same, the result is positive.
                                                            • \n
                                                          • If the signs are different, the result is negative.
                                                            • \n
                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                          Same signsResult
                                                          Two positivesPositive
                                                          Two negativesPositive
                                                          If the signs are the same, the result is positive.
                                                          Table \\(\\PageIndex{3}\\)
                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                          Different signsResult
                                                          Positive and negativeNegative
                                                          Negative and positiveNegative
                                                          If the signs are different, the result is negative.
                                                          Table \\(\\PageIndex{4}\\)
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{7}\\)
                                                          \n\n
                                                            \n
                                                          1. \\(-27\\div 3\\)
                                                            2. \n
                                                          2. \\(-100\\div (-4)\\)
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]
                                                            3. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{8}\\)
                                                          \n\n

                                                          Divide:

                                                          \n\n
                                                            \n
                                                          1. \\(-42\\div 6\\)
                                                            2. \n
                                                          2. \\(-117\\div (-3)\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-7\\)
                                                            2. \n
                                                          2. \\(39\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{9}\\)
                                                          \n\n

                                                          Divide:

                                                          \n\n
                                                            \n
                                                          1. \\(-63\\div 7\\)
                                                            2. \n
                                                          2. \\(-115\\div (-5)\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-9\\)
                                                            2. \n
                                                          2. \\(23\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Simplify Expressions with Integers

                                                          \n\n

                                                          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

                                                          \n\n

                                                          Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{10}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(7(-2)+4(-7)-6\\)

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{11}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(8(-3)+5(-7)-4\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-63\\)

                                                          \n
                                                          \n
                                                          \n
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                                                          \n
                                                          Try It \\(\\PageIndex{12}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(9(-3)+7(-8)-1\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-84\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{13}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n
                                                            \n
                                                          1. \\((-2)^{4}\\)
                                                            2. \n
                                                          2. \\(-2^{4}\\)
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]
                                                            3. \n
                                                          \n\n

                                                          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \\((−2)\\) to the \\(4^{th}\\) power. In part (2), the exponent means to raise just the \\(2\\) to the \\(4^{th}\\) power and then take the opposite.

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{14}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n
                                                            \n
                                                          1. \\((-3)^{4}\\)
                                                            2. \n
                                                          2. \\(-3^{4}\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(81\\)
                                                            2. \n
                                                          2. \\(-81\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{15}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n
                                                            \n
                                                          1. \\((-7)^{2}\\)
                                                            2. \n
                                                          2. \\(-7^{2}\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(49\\)
                                                            2. \n
                                                          2. \\(-49\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          The next example reminds us to simplify inside parentheses first.

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{16}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(12-3(9 - 12)\\)

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{17}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(17 - 4(8 - 11)\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(29\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{18}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(16 - 6(7 - 13)\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(52\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{19}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(8(-9)\\div (-2)^{3}\\)

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{20}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(12(-9)\\div (-3)^{3}\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(4\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{21}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(18(-4)\\div (-2)^{3}\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(9\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{22}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(-30\\div 2 + (-3)(-7)\\)

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{23}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(-27\\div 3 + (-5)(-6)\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(21\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{24}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(-32\\div 4 + (-2)(-7)\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(6\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Evaluate Variable Expressions with Integers

                                                          \n\n

                                                          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{25}\\)
                                                          \n\n

                                                          When \\(n=−5\\), evaluate:

                                                          \n\n
                                                            \n
                                                          1. \\(n+1\\)
                                                            2. \n
                                                          2. \\(−n+1\\).
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]
                                                            3. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{26}\\)
                                                          \n\n

                                                          When \\(n=−8\\), evaluate:

                                                          \n\n
                                                            \n
                                                          1. \\(n+2\\)
                                                            2. \n
                                                          2. \\(−n+2\\).
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-6\\)
                                                            2. \n
                                                          2. \\(10\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{27}\\)
                                                          \n\n

                                                          When \\(y=−9\\), evaluate:

                                                          \n\n
                                                            \n
                                                          1. \\(y+8\\)
                                                            2. \n
                                                          2. \\(−y+8\\).
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-1\\)
                                                            2. \n
                                                          2. \\(17\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{28}\\)
                                                          \n\n

                                                          Evaluate \\((x+y)^{2}\\) when \\(x = -18\\) and \\(y = 24\\).

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{29}\\)
                                                          \n\n

                                                          Evaluate \\((x+y)^{2}\\) when \\(x = -15\\) and \\(y = 29\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(196\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{30}\\)
                                                          \n\n

                                                          Evaluate \\((x+y)^{3}\\) when \\(x = -8\\) and \\(y = 10\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(8\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{31}\\)
                                                          \n\n

                                                          Evaluate \\(20 -z \\) when

                                                          \n\n
                                                            \n
                                                          1. \\(z = 12\\)
                                                            2. \n
                                                          2. \\(z = -12\\)
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]
                                                            3. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{32}\\)
                                                          \n\n

                                                          Evaluate \\(17 - k\\) when

                                                          \n\n
                                                            \n
                                                          1. \\(k = 19\\)
                                                            2. \n
                                                          2. \\(k = -19\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-2\\)
                                                            2. \n
                                                          2. \\(36\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{33}\\)
                                                          \n\n

                                                          Evaluate \\(-5 - b\\) when

                                                          \n\n
                                                            \n
                                                          1. \\(b = 14\\)
                                                            2. \n
                                                          2. \\(b = -14\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-19\\)
                                                            2. \n
                                                          2. \\(9\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{34}\\)
                                                          \n\n

                                                          Evaluate:

                                                          \n\n

                                                          \\(2x^{2} + 3x + 8\\) when \\(x = 4\\).

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          Substitute \\(4\\) for \\(x\\). Use parentheses to show multiplication.

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{35}\\)
                                                          \n\n

                                                          Evaluate:

                                                          \n\n

                                                          \\(3x^{2} - 2x + 6\\) when \\(x =-3\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(39\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{36}\\)
                                                          \n\n

                                                          Evaluate:

                                                          \n\n

                                                          \\(4x^{2} - x - 5\\) when \\(x = -2\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(13\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Translate Phrases to Expressions with Integers

                                                          \n\n

                                                          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{37}\\)
                                                          \n\n

                                                          Translate and simplify: the sum of \\(8\\) and \\(−12\\), increased by \\(3\\).

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{38}\\)
                                                          \n\n

                                                          Translate and simplify: the sum of \\(9\\) and \\(−16\\), increased by \\(4\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\((9 + (-16)) + 4 - 3\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{39}\\)
                                                          \n\n

                                                          Translate and simplify: the sum of \\(-8\\) and \\(−12\\), increased by \\(7\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\((-8 + (-12)) + 7 - 13\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n
                                                          \\(a−b\\)
                                                          \\(a\\) minus \\(b\\)
                                                          \n the difference of \\(a\\) and \\(b\\)
                                                          \n \\(b\\) subtracted from \\(a\\)
                                                          \n \\(b\\) less than \\(a\\)
                                                          Table \\(\\PageIndex{5}\\)
                                                          \n\n

                                                          Be careful to get a and b in the right order!

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{40}\\)
                                                          \n\n

                                                          Translate and then simplify

                                                          \n\n
                                                            \n
                                                          1. the difference of \\(13\\) and \\(−21\\)
                                                            2. \n
                                                          2. subtract \\(24\\) from \\(−19\\).
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]
                                                            3. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{41}\\)
                                                          \n\n

                                                          Translate and simplify

                                                          \n\n
                                                            \n
                                                          1. the difference of \\(14\\) and \\(−23\\)
                                                            2. \n
                                                          2. subtract \\(21\\) from \\(−17\\).
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(14 - (-23); 37\\)
                                                            2. \n
                                                          2. \\(-17 - 21; -38\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{42}\\)
                                                          \n\n

                                                          Translate and simplify

                                                          \n\n
                                                            \n
                                                          1. the difference of \\(11\\) and \\(−19\\)
                                                            2. \n
                                                          2. subtract \\(18\\) from \\(−11\\).
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(11 - (-19); 30\\)
                                                            2. \n
                                                          2. \\(-11 - 18; -29\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{43}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(−2\\) and \\(14\\).

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{44}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(−5\\) and \\(12\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-5(12); -60\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{45}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(8\\) and \\(-13\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-8(13); -104\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{46}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(−56\\) and \\(−7\\).

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{47}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(−63\\) and \\(−9\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-63\\div (-9); 7\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{48}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(−72\\) and \\(−9\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-72\\div (-9); 8\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Use Integers in Applications

                                                          \n\n

                                                          We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

                                                          \n\n

                                                          How to Apply a Strategy to Solve Applications with Integers

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{49}\\)
                                                          \n\n

                                                          The temperature in Urbana, Illinois one morning was \\(11\\) degrees. By mid-afternoon, the temperature had dropped to \\(−9\\) degrees. What was the difference of the morning and afternoon temperatures?

                                                          \n\n

                                                          Solution

                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                          Step 1. Read the problem. Make sure all the words and ideas are understood. 
                                                          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
                                                          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
                                                          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
                                                          Step 5. Simplify the expression.\\(20\\)
                                                          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{50}\\)
                                                          \n\n

                                                          The temperature in Anchorage, Alaska one morning was \\(15\\) degrees. By mid-afternoon the temperature had dropped to \\(30\\) degrees below zero. What was the difference in the morning and afternoon temperatures?

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          The difference in temperatures was \\(45\\) degrees.

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{51}\\)
                                                          \n\n

                                                          The temperature in Denver was \\(−6\\) degrees at lunchtime. By sunset the temperature had dropped to \\(−15\\) degrees. What was the difference in the lunchtime and sunset temperatures?

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          The difference in temperatures was \\(9\\) degrees.

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          APPLY A STRATEGY TO SOLVE APPLICATIONS WITH INTEGERS.
                                                          \n\n
                                                            \n
                                                          1. Read the problem. Make sure all the words and ideas are understood
                                                            2. \n
                                                          2. Identify what we are asked to find.
                                                            3. \n
                                                          3. Write a phrase that gives the information to find it.
                                                            4. \n
                                                          4. Translate the phrase to an expression.
                                                            5. \n
                                                          5. Simplify the expression.
                                                            6. \n
                                                          6. Answer the question with a complete sentence.
                                                            7. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{52}\\)
                                                          \n\n

                                                          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

                                                          \n\n

                                                          Solution

                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                          Step 1. Read the problem. Make sure all the words and ideas are understood. 
                                                          Step 2. Identify what we are asked to find.the number of yards lost
                                                          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
                                                          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
                                                          Step 5. Simplify the expression.\\(-45\\)
                                                          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{53}\\)
                                                          \n\n

                                                          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \\(15\\) yards. What is the number of yards lost due to penalties?

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          The Bears lost \\(105\\) yards.

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{54}\\)
                                                          \n\n

                                                          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          A $16 fee was deducted from his checking account.

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Key Concepts

                                                          \n\n
                                                            \n
                                                          • Multiplication and Division of Two Signed Numbers\n\n
                                                              \n
                                                            • Same signs—Product is positive
                                                              • \n
                                                            • Different signs—Product is negative
                                                              • \n
                                                            \n
                                                            • \n
                                                          • Strategy for Applications\n
                                                              \n
                                                            1. Identify what you are asked to find.
                                                              2. \n
                                                            2. Write a phrase that gives the information to find it.
                                                              3. \n
                                                            3. Translate the phrase to an expression.
                                                              4. \n
                                                            4. Simplify the expression.
                                                              5. \n
                                                            5. Answer the question with a complete sentence.
                                                              6. \n
                                                            \n
                                                            • \n
                                                          \n
                                                          \n\t\t\t\t
                                                          \n\n\n\n\n
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                                                          1.5: Multiply and Divide Integers is shared under a CC BY license and was authored, remixed, and/or curated by LibreTexts.

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                                                          \n 1.5: Multiply and Divide Integers\n

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\\(\\newcommand{\\lt}{<}\\) \\(\\newcommand{\\gt}{>}\\) \\(\\newcommand{\\amp}{&}\\) \\(\\definecolor{fillinmathshade}{gray}{0.9}\\)", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          By the end of this section, you will be able to:

                                                          ", "content": [{"c": "By the end of this section, you will be able to:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          • Multiply integers
                                                          • Divide integers
                                                          • Simplify expressions with integers
                                                          • Evaluate variable expressions with integers
                                                          • Translate English phrases to algebraic expressions
                                                          • Use integers in applications
                                                          ", "content": {"items": [[[{"c": "Multiply integers", "t": "text"}]], [[{"c": "Divide integers", "t": "text"}]], [[{"c": "Simplify expressions with integers", "t": "text"}]], [[{"c": "Evaluate variable expressions with integers", "t": "text"}]], [[{"c": "Translate English phrases to algebraic expressions", "t": "text"}]], [[{"c": "Use integers in applications", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

                                                          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

                                                          ", "content": [{"c": "A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Multiply Integers

                                                          ", "content": {"title_content": "Multiply Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

                                                          ", "content": [{"c": "Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          We remember that a\\cdot b means add a,\\, b times. Here, we are using the model just to help us discover the pattern.

                                                          ", "content": [{"c": "We remember that", "t": "text"}, {"c": "a\\cdot b", "t": "equation-inline"}, {"c": "means add", "t": "text"}, {"c": "a,\\, b", "t": "equation-inline"}, {"c": "times. Here, we are using the model just to help us discover the pattern.", "t": "text"}]}, {"type": "image", "raw_content": "
                                                          \"Two
                                                          Figure \\(\\PageIndex{1}\\)
                                                          ", "content": {"url": "https://math.libretexts.org/@api/deki/files/17395/CNX_ElemAlg_Figure_01_04_001_img_new.jpg?revision=1", "data": null, "alt": "Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads \u201cadd 5, 3 times.\u201d Below this depicts three rows of blue counters, with five counters in each row. Under this, it says \u201c15 positives.\u201d Under thisis the equation\u201c5 times 3 equals 15.\u201d The image on the right reads \u201cnegative 5 times three. The three is in parentheses. Below this it reads, \u201cadd negative five, three times.\u201d Under this are fifteen red counters in three rows of five. Below this it reads\u201d \u201c15 negatives\u201d. Below this is the equation negative five times 3 equals negative 15.\u201d", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

                                                          The next two examples are more interesting.

                                                          ", "content": [{"c": "The next two examples are more interesting.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          What does it mean to multiply 5 by \u22123? It means subtract 5, 3 times. Looking at subtraction as \u201ctaking away,\u201d it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.

                                                          ", "content": [{"c": "What does it mean to multiply", "t": "text"}, {"c": "5", "t": "equation-inline"}, {"c": "by", "t": "text"}, {"c": "\u22123", "t": "equation-inline"}, {"c": "? It means subtract", "t": "text"}, {"c": "5, 3", "t": "equation-inline"}, {"c": "times. Looking at subtraction as \u201ctaking away,\u201d it means to take away", "t": "text"}, {"c": "5, 3", "t": "equation-inline"}, {"c": "times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away", "t": "text"}, {"c": "5", "t": "equation-inline"}, {"c": "three times.", "t": "text"}]}, {"type": "image", "raw_content": "
                                                          \"This
                                                          Figure \\(\\PageIndex{2}\\)
                                                          ", "content": {"url": "https://math.libretexts.org/@api/deki/files/17306/CNX_ElemAlg_Figure_01_04_002_img_new.jpg?revision=1", "data": null, "alt": "This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

                                                          In summary:

                                                          ", "content": [{"c": "In summary:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}", "content": {"math_content": "\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Notice that for multiplication of two signed numbers, when the:

                                                          ", "content": [{"c": "Notice that for multiplication of two signed numbers, when the:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          • signs are the same, the product is positive.
                                                          • signs are different, the product is negative.
                                                          ", "content": {"items": [[[{"c": "signs are the ", "t": "text"}, {"c": "same", "t": "text"}, {"c": ", the product is ", "t": "text"}, {"c": "positive", "t": "text"}, {"c": ".", "t": "text"}]], [[{"c": "signs are ", "t": "text"}, {"c": "different", "t": "text"}, {"c": ", the product is ", "t": "text"}, {"c": "negative", "t": "text"}, {"c": ".", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

                                                          We\u2019ll put this all together in the chart below.

                                                          ", "content": [{"c": "We\u2019ll put this all together in the chart below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          For multiplication of two signed numbers:

                                                          ", "content": [{"c": "For multiplication of two signed numbers:", "t": "text"}]}, {"type": "table", "raw_content": "
                                                          Same signsProductExample
                                                          Two positivesPositive\\(7\\cdot 4 = 28\\)
                                                          Two negativesPositive\\(-8(-6) = 48\\)
                                                          Table \\(\\PageIndex{1}\\)
                                                          ", "content": {"html": "
                                                          Same signsProductExample
                                                          Two positivesPositive\\(7\\cdot 4 = 28\\)
                                                          Two negativesPositive\\(-8(-6) = 48\\)
                                                          Table \\(\\PageIndex{1}\\)
                                                          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "table", "raw_content": "
                                                          Different signsProductExample
                                                          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
                                                          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
                                                          Table \\(\\PageIndex{2}\\)
                                                          ", "content": {"html": "
                                                          Different signsProductExample
                                                          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
                                                          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
                                                          Table \\(\\PageIndex{2}\\)
                                                          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-9\\cdot 3\\)
                                                          2. \\(-2(-5)\\)
                                                          3. \\(4(-8)\\)
                                                          4. \\(7\\cdot 6\\)
                                                          ", "content": {"items": [[[{"c": "\\(-9\\cdot 3\\)", "t": "text"}]], [[{"c": "\\(-2(-5)\\)", "t": "text"}]], [[{"c": "\\(4(-8)\\)", "t": "text"}]], [[{"c": "\\(7\\cdot 6\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]
                                                          3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]
                                                          4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-6\\cdot 8\\)
                                                          2. \\(-4(-7)\\)
                                                          3. \\(9(-7)\\)
                                                          4. \\(5\\cdot 12\\)
                                                          ", "content": {"items": [[[{"c": "\\(-6\\cdot 8\\)", "t": "text"}]], [[{"c": "\\(-4(-7)\\)", "t": "text"}]], [[{"c": "\\(9(-7)\\)", "t": "text"}]], [[{"c": "\\(5\\cdot 12\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-48\\)
                                                          2. \\(28\\)
                                                          3. \\(-63\\)
                                                          4. \\(60\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-48\\)", "t": "text"}, {"c": "\\(28\\)", "t": "text"}, {"c": "\\(-63\\)", "t": "text"}, {"c": "\\(60\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-8\\cdot 7\\)
                                                          2. \\(-6(-9)\\)
                                                          3. \\(7(-4)\\)
                                                          4. \\(3\\cdot 13\\)
                                                          ", "content": {"items": [[[{"c": "\\(-8\\cdot 7\\)", "t": "text"}]], [[{"c": "\\(-6(-9)\\)", "t": "text"}]], [[{"c": "\\(7(-4)\\)", "t": "text"}]], [[{"c": "\\(3\\cdot 13\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-56\\)
                                                          2. \\(54\\)
                                                          3. \\(-28\\)
                                                          4. \\(39\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-56\\)", "t": "text"}, {"c": "\\(54\\)", "t": "text"}, {"c": "\\(-28\\)", "t": "text"}, {"c": "\\(39\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          When we multiply a number by 1, the result is the same number. What happens when we multiply a number by \u22121? Let\u2019s multiply a positive number and then a negative number by \u22121 to see what we get.

                                                          ", "content": [{"c": "When we multiply a number by", "t": "text"}, {"c": "1", "t": "equation-inline"}, {"c": ", the result is the same number. What happens when we multiply a number by", "t": "text"}, {"c": "\u22121", "t": "equation-inline"}, {"c": "? Let\u2019s multiply a positive number and then a negative number by", "t": "text"}, {"c": "\u22121", "t": "equation-inline"}, {"c": "to see what we get.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]
                                                          \nEach time we multiply a number by \\(\u22121\\), we get its opposite!

                                                          ", "content": [{"c": "\\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\] Each time we multiply a number by \\(\u22121\\), we get its opposite!", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          MULTIPLICATION BY \u22121

                                                          ", "content": [{"c": "MULTIPLICATION BY \u22121", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\u22121a=\u2212a", "content": {"math_content": "\u22121a=\u2212a", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Multiplying a number by \u22121 gives its opposite.

                                                          ", "content": [{"c": "Multiplying a number by", "t": "text"}, {"c": "\u22121", "t": "equation-inline"}, {"c": "gives its opposite.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-1 \\cdot 7\\)
                                                          2. \\(-1(-11)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-1 \\cdot 7\\)", "t": "text"}]], [[{"c": "\\(-1(-11)\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-1\\cdot 9\\)
                                                          2. \\(-1\\cdot(-17)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-1\\cdot 9\\)", "t": "text"}]], [[{"c": "\\(-1\\cdot(-17)\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-9\\)
                                                          2. \\(17\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-9\\)", "t": "text"}, {"c": "\\(17\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Multiply:

                                                          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-1\\cdot 8\\)
                                                          2. \\(-1\\cdot(-16)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-1\\cdot 8\\)", "t": "text"}]], [[{"c": "\\(-1\\cdot(-16)\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-8\\)
                                                          2. \\(16\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-8\\)", "t": "text"}, {"c": "\\(16\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Divide Integers

                                                          ", "content": {"title_content": "Divide Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.

                                                          ", "content": [{"c": "What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}", "content": {"math_content": "\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Division follows the same rules as multiplication!

                                                          ", "content": [{"c": "Division follows the same rules as multiplication!", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          For division of two signed numbers, when the:

                                                          ", "content": [{"c": "For division of two signed numbers, when the:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          • signs are the same, the quotient is positive.
                                                          • signs are different, the quotient is negative.
                                                          ", "content": {"items": [[[{"c": "signs are the ", "t": "text"}, {"c": "same", "t": "text"}, {"c": ", the quotient is ", "t": "text"}, {"c": "positive", "t": "text"}, {"c": ".", "t": "text"}]], [[{"c": "signs are ", "t": "text"}, {"c": "different", "t": "text"}, {"c": ", the quotient is ", "t": "text"}, {"c": "negative", "t": "text"}, {"c": ".", "t": "text"}]]], "ordered": false}}, {"type": "paragraph", "raw_content": "

                                                          And remember that we can always check the answer of a division problem by multiplying.

                                                          ", "content": [{"c": "And remember that we can always check the answer of a division problem by multiplying.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          For multiplication and division of two signed numbers:

                                                          ", "content": [{"c": "For multiplication and division of two signed numbers:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          • If the signs are the same, the result is positive.
                                                          • If the signs are different, the result is negative.
                                                          ", "content": {"items": [[[{"c": "If the signs are the same, the result is positive.", "t": "text"}]], [[{"c": "If the signs are different, the result is negative.", "t": "text"}]]], "ordered": false}}, {"type": "table", "raw_content": "
                                                          Same signsResult
                                                          Two positivesPositive
                                                          Two negativesPositive
                                                          If the signs are the same, the result is positive.
                                                          Table \\(\\PageIndex{3}\\)
                                                          ", "content": {"html": "
                                                          Same signsResult
                                                          Two positivesPositive
                                                          Two negativesPositive
                                                          If the signs are the same, the result is positive.
                                                          Table \\(\\PageIndex{3}\\)
                                                          ", "is_complex": true, "table_nest_level": "1"}}, {"type": "table", "raw_content": "
                                                          Different signsResult
                                                          Positive and negativeNegative
                                                          Negative and positiveNegative
                                                          If the signs are different, the result is negative.
                                                          Table \\(\\PageIndex{4}\\)
                                                          ", "content": {"html": "
                                                          Different signsResult
                                                          Positive and negativeNegative
                                                          Negative and positiveNegative
                                                          If the signs are different, the result is negative.
                                                          Table \\(\\PageIndex{4}\\)
                                                          ", "is_complex": true, "table_nest_level": "1"}}, {"type": "list", "raw_content": "
                                                          1. \\(-27\\div 3\\)
                                                          2. \\(-100\\div (-4)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-27\\div 3\\)", "t": "text"}]], [[{"c": "\\(-100\\div (-4)\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Divide:

                                                          ", "content": [{"c": "Divide:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-42\\div 6\\)
                                                          2. \\(-117\\div (-3)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-42\\div 6\\)", "t": "text"}]], [[{"c": "\\(-117\\div (-3)\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-7\\)
                                                          2. \\(39\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-7\\)", "t": "text"}, {"c": "\\(39\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Divide:

                                                          ", "content": [{"c": "Divide:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(-63\\div 7\\)
                                                          2. \\(-115\\div (-5)\\)
                                                          ", "content": {"items": [[[{"c": "\\(-63\\div 7\\)", "t": "text"}]], [[{"c": "\\(-115\\div (-5)\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-9\\)
                                                          2. \\(23\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-9\\)", "t": "text"}, {"c": "\\(23\\)", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Simplify Expressions with Integers

                                                          ", "content": {"title_content": "Simplify Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

                                                          ", "content": [{"c": "What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Let\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.

                                                          ", "content": [{"c": "Let\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          7(-2)+4(-7)-6

                                                          ", "content": [{"c": "7(-2)+4(-7)-6", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}", "content": {"math_content": "\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          8(-3)+5(-7)-4

                                                          ", "content": [{"c": "8(-3)+5(-7)-4", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          -63

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-63", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          9(-3)+7(-8)-1

                                                          ", "content": [{"c": "9(-3)+7(-8)-1", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          -84

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-84", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\((-2)^{4}\\)
                                                          2. \\(-2^{4}\\)
                                                          ", "content": {"items": [[[{"c": "\\((-2)^{4}\\)", "t": "text"}]], [[{"c": "\\(-2^{4}\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the (\u22122) to the 4^{th} power. In part (2), the exponent means to raise just the 2 to the 4^{th} power and then take the opposite.

                                                          ", "content": [{"c": "Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the", "t": "text"}, {"c": "(\u22122)", "t": "equation-inline"}, {"c": "to the", "t": "text"}, {"c": "4^{th}", "t": "equation-inline"}, {"c": "power. In part (2), the exponent means to raise just the", "t": "text"}, {"c": "2", "t": "equation-inline"}, {"c": "to the", "t": "text"}, {"c": "4^{th}", "t": "equation-inline"}, {"c": "power and then take the opposite.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\((-3)^{4}\\)
                                                          2. \\(-3^{4}\\)
                                                          ", "content": {"items": [[[{"c": "\\((-3)^{4}\\)", "t": "text"}]], [[{"c": "\\(-3^{4}\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(81\\)
                                                          2. \\(-81\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(81\\)", "t": "text"}, {"c": "\\(-81\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\((-7)^{2}\\)
                                                          2. \\(-7^{2}\\)
                                                          ", "content": {"items": [[[{"c": "\\((-7)^{2}\\)", "t": "text"}]], [[{"c": "\\(-7^{2}\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(49\\)
                                                          2. \\(-49\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(49\\)", "t": "text"}, {"c": "\\(-49\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          The next example reminds us to simplify inside parentheses first.

                                                          ", "content": [{"c": "The next example reminds us to simplify inside parentheses first.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          12-3(9 - 12)

                                                          ", "content": [{"c": "12-3(9 - 12)", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}", "content": {"math_content": "\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          17 - 4(8 - 11)

                                                          ", "content": [{"c": "17 - 4(8 - 11)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          29

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "29", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          16 - 6(7 - 13)

                                                          ", "content": [{"c": "16 - 6(7 - 13)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          52

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "52", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          8(-9)\\div (-2)^{3}

                                                          ", "content": [{"c": "8(-9)\\div (-2)^{3}", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}", "content": {"math_content": "\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          12(-9)\\div (-3)^{3}

                                                          ", "content": [{"c": "12(-9)\\div (-3)^{3}", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          4

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "4", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          18(-4)\\div (-2)^{3}

                                                          ", "content": [{"c": "18(-4)\\div (-2)^{3}", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          9

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "9", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          -30\\div 2 + (-3)(-7)

                                                          ", "content": [{"c": "-30\\div 2 + (-3)(-7)", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}", "content": {"math_content": "\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          -27\\div 3 + (-5)(-6)

                                                          ", "content": [{"c": "-27\\div 3 + (-5)(-6)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          21

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "21", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Simplify:

                                                          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          -32\\div 4 + (-2)(-7)

                                                          ", "content": [{"c": "-32\\div 4 + (-2)(-7)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          6

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "6", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Evaluate Variable Expressions with Integers

                                                          ", "content": {"title_content": "Evaluate Variable Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

                                                          ", "content": [{"c": "Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          When n=\u22125, evaluate:

                                                          ", "content": [{"c": "When", "t": "text"}, {"c": "n=\u22125", "t": "equation-inline"}, {"c": ", evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(n+1\\)
                                                          2. \\(\u2212n+1\\).
                                                          ", "content": {"items": [[[{"c": "\\(n+1\\)", "t": "text"}]], [[{"c": "\\(\u2212n+1\\).", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1\u00a0\\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{-\u00a0{\\color{red}{(-5)}}\u00a0+1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1\u00a0\\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{-\u00a0{\\color{red}{(-5)}}\u00a0+1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          When n=\u22128, evaluate:

                                                          ", "content": [{"c": "When", "t": "text"}, {"c": "n=\u22128", "t": "equation-inline"}, {"c": ", evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(n+2\\)
                                                          2. \\(\u2212n+2\\).
                                                          ", "content": {"items": [[[{"c": "\\(n+2\\)", "t": "text"}]], [[{"c": "\\(\u2212n+2\\).", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-6\\)
                                                          2. \\(10\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-6\\)", "t": "text"}, {"c": "\\(10\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          When y=\u22129, evaluate:

                                                          ", "content": [{"c": "When", "t": "text"}, {"c": "y=\u22129", "t": "equation-inline"}, {"c": ", evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(y+8\\)
                                                          2. \\(\u2212y+8\\).
                                                          ", "content": {"items": [[[{"c": "\\(y+8\\)", "t": "text"}]], [[{"c": "\\(\u2212y+8\\).", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-1\\)
                                                          2. \\(17\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-1\\)", "t": "text"}, {"c": "\\(17\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate (x+y)^{2} when x = -18 and y = 24.

                                                          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "(x+y)^{2}", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -18", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "y = 24", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}", "content": {"math_content": "\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate (x+y)^{2} when x = -15 and y = 29.

                                                          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "(x+y)^{2}", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -15", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "y = 29", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          196

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "196", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate (x+y)^{3} when x = -8 and y = 10.

                                                          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "(x+y)^{3}", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "y = 10", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          8

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "8", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate 20 -z when

                                                          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "20 -z ", "t": "equation-inline"}, {"c": "when", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(z = 12\\)
                                                          2. \\(z = -12\\)
                                                          ", "content": {"items": [[[{"c": "\\(z = 12\\)", "t": "text"}]], [[{"c": "\\(z = -12\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate 17 - k when

                                                          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "17 - k", "t": "equation-inline"}, {"c": "when", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(k = 19\\)
                                                          2. \\(k = -19\\)
                                                          ", "content": {"items": [[[{"c": "\\(k = 19\\)", "t": "text"}]], [[{"c": "\\(k = -19\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-2\\)
                                                          2. \\(36\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-2\\)", "t": "text"}, {"c": "\\(36\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate -5 - b when

                                                          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "-5 - b", "t": "equation-inline"}, {"c": "when", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\(b = 14\\)
                                                          2. \\(b = -14\\)
                                                          ", "content": {"items": [[[{"c": "\\(b = 14\\)", "t": "text"}]], [[{"c": "\\(b = -14\\)", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(-19\\)
                                                          2. \\(9\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(-19\\)", "t": "text"}, {"c": "\\(9\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate:

                                                          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          2x^{2} + 3x + 8 when x = 4.

                                                          ", "content": [{"c": "2x^{2} + 3x + 8", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = 4", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Substitute 4 for x. Use parentheses to show multiplication.

                                                          ", "content": [{"c": "Substitute", "t": "text"}, {"c": "4", "t": "equation-inline"}, {"c": "for", "t": "text"}, {"c": "x", "t": "equation-inline"}, {"c": ". Use parentheses to show multiplication.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}", "content": {"math_content": "\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate:

                                                          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          3x^{2} - 2x + 6 when x =-3.

                                                          ", "content": [{"c": "3x^{2} - 2x + 6", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x =-3", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          39

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "39", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Evaluate:

                                                          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          4x^{2} - x - 5 when x = -2.

                                                          ", "content": [{"c": "4x^{2} - x - 5", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -2", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          13

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "13", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Translate Phrases to Expressions with Integers

                                                          ", "content": {"title_content": "Translate Phrases to Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

                                                          ", "content": [{"c": "Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Translate and simplify: the sum of 8 and \u221212, increased by 3.

                                                          ", "content": [{"c": "Translate and simplify: the sum of", "t": "text"}, {"c": "8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "\u221212", "t": "equation-inline"}, {"c": ", increased by", "t": "text"}, {"c": "3", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Translate and simplify: the sum of 9 and \u221216, increased by 4.

                                                          ", "content": [{"c": "Translate and simplify: the sum of", "t": "text"}, {"c": "9", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "\u221216", "t": "equation-inline"}, {"c": ", increased by", "t": "text"}, {"c": "4", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          (9 + (-16)) + 4 - 3

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "(9 + (-16)) + 4 - 3", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Translate and simplify: the sum of -8 and \u221212, increased by 7.

                                                          ", "content": [{"c": "Translate and simplify: the sum of", "t": "text"}, {"c": "-8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "\u221212", "t": "equation-inline"}, {"c": ", increased by", "t": "text"}, {"c": "7", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          (-8 + (-12)) + 7 - 13

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "(-8 + (-12)) + 7 - 13", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

                                                          ", "content": [{"c": "When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.", "t": "text"}]}, {"type": "table", "raw_content": "
                                                          \\(a\u2212b\\)
                                                          \\(a\\) minus \\(b\\)
                                                          \n the difference of \\(a\\) and \\(b\\)
                                                          \n \\(b\\) subtracted from \\(a\\)
                                                          \n \\(b\\) less than \\(a\\)
                                                          Table \\(\\PageIndex{5}\\)
                                                          ", "content": {"html": "
                                                          \\(a\u2212b\\)
                                                          \\(a\\) minus \\(b\\)
                                                          Table \\(\\PageIndex{5}\\)
                                                          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

                                                          Be careful to get a and b in the right order!

                                                          ", "content": [{"c": "Be careful to get a and b in the right order!", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Translate and then simplify

                                                          ", "content": [{"c": "Translate and then simplify", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. the difference of \\(13\\) and \\(\u221221\\)
                                                          2. subtract \\(24\\) from \\(\u221219\\).
                                                          ", "content": {"items": [[[{"c": "the difference of \\(13\\) and \\(\u221221\\)", "t": "text"}]], [[{"c": "subtract \\(24\\) from \\(\u221219\\).", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]
                                                          ", "content": {"items": [[[{"c": "\\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]", "t": "text"}]], [[{"c": "\\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Translate and simplify

                                                          ", "content": [{"c": "Translate and simplify", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. the difference of \\(14\\) and \\(\u221223\\)
                                                          2. subtract \\(21\\) from \\(\u221217\\).
                                                          ", "content": {"items": [[[{"c": "the difference of \\(14\\) and \\(\u221223\\)", "t": "text"}]], [[{"c": "subtract \\(21\\) from \\(\u221217\\).", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(14 - (-23); 37\\)
                                                          2. \\(-17 - 21; -38\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(14 - (-23); 37\\)", "t": "text"}, {"c": "\\(-17 - 21; -38\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Translate and simplify

                                                          ", "content": [{"c": "Translate and simplify", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          1. the difference of \\(11\\) and \\(\u221219\\)
                                                          2. subtract \\(18\\) from \\(\u221211\\).
                                                          ", "content": {"items": [[[{"c": "the difference of \\(11\\) and \\(\u221219\\)", "t": "text"}]], [[{"c": "subtract \\(18\\) from \\(\u221211\\).", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          Answer
                                                          1. \\(11 - (-19); 30\\)
                                                          2. \\(-11 - 18; -29\\)
                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "\\(11 - (-19); 30\\)", "t": "text"}, {"c": "\\(-11 - 18; -29\\)", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is \u201cproduct\u201d and for division is \u201cquotient.\u201d

                                                          ", "content": [{"c": "Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is \u201c product \u201d and for division is \u201c quotient.\u201d", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the product of \u22122 and 14.

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of", "t": "text"}, {"c": "\u22122", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "14", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the product of \u22125 and 12.

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of", "t": "text"}, {"c": "\u22125", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "12", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          -5(12); -60

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-5(12); -60", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the product of 8 and -13.

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of", "t": "text"}, {"c": "8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "-13", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          -8(13); -104

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-8(13); -104", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \u221256 and \u22127.

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of", "t": "text"}, {"c": "\u221256", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "\u22127", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \u221263 and \u22129.

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of", "t": "text"}, {"c": "\u221263", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "\u22129", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          -63\\div (-9); 7

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-63\\div (-9); 7", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \u221272 and \u22129.

                                                          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of", "t": "text"}, {"c": "\u221272", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "\u22129", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          -72\\div (-9); 8

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-72\\div (-9); 8", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Use Integers in Applications

                                                          ", "content": {"title_content": "Use Integers in Applications", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          We\u2019ll outline a plan to solve applications. It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we\u2019ll write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

                                                          ", "content": [{"c": "We\u2019ll outline a plan to solve applications. It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we\u2019ll write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          How to Apply a Strategy to Solve Applications with Integers

                                                          ", "content": [{"c": "How to Apply a Strategy to Solve Applications with Integers", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The temperature in Urbana, Illinois one morning was 11 degrees. By mid-afternoon, the temperature had dropped to \u22129 degrees. What was the difference of the morning and afternoon temperatures?

                                                          ", "content": [{"c": "The temperature in Urbana, Illinois one morning was", "t": "text"}, {"c": "11", "t": "equation-inline"}, {"c": "degrees. By mid-afternoon, the temperature had dropped to", "t": "text"}, {"c": "\u22129", "t": "equation-inline"}, {"c": "degrees. What was the difference of the morning and afternoon temperatures?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "table", "raw_content": "
                                                          Step 1. Read the problem. Make sure all the words and ideas are understood.
                                                          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
                                                          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
                                                          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
                                                          Step 5. Simplify the expression.\\(20\\)
                                                          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
                                                          ", "content": {"html": "
                                                          Step 1
                                                          Step 2the difference of the morning and afternoon temperatures
                                                          Step 3the difference of \\(11\\) and \\(-9\\)
                                                          Step 4.\\(11 - (-9)\\)
                                                          Step 5\\(20\\)
                                                          Step 6The difference in temperatures was 20 degrees.
                                                          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

                                                          The temperature in Anchorage, Alaska one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

                                                          ", "content": [{"c": "The temperature in Anchorage, Alaska one morning was", "t": "text"}, {"c": "15", "t": "equation-inline"}, {"c": "degrees. By mid-afternoon the temperature had dropped to", "t": "text"}, {"c": "30", "t": "equation-inline"}, {"c": "degrees below zero. What was the difference in the morning and afternoon temperatures?", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          The difference in temperatures was 45 degrees.

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "The difference in temperatures was ", "t": "text"}, {"c": "45", "t": "equation-inline"}, {"c": " degrees.", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          The temperature in Denver was \u22126 degrees at lunchtime. By sunset the temperature had dropped to \u221215 degrees. What was the difference in the lunchtime and sunset temperatures?

                                                          ", "content": [{"c": "The temperature in Denver was", "t": "text"}, {"c": "\u22126", "t": "equation-inline"}, {"c": "degrees at lunchtime. By sunset the temperature had dropped to", "t": "text"}, {"c": "\u221215", "t": "equation-inline"}, {"c": "degrees. What was the difference in the lunchtime and sunset temperatures?", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          The difference in temperatures was 9 degrees.

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "The difference in temperatures was ", "t": "text"}, {"c": "9", "t": "equation-inline"}, {"c": " degrees.", "t": "text"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          1. Read the problem. Make sure all the words and ideas are understood
                                                          2. Identify what we are asked to find.
                                                          3. Write a phrase that gives the information to find it.
                                                          4. Translate the phrase to an expression.
                                                          5. Simplify the expression.
                                                          6. Answer the question with a complete sentence.
                                                          ", "content": {"items": [[[{"c": "Read the problem. Make sure all the words and ideas are understood", "t": "text"}]], [[{"c": "Identify what we are asked to find.", "t": "text"}]], [[{"c": "Write a phrase that gives the information to find it.", "t": "text"}]], [[{"c": "Translate the phrase to an expression.", "t": "text"}]], [[{"c": "Simplify the expression.", "t": "text"}]], [[{"c": "Answer the question with a complete sentence.", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

                                                          ", "content": [{"c": "The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Solution

                                                          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "table", "raw_content": "
                                                          Step 1. Read the problem. Make sure all the words and ideas are understood.
                                                          Step 2. Identify what we are asked to find.the number of yards lost
                                                          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
                                                          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
                                                          Step 5. Simplify the expression.\\(-45\\)
                                                          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.
                                                          ", "content": {"html": "
                                                          Step 1
                                                          Step 2the number of yards lost
                                                          Step 3three times a \\(15\\)-yard penalty
                                                          Step 4.\\(3(-15)\\)
                                                          Step 5\\(-45\\)
                                                          Step 6The team lost \\(45\\) yards.
                                                          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

                                                          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of 15 yards. What is the number of yards lost due to penalties?

                                                          ", "content": [{"c": "The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of", "t": "text"}, {"c": "15", "t": "equation-inline"}, {"c": "yards. What is the number of yards lost due to penalties?", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          The Bears lost 105 yards.

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "The Bears lost ", "t": "text"}, {"c": "105", "t": "equation-inline"}, {"c": " yards.", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

                                                          ", "content": [{"c": "Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          Answer

                                                          A $16 fee was deducted from his checking account.

                                                          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "A $16 fee was deducted from his checking account.", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Key Concepts

                                                          ", "content": {"title_content": "Key Concepts", "level": "2"}}, {"type": "list", "raw_content": "
                                                          • Multiplication and Division of Two Signed Numbers
                                                            • Same signs\u2014Product is positive
                                                            • Different signs\u2014Product is negative
                                                          • Strategy for Applications
                                                            1. Identify what you are asked to find.
                                                            2. Write a phrase that gives the information to find it.
                                                            3. Translate the phrase to an expression.
                                                            4. Simplify the expression.
                                                            5. Answer the question with a complete sentence.
                                                          ", "content": {"items": [[[{"c": "Multiplication and Division of Two Signed Numbers", "t": "text"}, {"c": "Same signs\u2014Product is positive", "t": "text"}, {"c": "Different signs\u2014Product is negative", "t": "text"}]], [[{"c": "Strategy for Applications", "t": "text"}, {"c": "Identify what you are asked to find.", "t": "text"}, {"c": "Write a phrase that gives the information to find it.", "t": "text"}, {"c": "Translate the phrase to an expression.", "t": "text"}, {"c": "Simplify the expression.", "t": "text"}, {"c": "Answer the question with a complete sentence.", "t": "text"}]]], "ordered": false}}]], "main_html": "

                                                          \n 1.5: Multiply and Divide Integers\n

                                                          1. Last updated
                                                          Page ID
                                                          30345

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                                                          \n\\(\\newcommand{\\avec}{\\mathbf a}\\) \\(\\newcommand{\\bvec}{\\mathbf b}\\) \\(\\newcommand{\\cvec}{\\mathbf c}\\) \\(\\newcommand{\\dvec}{\\mathbf d}\\) \\(\\newcommand{\\dtil}{\\widetilde{\\mathbf d}}\\) \\(\\newcommand{\\evec}{\\mathbf e}\\) \\(\\newcommand{\\fvec}{\\mathbf f}\\) \\(\\newcommand{\\nvec}{\\mathbf n}\\) \\(\\newcommand{\\pvec}{\\mathbf p}\\) \\(\\newcommand{\\qvec}{\\mathbf q}\\) \\(\\newcommand{\\svec}{\\mathbf s}\\) \\(\\newcommand{\\tvec}{\\mathbf t}\\) \\(\\newcommand{\\uvec}{\\mathbf u}\\) \\(\\newcommand{\\vvec}{\\mathbf v}\\) \\(\\newcommand{\\wvec}{\\mathbf w}\\) \\(\\newcommand{\\xvec}{\\mathbf x}\\) \\(\\newcommand{\\yvec}{\\mathbf y}\\) \\(\\newcommand{\\zvec}{\\mathbf z}\\) \\(\\newcommand{\\rvec}{\\mathbf r}\\) \\(\\newcommand{\\mvec}{\\mathbf m}\\) \\(\\newcommand{\\zerovec}{\\mathbf 0}\\) \\(\\newcommand{\\onevec}{\\mathbf 1}\\) \\(\\newcommand{\\real}{\\mathbb R}\\) \\(\\newcommand{\\twovec}[2]{\\left[\\begin{array}{r}#1 \\\\ #2 \\end{array}\\right]}\\) \\(\\newcommand{\\ctwovec}[2]{\\left[\\begin{array}{c}#1 \\\\ #2 \\end{array}\\right]}\\) \\(\\newcommand{\\threevec}[3]{\\left[\\begin{array}{r}#1 \\\\ #2 \\\\ #3 \\end{array}\\right]}\\) \\(\\newcommand{\\cthreevec}[3]{\\left[\\begin{array}{c}#1 \\\\ #2 \\\\ #3 \\end{array}\\right]}\\) \\(\\newcommand{\\fourvec}[4]{\\left[\\begin{array}{r}#1 \\\\ #2 \\\\ #3 \\\\ #4 \\end{array}\\right]}\\) \\(\\newcommand{\\cfourvec}[4]{\\left[\\begin{array}{c}#1 \\\\ #2 \\\\ #3 \\\\ #4 \\end{array}\\right]}\\) \\(\\newcommand{\\fivevec}[5]{\\left[\\begin{array}{r}#1 \\\\ #2 \\\\ #3 \\\\ #4 \\\\ #5 \\\\ \\end{array}\\right]}\\) \\(\\newcommand{\\cfivevec}[5]{\\left[\\begin{array}{c}#1 \\\\ #2 \\\\ #3 \\\\ #4 \\\\ #5 \\\\ \\end{array}\\right]}\\) \\(\\newcommand{\\mattwo}[4]{\\left[\\begin{array}{rr}#1 \\amp #2 \\\\ #3 \\amp #4 \\\\ \\end{array}\\right]}\\) \\(\\newcommand{\\laspan}[1]{\\text{Span}\\{#1\\}}\\) \\(\\newcommand{\\bcal}{\\cal B}\\) \\(\\newcommand{\\ccal}{\\cal C}\\) \\(\\newcommand{\\scal}{\\cal S}\\) \\(\\newcommand{\\wcal}{\\cal W}\\) \\(\\newcommand{\\ecal}{\\cal E}\\) \\(\\newcommand{\\coords}[2]{\\left\\{#1\\right\\}_{#2}}\\) \\(\\newcommand{\\gray}[1]{\\color{gray}{#1}}\\) \\(\\newcommand{\\lgray}[1]{\\color{lightgray}{#1}}\\) \\(\\newcommand{\\rank}{\\operatorname{rank}}\\) \\(\\newcommand{\\row}{\\text{Row}}\\) \\(\\newcommand{\\col}{\\text{Col}}\\) \\(\\renewcommand{\\row}{\\text{Row}}\\) \\(\\newcommand{\\nul}{\\text{Nul}}\\) \\(\\newcommand{\\var}{\\text{Var}}\\) \\(\\newcommand{\\corr}{\\text{corr}}\\) \\(\\newcommand{\\len}[1]{\\left|#1\\right|}\\) \\(\\newcommand{\\bbar}{\\overline{\\bvec}}\\) \\(\\newcommand{\\bhat}{\\widehat{\\bvec}}\\) \\(\\newcommand{\\bperp}{\\bvec^\\perp}\\) \\(\\newcommand{\\xhat}{\\widehat{\\xvec}}\\) \\(\\newcommand{\\vhat}{\\widehat{\\vvec}}\\) \\(\\newcommand{\\uhat}{\\widehat{\\uvec}}\\) \\(\\newcommand{\\what}{\\widehat{\\wvec}}\\) \\(\\newcommand{\\Sighat}{\\widehat{\\Sigma}}\\) \\(\\newcommand{\\lt}{<}\\) \\(\\newcommand{\\gt}{>}\\) \\(\\newcommand{\\amp}{&}\\) \\(\\definecolor{fillinmathshade}{gray}{0.9}\\)

                                                          • By the end of this section, you will be able to:

                                                          • Multiply integers
                                                          • Divide integers
                                                          • Simplify expressions with integers
                                                          • Evaluate variable expressions with integers
                                                          • Translate English phrases to algebraic expressions
                                                          • Use integers in applications

                                                          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

                                                          Multiply Integers

                                                          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

                                                          We remember that a\\cdot b means add a,\\, b times. Here, we are using the model just to help us discover the pattern.

                                                          \"Two
                                                          Figure \\(\\PageIndex{1}\\)

                                                          The next two examples are more interesting.

                                                          What does it mean to multiply 5 by \u22123? It means subtract 5, 3 times. Looking at subtraction as \u201ctaking away,\u201d it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.

                                                          \"This
                                                          Figure \\(\\PageIndex{2}\\)

                                                          In summary:

                                                          \\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}

                                                          Notice that for multiplication of two signed numbers, when the:

                                                          • signs are the same, the product is positive.
                                                          • signs are different, the product is negative.

                                                          We\u2019ll put this all together in the chart below.

                                                          For multiplication of two signed numbers:

                                                          Same signsProductExample
                                                          Two positivesPositive\\(7\\cdot 4 = 28\\)
                                                          Two negativesPositive\\(-8(-6) = 48\\)
                                                          Table \\(\\PageIndex{1}\\)
                                                          Different signsProductExample
                                                          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
                                                          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
                                                          Table \\(\\PageIndex{2}\\)

                                                          Multiply:

                                                          1. \\(-9\\cdot 3\\)
                                                          2. \\(-2(-5)\\)
                                                          3. \\(4(-8)\\)
                                                          4. \\(7\\cdot 6\\)

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]
                                                          3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]
                                                          4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]

                                                          Multiply:

                                                          1. \\(-6\\cdot 8\\)
                                                          2. \\(-4(-7)\\)
                                                          3. \\(9(-7)\\)
                                                          4. \\(5\\cdot 12\\)
                                                          Answer
                                                          1. \\(-48\\)
                                                          2. \\(28\\)
                                                          3. \\(-63\\)
                                                          4. \\(60\\)

                                                          Multiply:

                                                          1. \\(-8\\cdot 7\\)
                                                          2. \\(-6(-9)\\)
                                                          3. \\(7(-4)\\)
                                                          4. \\(3\\cdot 13\\)
                                                          Answer
                                                          1. \\(-56\\)
                                                          2. \\(54\\)
                                                          3. \\(-28\\)
                                                          4. \\(39\\)

                                                          When we multiply a number by 1, the result is the same number. What happens when we multiply a number by \u22121? Let\u2019s multiply a positive number and then a negative number by \u22121 to see what we get.

                                                          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]
                                                          \nEach time we multiply a number by \\(\u22121\\), we get its opposite!

                                                          MULTIPLICATION BY \u22121

                                                          \u22121a=\u2212a

                                                          Multiplying a number by \u22121 gives its opposite.

                                                          Multiply:

                                                          1. \\(-1 \\cdot 7\\)
                                                          2. \\(-1(-11)\\)

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]

                                                          Multiply:

                                                          1. \\(-1\\cdot 9\\)
                                                          2. \\(-1\\cdot(-17)\\)
                                                          Answer
                                                          1. \\(-9\\)
                                                          2. \\(17\\)

                                                          Multiply:

                                                          1. \\(-1\\cdot 8\\)
                                                          2. \\(-1\\cdot(-16)\\)
                                                          Answer
                                                          1. \\(-8\\)
                                                          2. \\(16\\)

                                                          Divide Integers

                                                          What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.

                                                          \\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}

                                                          Division follows the same rules as multiplication!

                                                          For division of two signed numbers, when the:

                                                          • signs are the same, the quotient is positive.
                                                          • signs are different, the quotient is negative.

                                                          And remember that we can always check the answer of a division problem by multiplying.

                                                          For multiplication and division of two signed numbers:

                                                          • If the signs are the same, the result is positive.
                                                          • If the signs are different, the result is negative.
                                                          Same signsResult
                                                          Two positivesPositive
                                                          Two negativesPositive
                                                          If the signs are the same, the result is positive.
                                                          Table \\(\\PageIndex{3}\\)
                                                          Different signsResult
                                                          Positive and negativeNegative
                                                          Negative and positiveNegative
                                                          If the signs are different, the result is negative.
                                                          Table \\(\\PageIndex{4}\\)
                                                          1. \\(-27\\div 3\\)
                                                          2. \\(-100\\div (-4)\\)

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]

                                                          Divide:

                                                          1. \\(-42\\div 6\\)
                                                          2. \\(-117\\div (-3)\\)
                                                          Answer
                                                          1. \\(-7\\)
                                                          2. \\(39\\)

                                                          Divide:

                                                          1. \\(-63\\div 7\\)
                                                          2. \\(-115\\div (-5)\\)
                                                          Answer
                                                          1. \\(-9\\)
                                                          2. \\(23\\)

                                                          Simplify Expressions with Integers

                                                          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

                                                          Let\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.

                                                          Simplify:

                                                          7(-2)+4(-7)-6

                                                          Solution

                                                          \\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}

                                                          Simplify:

                                                          8(-3)+5(-7)-4

                                                          Answer

                                                          -63

                                                          Simplify:

                                                          9(-3)+7(-8)-1

                                                          Answer

                                                          -84

                                                          Simplify:

                                                          1. \\((-2)^{4}\\)
                                                          2. \\(-2^{4}\\)

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]

                                                          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the (\u22122) to the 4^{th} power. In part (2), the exponent means to raise just the 2 to the 4^{th} power and then take the opposite.

                                                          Simplify:

                                                          1. \\((-3)^{4}\\)
                                                          2. \\(-3^{4}\\)
                                                          Answer
                                                          1. \\(81\\)
                                                          2. \\(-81\\)

                                                          Simplify:

                                                          1. \\((-7)^{2}\\)
                                                          2. \\(-7^{2}\\)
                                                          Answer
                                                          1. \\(49\\)
                                                          2. \\(-49\\)

                                                          The next example reminds us to simplify inside parentheses first.

                                                          Simplify:

                                                          12-3(9 - 12)

                                                          Solution

                                                          \\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}

                                                          Simplify:

                                                          17 - 4(8 - 11)

                                                          Answer

                                                          29

                                                          Simplify:

                                                          16 - 6(7 - 13)

                                                          Answer

                                                          52

                                                          Simplify:

                                                          8(-9)\\div (-2)^{3}

                                                          Solution

                                                          \\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}

                                                          Simplify:

                                                          12(-9)\\div (-3)^{3}

                                                          Answer

                                                          4

                                                          Simplify:

                                                          18(-4)\\div (-2)^{3}

                                                          Answer

                                                          9

                                                          Simplify:

                                                          -30\\div 2 + (-3)(-7)

                                                          Solution

                                                          \\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}

                                                          Simplify:

                                                          -27\\div 3 + (-5)(-6)

                                                          Answer

                                                          21

                                                          Simplify:

                                                          -32\\div 4 + (-2)(-7)

                                                          Answer

                                                          6

                                                          Evaluate Variable Expressions with Integers

                                                          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

                                                          When n=\u22125, evaluate:

                                                          1. \\(n+1\\)
                                                          2. \\(\u2212n+1\\).

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1\u00a0\\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{-\u00a0{\\color{red}{(-5)}}\u00a0+1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]

                                                          When n=\u22128, evaluate:

                                                          1. \\(n+2\\)
                                                          2. \\(\u2212n+2\\).
                                                          Answer
                                                          1. \\(-6\\)
                                                          2. \\(10\\)

                                                          When y=\u22129, evaluate:

                                                          1. \\(y+8\\)
                                                          2. \\(\u2212y+8\\).
                                                          Answer
                                                          1. \\(-1\\)
                                                          2. \\(17\\)

                                                          Evaluate (x+y)^{2} when x = -18 and y = 24.

                                                          Solution

                                                          \\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}

                                                          Evaluate (x+y)^{2} when x = -15 and y = 29.

                                                          Answer

                                                          196

                                                          Evaluate (x+y)^{3} when x = -8 and y = 10.

                                                          Answer

                                                          8

                                                          Evaluate 20 -z when

                                                          1. \\(z = 12\\)
                                                          2. \\(z = -12\\)

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]

                                                          Evaluate 17 - k when

                                                          1. \\(k = 19\\)
                                                          2. \\(k = -19\\)
                                                          Answer
                                                          1. \\(-2\\)
                                                          2. \\(36\\)

                                                          Evaluate -5 - b when

                                                          1. \\(b = 14\\)
                                                          2. \\(b = -14\\)
                                                          Answer
                                                          1. \\(-19\\)
                                                          2. \\(9\\)

                                                          Evaluate:

                                                          2x^{2} + 3x + 8 when x = 4.

                                                          Solution

                                                          Substitute 4 for x. Use parentheses to show multiplication.

                                                          \\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}

                                                          Evaluate:

                                                          3x^{2} - 2x + 6 when x =-3.

                                                          Answer

                                                          39

                                                          Evaluate:

                                                          4x^{2} - x - 5 when x = -2.

                                                          Answer

                                                          13

                                                          Translate Phrases to Expressions with Integers

                                                          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

                                                          Translate and simplify: the sum of 8 and \u221212, increased by 3.

                                                          Solution

                                                          \\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}

                                                          Translate and simplify: the sum of 9 and \u221216, increased by 4.

                                                          Answer

                                                          (9 + (-16)) + 4 - 3

                                                          Translate and simplify: the sum of -8 and \u221212, increased by 7.

                                                          Answer

                                                          (-8 + (-12)) + 7 - 13

                                                          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

                                                          \\(a\u2212b\\)
                                                          \\(a\\) minus \\(b\\)
                                                          \n the difference of \\(a\\) and \\(b\\)
                                                          \n \\(b\\) subtracted from \\(a\\)
                                                          \n \\(b\\) less than \\(a\\)
                                                          Table \\(\\PageIndex{5}\\)

                                                          Be careful to get a and b in the right order!

                                                          Translate and then simplify

                                                          1. the difference of \\(13\\) and \\(\u221221\\)
                                                          2. subtract \\(24\\) from \\(\u221219\\).

                                                          Solution

                                                          1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]
                                                          2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]

                                                          Translate and simplify

                                                          1. the difference of \\(14\\) and \\(\u221223\\)
                                                          2. subtract \\(21\\) from \\(\u221217\\).
                                                          Answer
                                                          1. \\(14 - (-23); 37\\)
                                                          2. \\(-17 - 21; -38\\)

                                                          Translate and simplify

                                                          1. the difference of \\(11\\) and \\(\u221219\\)
                                                          2. subtract \\(18\\) from \\(\u221211\\).
                                                          Answer
                                                          1. \\(11 - (-19); 30\\)
                                                          2. \\(-11 - 18; -29\\)

                                                          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is \u201cproduct\u201d and for division is \u201cquotient.\u201d

                                                          Translate to an algebraic expression and simplify if possible: the product of \u22122 and 14.

                                                          Solution

                                                          \\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}

                                                          Translate to an algebraic expression and simplify if possible: the product of \u22125 and 12.

                                                          Answer

                                                          -5(12); -60

                                                          Translate to an algebraic expression and simplify if possible: the product of 8 and -13.

                                                          Answer

                                                          -8(13); -104

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \u221256 and \u22127.

                                                          Solution

                                                          \\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \u221263 and \u22129.

                                                          Answer

                                                          -63\\div (-9); 7

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \u221272 and \u22129.

                                                          Answer

                                                          -72\\div (-9); 8

                                                          Use Integers in Applications

                                                          We\u2019ll outline a plan to solve applications. It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we\u2019ll write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

                                                          How to Apply a Strategy to Solve Applications with Integers

                                                          The temperature in Urbana, Illinois one morning was 11 degrees. By mid-afternoon, the temperature had dropped to \u22129 degrees. What was the difference of the morning and afternoon temperatures?

                                                          Solution

                                                          Step 1. Read the problem. Make sure all the words and ideas are understood.
                                                          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
                                                          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
                                                          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
                                                          Step 5. Simplify the expression.\\(20\\)
                                                          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.

                                                          The temperature in Anchorage, Alaska one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

                                                          Answer

                                                          The difference in temperatures was 45 degrees.

                                                          The temperature in Denver was \u22126 degrees at lunchtime. By sunset the temperature had dropped to \u221215 degrees. What was the difference in the lunchtime and sunset temperatures?

                                                          Answer

                                                          The difference in temperatures was 9 degrees.

                                                          1. Read the problem. Make sure all the words and ideas are understood
                                                          2. Identify what we are asked to find.
                                                          3. Write a phrase that gives the information to find it.
                                                          4. Translate the phrase to an expression.
                                                          5. Simplify the expression.
                                                          6. Answer the question with a complete sentence.

                                                          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

                                                          Solution

                                                          Step 1. Read the problem. Make sure all the words and ideas are understood.
                                                          Step 2. Identify what we are asked to find.the number of yards lost
                                                          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
                                                          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
                                                          Step 5. Simplify the expression.\\(-45\\)
                                                          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.

                                                          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of 15 yards. What is the number of yards lost due to penalties?

                                                          Answer

                                                          The Bears lost 105 yards.

                                                          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

                                                          Answer

                                                          A $16 fee was deducted from his checking account.

                                                          Key Concepts

                                                          • Multiplication and Division of Two Signed Numbers
                                                            • Same signs\u2014Product is positive
                                                            • Different signs\u2014Product is negative
                                                          • Strategy for Applications
                                                            1. Identify what you are asked to find.
                                                            2. Write a phrase that gives the information to find it.
                                                            3. Translate the phrase to an expression.
                                                            4. Simplify the expression.
                                                            5. Answer the question with a complete sentence.
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\\(\\definecolor{fillinmathshade}{gray}{0.9}\\)\n\nBy the end of this section, you will be able to:\n\n- Multiply integers\n- Divide integers\n- Simplify expressions with integers\n- Evaluate variable expressions with integers\n- Translate English phrases to algebraic expressions\n- Use integers in applications\n\nA more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.\n\n## Multiply Integers\n\nSince multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.\n\nWe remember that $a\\cdot b$ means add $a,\\, b$ times. Here, we are using the model just to help us discover the pattern.\n\nThe next two examples are more interesting.\n\nWhat does it mean to multiply $5$ by $\u22123$ ? It means subtract $5, 3$ times. Looking at subtraction as \u201ctaking away,\u201d it means to take away $5, 3$ times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away $5$ three times.\n\nIn summary:\n\n$$\n\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\n$$\n\nNotice that for multiplication of two signed numbers, when the:\n\n- signs are the same , the product is positive .\n- signs are different , the product is negative .\n\nWe\u2019ll put this all together in the chart below.\n\nFor multiplication of two signed numbers:\n\n| Same signs | Product | Example |\n|---|---|---|\n| Two positives | Positive | \\(7\\cdot 4 = 28\\) |\n| Two negatives | Positive | \\(-8(-6) = 48\\) |\n\n| Different signs | Product | Example |\n|---|---|---|\n| Positives \\(\\cdot\\) negative | Negative | \\(7(-9) = -63\\) |\n| Negative \\(\\cdot\\) positives | Negative | \\(-5\\cdot 10= -50\\) |\n\nMultiply:\n\n1. \\(-9\\cdot 3\\)\n2. \\(-2(-5)\\)\n3. \\(4(-8)\\)\n4. \\(7\\cdot 6\\)\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]\n3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]\n4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]\n\nMultiply:\n\n1. \\(-6\\cdot 8\\)\n2. \\(-4(-7)\\)\n3. \\(9(-7)\\)\n4. \\(5\\cdot 12\\)\n\n1. Answer\n2. \\(-48\\) \\(28\\) \\(-63\\) \\(60\\)\n\nMultiply:\n\n1. \\(-8\\cdot 7\\)\n2. \\(-6(-9)\\)\n3. \\(7(-4)\\)\n4. \\(3\\cdot 13\\)\n\n1. Answer\n2. \\(-56\\) \\(54\\) \\(-28\\) \\(39\\)\n\nWhen we multiply a number by $1$ , the result is the same number. What happens when we multiply a number by $\u22121$ ? Let\u2019s multiply a positive number and then a negative number by $\u22121$ to see what we get.\n\n\\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\] Each time we multiply a number by \\(\u22121\\), we get its opposite!\n\nMULTIPLICATION BY \u22121\n\n$$\n\u22121a=\u2212a\n$$\n\nMultiplying a number by $\u22121$ gives its opposite.\n\nMultiply:\n\n1. \\(-1 \\cdot 7\\)\n2. \\(-1(-11)\\)\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]\n\nMultiply:\n\n1. \\(-1\\cdot 9\\)\n2. \\(-1\\cdot(-17)\\)\n\n1. Answer\n2. \\(-9\\) \\(17\\)\n\nMultiply:\n\n1. \\(-1\\cdot 8\\)\n2. \\(-1\\cdot(-16)\\)\n\n1. Answer\n2. \\(-8\\) \\(16\\)\n\n## Divide Integers\n\nWhat about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.\n\n$$\n\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\n$$\n\nDivision follows the same rules as multiplication!\n\nFor division of two signed numbers, when the:\n\n- signs are the same , the quotient is positive .\n- signs are different , the quotient is negative .\n\nAnd remember that we can always check the answer of a division problem by multiplying.\n\nFor multiplication and division of two signed numbers:\n\n- If the signs are the same, the result is positive.\n- If the signs are different, the result is negative.\n\n
                                                          Same signsResult
                                                          Two positivesPositive
                                                          Two negativesPositive
                                                          If the signs are the same, the result is positive.
                                                          Table \\(\\PageIndex{3}\\)
                                                          \n\n
                                                          Different signsResult
                                                          Positive and negativeNegative
                                                          Negative and positiveNegative
                                                          If the signs are different, the result is negative.
                                                          Table \\(\\PageIndex{4}\\)
                                                          \n\n1. \\(-27\\div 3\\)\n2. \\(-100\\div (-4)\\)\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]\n\nDivide:\n\n1. \\(-42\\div 6\\)\n2. \\(-117\\div (-3)\\)\n\n1. Answer\n2. \\(-7\\) \\(39\\)\n\nDivide:\n\n1. \\(-63\\div 7\\)\n2. \\(-115\\div (-5)\\)\n\n1. Answer\n2. \\(-9\\) \\(23\\)\n\n## Simplify Expressions with Integers\n\nWhat happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?\n\nLet\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.\n\nSimplify:\n\n$7(-2)+4(-7)-6$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\n$$\n\nSimplify:\n\n$8(-3)+5(-7)-4$\n\n1. Answer\n2. $-63$\n\nSimplify:\n\n$9(-3)+7(-8)-1$\n\n1. Answer\n2. $-84$\n\nSimplify:\n\n1. \\((-2)^{4}\\)\n2. \\(-2^{4}\\)\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]\n\nNotice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the $(\u22122)$ to the $4^{th}$ power. In part (2), the exponent means to raise just the $2$ to the $4^{th}$ power and then take the opposite.\n\nSimplify:\n\n1. \\((-3)^{4}\\)\n2. \\(-3^{4}\\)\n\n1. Answer\n2. \\(81\\) \\(-81\\)\n\nSimplify:\n\n1. \\((-7)^{2}\\)\n2. \\(-7^{2}\\)\n\n1. Answer\n2. \\(49\\) \\(-49\\)\n\nThe next example reminds us to simplify inside parentheses first.\n\nSimplify:\n\n$12-3(9 - 12)$\n\nSolution\n\n$$\n\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\n$$\n\nSimplify:\n\n$17 - 4(8 - 11)$\n\n1. Answer\n2. $29$\n\nSimplify:\n\n$16 - 6(7 - 13)$\n\n1. Answer\n2. $52$\n\nSimplify:\n\n$8(-9)\\div (-2)^{3}$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\n$$\n\nSimplify:\n\n$12(-9)\\div (-3)^{3}$\n\n1. Answer\n2. $4$\n\nSimplify:\n\n$18(-4)\\div (-2)^{3}$\n\n1. Answer\n2. $9$\n\nSimplify:\n\n$-30\\div 2 + (-3)(-7)$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\n$$\n\nSimplify:\n\n$-27\\div 3 + (-5)(-6)$\n\n1. Answer\n2. $21$\n\nSimplify:\n\n$-32\\div 4 + (-2)(-7)$\n\n1. Answer\n2. $6$\n\n## Evaluate Variable Expressions with Integers\n\nRemember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.\n\nWhen $n=\u22125$ , evaluate:\n\n1. \\(n+1\\)\n2. \\(\u2212n+1\\).\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1\u00a0\\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{-\u00a0{\\color{red}{(-5)}}\u00a0+1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]\n\nWhen $n=\u22128$ , evaluate:\n\n1. \\(n+2\\)\n2. \\(\u2212n+2\\).\n\n1. Answer\n2. \\(-6\\) \\(10\\)\n\nWhen $y=\u22129$ , evaluate:\n\n1. \\(y+8\\)\n2. \\(\u2212y+8\\).\n\n1. Answer\n2. \\(-1\\) \\(17\\)\n\nEvaluate $(x+y)^{2}$ when $x = -18$ and $y = 24$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\n$$\n\nEvaluate $(x+y)^{2}$ when $x = -15$ and $y = 29$ .\n\n1. Answer\n2. $196$\n\nEvaluate $(x+y)^{3}$ when $x = -8$ and $y = 10$ .\n\n1. Answer\n2. $8$\n\nEvaluate $20 -z$ when\n\n1. \\(z = 12\\)\n2. \\(z = -12\\)\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]\n\nEvaluate $17 - k$ when\n\n1. \\(k = 19\\)\n2. \\(k = -19\\)\n\n1. Answer\n2. \\(-2\\) \\(36\\)\n\nEvaluate $-5 - b$ when\n\n1. \\(b = 14\\)\n2. \\(b = -14\\)\n\n1. Answer\n2. \\(-19\\) \\(9\\)\n\nEvaluate:\n\n$2x^{2} + 3x + 8$ when $x = 4$ .\n\nSolution\n\nSubstitute $4$ for $x$ . Use parentheses to show multiplication.\n\n$$\n\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\n$$\n\nEvaluate:\n\n$3x^{2} - 2x + 6$ when $x =-3$ .\n\n1. Answer\n2. $39$\n\nEvaluate:\n\n$4x^{2} - x - 5$ when $x = -2$ .\n\n1. Answer\n2. $13$\n\n## Translate Phrases to Expressions with Integers\n\nOur earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.\n\nTranslate and simplify: the sum of $8$ and $\u221212$ , increased by $3$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\n$$\n\nTranslate and simplify: the sum of $9$ and $\u221216$ , increased by $4$ .\n\n1. Answer\n2. $(9 + (-16)) + 4 - 3$\n\nTranslate and simplify: the sum of $-8$ and $\u221212$ , increased by $7$ .\n\n1. Answer\n2. $(-8 + (-12)) + 7 - 13$\n\nWhen we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.\n\n| \\(a\u2212b\\) |\n|---|\n| \\(a\\) minus \\(b\\) |\n\nBe careful to get a and b in the right order!\n\nTranslate and then simplify\n\n1. the difference of \\(13\\) and \\(\u221221\\)\n2. subtract \\(24\\) from \\(\u221219\\).\n\nSolution\n\n1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]\n2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]\n\nTranslate and simplify\n\n1. the difference of \\(14\\) and \\(\u221223\\)\n2. subtract \\(21\\) from \\(\u221217\\).\n\n1. Answer\n2. \\(14 - (-23); 37\\) \\(-17 - 21; -38\\)\n\nTranslate and simplify\n\n1. the difference of \\(11\\) and \\(\u221219\\)\n2. subtract \\(18\\) from \\(\u221211\\).\n\n1. Answer\n2. \\(11 - (-19); 30\\) \\(-11 - 18; -29\\)\n\nOnce again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is \u201c product \u201d and for division is \u201c quotient.\u201d\n\nTranslate to an algebraic expression and simplify if possible: the product of $\u22122$ and $14$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\n$$\n\nTranslate to an algebraic expression and simplify if possible: the product of $\u22125$ and $12$ .\n\n1. Answer\n2. $-5(12); -60$\n\nTranslate to an algebraic expression and simplify if possible: the product of $8$ and $-13$ .\n\n1. Answer\n2. $-8(13); -104$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of $\u221256$ and $\u22127$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\n$$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of $\u221263$ and $\u22129$ .\n\n1. Answer\n2. $-63\\div (-9); 7$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of $\u221272$ and $\u22129$ .\n\n1. Answer\n2. $-72\\div (-9); 8$\n\n## Use Integers in Applications\n\nWe\u2019ll outline a plan to solve applications. It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we\u2019ll write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.\n\nHow to Apply a Strategy to Solve Applications with Integers\n\nThe temperature in Urbana, Illinois one morning was $11$ degrees. By mid-afternoon, the temperature had dropped to $\u22129$ degrees. What was the difference of the morning and afternoon temperatures?\n\nSolution\n\n| Step 1 | |\n|---|---|\n| Step 2 | the difference of the morning and afternoon temperatures |\n| Step 3 | the difference of \\(11\\) and \\(-9\\) |\n| Step 4. | \\(11 - (-9)\\) |\n| Step 5 | \\(20\\) |\n| Step 6 | The difference in temperatures was 20 degrees. |\n\nThe temperature in Anchorage, Alaska one morning was $15$ degrees. By mid-afternoon the temperature had dropped to $30$ degrees below zero. What was the difference in the morning and afternoon temperatures?\n\n1. Answer\n2. The difference in temperatures was $45$ degrees.\n\nThe temperature in Denver was $\u22126$ degrees at lunchtime. By sunset the temperature had dropped to $\u221215$ degrees. What was the difference in the lunchtime and sunset temperatures?\n\n1. Answer\n2. The difference in temperatures was $9$ degrees.\n\n1. Read the problem. Make sure all the words and ideas are understood\n2. Identify what we are asked to find.\n3. Write a phrase that gives the information to find it.\n4. Translate the phrase to an expression.\n5. Simplify the expression.\n6. Answer the question with a complete sentence.\n\nThe Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?\n\nSolution\n\n| Step 1 | |\n|---|---|\n| Step 2 | the number of yards lost |\n| Step 3 | three times a \\(15\\)-yard penalty |\n| Step 4. | \\(3(-15)\\) |\n| Step 5 | \\(-45\\) |\n| Step 6 | The team lost \\(45\\) yards. |\n\nThe Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of $15$ yards. What is the number of yards lost due to penalties?\n\n1. Answer\n2. The Bears lost $105$ yards.\n\nBill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?\n\n1. Answer\n2. A $16 fee was deducted from his checking account.\n\n## Key Concepts\n\n- Multiplication and Division of Two Signed Numbers Same signs\u2014Product is positive Different signs\u2014Product is negative\n- Strategy for Applications Identify what you are asked to find. Write a phrase that gives the information to find it. Translate the phrase to an expression. Simplify the expression. Answer the question with a complete sentence.\n", "html": "\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n\n 1.5: Multiply and Divide Integers - Mathematics LibreTexts\n\n\n \n\n \n\n\n \n\n\n \n \n\n\n \n\n\n\n\n\n\nSkip to main content
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                                                          1: Foundations
                                                          MTH 098 Elementary Algebra
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                                                          Mon, 06 Jan 2020 03:19:01 GMT
                                                          1.5: Multiply and Divide Integers
                                                          30345
                                                          30345
                                                          Paul Seeburger
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                                                          [ "article:topic", "license:ccby", "showtoc:no", "transcluded:yes", "source[1]-math-15120" ]
                                                          [ "article:topic", "license:ccby", "showtoc:no", "transcluded:yes", "source[1]-math-15120" ]
                                                          https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FUnder_Construction%2FPurgatory%2FRemixer_University%2FUsername%253A_pseeburger%2FMTH_098_Elementary_Algebra%2F1%253A_Foundations%2F1.5%253A_Multiply_and_Divide_Integers
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                                                          \n 1.5: Multiply and Divide Integers\n

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                                                        • Page ID
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                                                          Learning Objectives
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                                                          By the end of this section, you will be able to:

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                                                          • Multiply integers
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                                                          • Divide integers
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                                                          • Simplify expressions with integers
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                                                          • Evaluate variable expressions with integers
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                                                          • Translate English phrases to algebraic expressions
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                                                          • Use integers in applications
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                                                          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

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                                                          Multiply Integers

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                                                          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

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                                                          We remember that \\(a\\cdot b\\) means add \\(a,\\, b\\) times. Here, we are using the model just to help us discover the pattern.

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                                                          Figure \\(\\PageIndex{1}\\)
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                                                          The next two examples are more interesting.

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                                                          What does it mean to multiply \\(5\\) by \\(−3\\)? It means subtract \\(5, 3\\) times. Looking at subtraction as “taking away,” it means to take away \\(5, 3\\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \\(5\\) three times.

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                                                          In summary:

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                                                          \\[\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\\]

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                                                          Notice that for multiplication of two signed numbers, when the:

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                                                          • signs are the same, the product is positive.
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                                                          • signs are different, the product is negative.
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                                                          We’ll put this all together in the chart below.

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                                                          MULTIPLICATION OF SIGNED NUMBERS
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                                                          For multiplication of two signed numbers:

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                                                          Same signsProductExample
                                                          Two positivesPositive\\(7\\cdot 4 = 28\\)
                                                          Two negativesPositive\\(-8(-6) = 48\\)
                                                          Table \\(\\PageIndex{1}\\)
                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                          Different signsProductExample
                                                          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
                                                          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
                                                          Table \\(\\PageIndex{2}\\)
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{1}\\)
                                                          \n\n

                                                          Multiply:

                                                          \n\n
                                                            \n
                                                          1. \\(-9\\cdot 3\\)
                                                            2. \n
                                                          2. \\(-2(-5)\\)
                                                            3. \n
                                                          3. \\(4(-8)\\)
                                                            4. \n
                                                          4. \\(7\\cdot 6\\)
                                                            5. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]
                                                            3. \n
                                                          3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]
                                                            4. \n
                                                          4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]
                                                            5. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{2}\\)
                                                          \n\n

                                                          Multiply:

                                                          \n\n
                                                            \n
                                                          1. \\(-6\\cdot 8\\)
                                                            2. \n
                                                          2. \\(-4(-7)\\)
                                                            3. \n
                                                          3. \\(9(-7)\\)
                                                            4. \n
                                                          4. \\(5\\cdot 12\\)
                                                            5. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-48\\)
                                                            2. \n
                                                          2. \\(28\\)
                                                            3. \n
                                                          3. \\(-63\\)
                                                            4. \n
                                                          4. \\(60\\)
                                                            5. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{3}\\)
                                                          \n\n

                                                          Multiply:

                                                          \n\n
                                                            \n
                                                          1. \\(-8\\cdot 7\\)
                                                            2. \n
                                                          2. \\(-6(-9)\\)
                                                            3. \n
                                                          3. \\(7(-4)\\)
                                                            4. \n
                                                          4. \\(3\\cdot 13\\)
                                                            5. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-56\\)
                                                            2. \n
                                                          2. \\(54\\)
                                                            3. \n
                                                          3. \\(-28\\)
                                                            4. \n
                                                          4. \\(39\\)
                                                            5. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          When we multiply a number by \\(1\\), the result is the same number. What happens when we multiply a number by \\(−1\\)? Let’s multiply a positive number and then a negative number by \\(−1\\) to see what we get.

                                                          \n\n

                                                          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]
                                                          \nEach time we multiply a number by \\(−1\\), we get its opposite!

                                                          \n\n
                                                          \n
                                                           
                                                          \n\n

                                                          MULTIPLICATION BY −1

                                                          \n\n

                                                          \\[−1a=−a\\]

                                                          \n\n

                                                          Multiplying a number by \\(−1\\) gives its opposite.

                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{4}\\)
                                                          \n\n

                                                          Multiply:

                                                          \n\n
                                                            \n
                                                          1. \\(-1 \\cdot 7\\)
                                                            2. \n
                                                          2. \\(-1(-11)\\)
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]
                                                            3. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{5}\\)
                                                          \n\n

                                                          Multiply:

                                                          \n\n
                                                            \n
                                                          1. \\(-1\\cdot 9\\)
                                                            2. \n
                                                          2. \\(-1\\cdot(-17)\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-9\\)
                                                            2. \n
                                                          2. \\(17\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{6}\\)
                                                          \n\n

                                                          Multiply:

                                                          \n\n
                                                            \n
                                                          1. \\(-1\\cdot 8\\)
                                                            2. \n
                                                          2. \\(-1\\cdot(-16)\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-8\\)
                                                            2. \n
                                                          2. \\(16\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Divide Integers

                                                          \n\n

                                                          What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.

                                                          \n\n

                                                          \\[\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\\]

                                                          \n\n

                                                          Division follows the same rules as multiplication!

                                                          \n\n

                                                          For division of two signed numbers, when the:

                                                          \n\n
                                                            \n
                                                          • signs are the same, the quotient is positive.
                                                            • \n
                                                          • signs are different, the quotient is negative.
                                                            • \n
                                                          \n\n

                                                          And remember that we can always check the answer of a division problem by multiplying.

                                                          \n\n
                                                          \n
                                                          MULTIPLICATION AND DIVISION OF SIGNED NUMBERS
                                                          \n\n

                                                          For multiplication and division of two signed numbers:

                                                          \n\n
                                                            \n
                                                          • If the signs are the same, the result is positive.
                                                            • \n
                                                          • If the signs are different, the result is negative.
                                                            • \n
                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                          Same signsResult
                                                          Two positivesPositive
                                                          Two negativesPositive
                                                          If the signs are the same, the result is positive.
                                                          Table \\(\\PageIndex{3}\\)
                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                          Different signsResult
                                                          Positive and negativeNegative
                                                          Negative and positiveNegative
                                                          If the signs are different, the result is negative.
                                                          Table \\(\\PageIndex{4}\\)
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{7}\\)
                                                          \n\n
                                                            \n
                                                          1. \\(-27\\div 3\\)
                                                            2. \n
                                                          2. \\(-100\\div (-4)\\)
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]
                                                            3. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{8}\\)
                                                          \n\n

                                                          Divide:

                                                          \n\n
                                                            \n
                                                          1. \\(-42\\div 6\\)
                                                            2. \n
                                                          2. \\(-117\\div (-3)\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-7\\)
                                                            2. \n
                                                          2. \\(39\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{9}\\)
                                                          \n\n

                                                          Divide:

                                                          \n\n
                                                            \n
                                                          1. \\(-63\\div 7\\)
                                                            2. \n
                                                          2. \\(-115\\div (-5)\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-9\\)
                                                            2. \n
                                                          2. \\(23\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Simplify Expressions with Integers

                                                          \n\n

                                                          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

                                                          \n\n

                                                          Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{10}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(7(-2)+4(-7)-6\\)

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{11}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(8(-3)+5(-7)-4\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-63\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{12}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(9(-3)+7(-8)-1\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-84\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{13}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n
                                                            \n
                                                          1. \\((-2)^{4}\\)
                                                            2. \n
                                                          2. \\(-2^{4}\\)
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]
                                                            3. \n
                                                          \n\n

                                                          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \\((−2)\\) to the \\(4^{th}\\) power. In part (2), the exponent means to raise just the \\(2\\) to the \\(4^{th}\\) power and then take the opposite.

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{14}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n
                                                            \n
                                                          1. \\((-3)^{4}\\)
                                                            2. \n
                                                          2. \\(-3^{4}\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(81\\)
                                                            2. \n
                                                          2. \\(-81\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{15}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n
                                                            \n
                                                          1. \\((-7)^{2}\\)
                                                            2. \n
                                                          2. \\(-7^{2}\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(49\\)
                                                            2. \n
                                                          2. \\(-49\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          The next example reminds us to simplify inside parentheses first.

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{16}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(12-3(9 - 12)\\)

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{17}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(17 - 4(8 - 11)\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(29\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{18}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(16 - 6(7 - 13)\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(52\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{19}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(8(-9)\\div (-2)^{3}\\)

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{20}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(12(-9)\\div (-3)^{3}\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(4\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{21}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(18(-4)\\div (-2)^{3}\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(9\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{22}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(-30\\div 2 + (-3)(-7)\\)

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{23}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(-27\\div 3 + (-5)(-6)\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(21\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{24}\\)
                                                          \n\n

                                                          Simplify:

                                                          \n\n

                                                          \\(-32\\div 4 + (-2)(-7)\\)

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(6\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Evaluate Variable Expressions with Integers

                                                          \n\n

                                                          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{25}\\)
                                                          \n\n

                                                          When \\(n=−5\\), evaluate:

                                                          \n\n
                                                            \n
                                                          1. \\(n+1\\)
                                                            2. \n
                                                          2. \\(−n+1\\).
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]
                                                            3. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{26}\\)
                                                          \n\n

                                                          When \\(n=−8\\), evaluate:

                                                          \n\n
                                                            \n
                                                          1. \\(n+2\\)
                                                            2. \n
                                                          2. \\(−n+2\\).
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-6\\)
                                                            2. \n
                                                          2. \\(10\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{27}\\)
                                                          \n\n

                                                          When \\(y=−9\\), evaluate:

                                                          \n\n
                                                            \n
                                                          1. \\(y+8\\)
                                                            2. \n
                                                          2. \\(−y+8\\).
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-1\\)
                                                            2. \n
                                                          2. \\(17\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{28}\\)
                                                          \n\n

                                                          Evaluate \\((x+y)^{2}\\) when \\(x = -18\\) and \\(y = 24\\).

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{29}\\)
                                                          \n\n

                                                          Evaluate \\((x+y)^{2}\\) when \\(x = -15\\) and \\(y = 29\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(196\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{30}\\)
                                                          \n\n

                                                          Evaluate \\((x+y)^{3}\\) when \\(x = -8\\) and \\(y = 10\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(8\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{31}\\)
                                                          \n\n

                                                          Evaluate \\(20 -z \\) when

                                                          \n\n
                                                            \n
                                                          1. \\(z = 12\\)
                                                            2. \n
                                                          2. \\(z = -12\\)
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]
                                                            3. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{32}\\)
                                                          \n\n

                                                          Evaluate \\(17 - k\\) when

                                                          \n\n
                                                            \n
                                                          1. \\(k = 19\\)
                                                            2. \n
                                                          2. \\(k = -19\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-2\\)
                                                            2. \n
                                                          2. \\(36\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{33}\\)
                                                          \n\n

                                                          Evaluate \\(-5 - b\\) when

                                                          \n\n
                                                            \n
                                                          1. \\(b = 14\\)
                                                            2. \n
                                                          2. \\(b = -14\\)
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(-19\\)
                                                            2. \n
                                                          2. \\(9\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{34}\\)
                                                          \n\n

                                                          Evaluate:

                                                          \n\n

                                                          \\(2x^{2} + 3x + 8\\) when \\(x = 4\\).

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          Substitute \\(4\\) for \\(x\\). Use parentheses to show multiplication.

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{35}\\)
                                                          \n\n

                                                          Evaluate:

                                                          \n\n

                                                          \\(3x^{2} - 2x + 6\\) when \\(x =-3\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(39\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{36}\\)
                                                          \n\n

                                                          Evaluate:

                                                          \n\n

                                                          \\(4x^{2} - x - 5\\) when \\(x = -2\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(13\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Translate Phrases to Expressions with Integers

                                                          \n\n

                                                          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{37}\\)
                                                          \n\n

                                                          Translate and simplify: the sum of \\(8\\) and \\(−12\\), increased by \\(3\\).

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{38}\\)
                                                          \n\n

                                                          Translate and simplify: the sum of \\(9\\) and \\(−16\\), increased by \\(4\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\((9 + (-16)) + 4 - 3\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{39}\\)
                                                          \n\n

                                                          Translate and simplify: the sum of \\(-8\\) and \\(−12\\), increased by \\(7\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\((-8 + (-12)) + 7 - 13\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n
                                                          \\(a−b\\)
                                                          \\(a\\) minus \\(b\\)
                                                          \n the difference of \\(a\\) and \\(b\\)
                                                          \n \\(b\\) subtracted from \\(a\\)
                                                          \n \\(b\\) less than \\(a\\)
                                                          Table \\(\\PageIndex{5}\\)
                                                          \n\n

                                                          Be careful to get a and b in the right order!

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{40}\\)
                                                          \n\n

                                                          Translate and then simplify

                                                          \n\n
                                                            \n
                                                          1. the difference of \\(13\\) and \\(−21\\)
                                                            2. \n
                                                          2. subtract \\(24\\) from \\(−19\\).
                                                            3. \n
                                                          \n\n

                                                          Solution

                                                          \n\n
                                                            \n
                                                          1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]
                                                            2. \n
                                                          2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]
                                                            3. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{41}\\)
                                                          \n\n

                                                          Translate and simplify

                                                          \n\n
                                                            \n
                                                          1. the difference of \\(14\\) and \\(−23\\)
                                                            2. \n
                                                          2. subtract \\(21\\) from \\(−17\\).
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(14 - (-23); 37\\)
                                                            2. \n
                                                          2. \\(-17 - 21; -38\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{42}\\)
                                                          \n\n

                                                          Translate and simplify

                                                          \n\n
                                                            \n
                                                          1. the difference of \\(11\\) and \\(−19\\)
                                                            2. \n
                                                          2. subtract \\(18\\) from \\(−11\\).
                                                            3. \n
                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n
                                                            \n
                                                          1. \\(11 - (-19); 30\\)
                                                            2. \n
                                                          2. \\(-11 - 18; -29\\)
                                                            3. \n
                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{43}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(−2\\) and \\(14\\).

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{44}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(−5\\) and \\(12\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-5(12); -60\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{45}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the product of \\(8\\) and \\(-13\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-8(13); -104\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{46}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(−56\\) and \\(−7\\).

                                                          \n\n

                                                          Solution

                                                          \n\n

                                                          \\[\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\\]

                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{47}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(−63\\) and \\(−9\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-63\\div (-9); 7\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{48}\\)
                                                          \n\n

                                                          Translate to an algebraic expression and simplify if possible: the quotient of \\(−72\\) and \\(−9\\).

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          \\(-72\\div (-9); 8\\)

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Use Integers in Applications

                                                          \n\n

                                                          We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

                                                          \n\n

                                                          How to Apply a Strategy to Solve Applications with Integers

                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{49}\\)
                                                          \n\n

                                                          The temperature in Urbana, Illinois one morning was \\(11\\) degrees. By mid-afternoon, the temperature had dropped to \\(−9\\) degrees. What was the difference of the morning and afternoon temperatures?

                                                          \n\n

                                                          Solution

                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                          Step 1. Read the problem. Make sure all the words and ideas are understood. 
                                                          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
                                                          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
                                                          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
                                                          Step 5. Simplify the expression.\\(20\\)
                                                          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{50}\\)
                                                          \n\n

                                                          The temperature in Anchorage, Alaska one morning was \\(15\\) degrees. By mid-afternoon the temperature had dropped to \\(30\\) degrees below zero. What was the difference in the morning and afternoon temperatures?

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          The difference in temperatures was \\(45\\) degrees.

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{51}\\)
                                                          \n\n

                                                          The temperature in Denver was \\(−6\\) degrees at lunchtime. By sunset the temperature had dropped to \\(−15\\) degrees. What was the difference in the lunchtime and sunset temperatures?

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          The difference in temperatures was \\(9\\) degrees.

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          APPLY A STRATEGY TO SOLVE APPLICATIONS WITH INTEGERS.
                                                          \n\n
                                                            \n
                                                          1. Read the problem. Make sure all the words and ideas are understood
                                                            2. \n
                                                          2. Identify what we are asked to find.
                                                            3. \n
                                                          3. Write a phrase that gives the information to find it.
                                                            4. \n
                                                          4. Translate the phrase to an expression.
                                                            5. \n
                                                          5. Simplify the expression.
                                                            6. \n
                                                          6. Answer the question with a complete sentence.
                                                            7. \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Example \\(\\PageIndex{52}\\)
                                                          \n\n

                                                          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

                                                          \n\n

                                                          Solution

                                                          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                          Step 1. Read the problem. Make sure all the words and ideas are understood. 
                                                          Step 2. Identify what we are asked to find.the number of yards lost
                                                          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
                                                          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
                                                          Step 5. Simplify the expression.\\(-45\\)
                                                          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{53}\\)
                                                          \n\n

                                                          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \\(15\\) yards. What is the number of yards lost due to penalties?

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          The Bears lost \\(105\\) yards.

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n
                                                          \n
                                                          Try It \\(\\PageIndex{54}\\)
                                                          \n\n

                                                          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

                                                          \n\n
                                                          \n
                                                          Answer
                                                          \n
                                                          \n

                                                          A $16 fee was deducted from his checking account.

                                                          \n
                                                          \n
                                                          \n
                                                          \n\n

                                                          Key Concepts

                                                          \n\n
                                                            \n
                                                          • Multiplication and Division of Two Signed Numbers\n\n
                                                              \n
                                                            • Same signs—Product is positive
                                                              • \n
                                                            • Different signs—Product is negative
                                                              • \n
                                                            \n
                                                            • \n
                                                          • Strategy for Applications\n
                                                              \n
                                                            1. Identify what you are asked to find.
                                                              2. \n
                                                            2. Write a phrase that gives the information to find it.
                                                              3. \n
                                                            3. Translate the phrase to an expression.
                                                              4. \n
                                                            4. Simplify the expression.
                                                              5. \n
                                                            5. Answer the question with a complete sentence.
                                                              6. \n
                                                            \n
                                                            • \n
                                                          \n
                                                          \n\t\t\t\t
                                                          \n\n\n\n\n
                                                          \n\n\n\n\n\n\n\n\n

                                                          1.5: Multiply and Divide Integers is shared under a CC BY license and was authored, remixed, and/or curated by LibreTexts.

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                                                          ", "content": {"title_content": "An Identity of Ramanujan", "level": "2"}}, {"type": "equation-interline", "raw_content": "1(\u03d55\u2212\u03d5)e25\u03c0=1+e\u22122\u03c01+e\u22124\u03c01+e\u22126\u03c01+e\u22128\u03c01+\u2026", "content": {"math_content": "\\frac{1}{\\left(\\sqrt{\\varphi \\sqrt{5}}-\\varphi \\right){e}^{\\frac{2}{5}\\pi }}=1+\\frac{{e}^{-2\\pi }}{1+\\frac{{e}^{-4\\pi }}{1+\\frac{{e}^{-6\\pi }}{1+\\frac{{e}^{-8\\pi }}{1+\\dots }}}}", "math_type": "mathml", "by": "mathjax"}}, {"type": "title", "raw_content": "

                                                          A Rogers-Ramanujan Identity

                                                          ", "content": {"title_content": "A Rogers-Ramanujan Identity", "level": "2"}}, {"type": "equation-interline", "raw_content": "1+q2(1\u2212q)+q6(1\u2212q)(1\u2212q2)+\u22ef=\u220fj=0\u221e1(1\u2212q5j+2)(1\u2212q5j+3),for\u00a0|q|<1.", "content": {"math_content": "1+\\frac{{q}^{2}}{\\left(1-q\\right)}+\\frac{{q}^{6}}{\\left(1-q\\right)\\left(1-{q}^{2}\\right)}+\\cdots =\\prod _{j=0}^{\\mathrm{\\infty }}\\frac{1}{\\left(1-{q}^{5j+2}\\right)\\left(1-{q}^{5j+3}\\right)},\\phantom{\\rule{1em}{0ex}}\\phantom{\\rule{1em}{0ex}}\\text{for\u00a0}|q|<1.", "math_type": "mathml", "by": "mathjax"}}, {"type": "title", "raw_content": "

                                                          Maxwell's Equations

                                                          ", "content": {"title_content": "Maxwell's Equations", "level": "2"}}, {"type": "equation-interline", "raw_content": "\u2207\u00d7B\u2192\u22121c\u2202E\u2192\u2202t=4\u03c0cj\u2192\u2207\u22c5E\u2192=4\u03c0\u03c1\u2207\u00d7E\u2192+1c\u2202B\u2192\u2202t=0\u2192\u2207\u22c5B\u2192=0", "content": {"math_content": "\\begin{array}{rl}\\mathrm{\\nabla }\u00d7\\stackrel{\\to }{\\mathbf{B}}-\\phantom{\\rule{thinmathspace}{0ex}}\\frac{1}{c}\\phantom{\\rule{thinmathspace}{0ex}}\\frac{\\mathrm{\\partial }\\stackrel{\\to }{\\mathbf{E}}}{\\mathrm{\\partial }t}& =\\frac{4\\pi }{c}\\stackrel{\\to }{\\mathbf{j}}\\\\ \\mathrm{\\nabla }\\cdot \\stackrel{\\to }{\\mathbf{E}}& =4\\pi \\rho \\\\ \\mathrm{\\nabla }\u00d7\\stackrel{\\to }{\\mathbf{E}}\\phantom{\\rule{thinmathspace}{0ex}}+\\phantom{\\rule{thinmathspace}{0ex}}\\frac{1}{c}\\phantom{\\rule{thinmathspace}{0ex}}\\frac{\\mathrm{\\partial }\\stackrel{\\to }{\\mathbf{B}}}{\\mathrm{\\partial }t}& =\\stackrel{\\to }{\\mathbf{0}}\\\\ \\mathrm{\\nabla }\\cdot \\stackrel{\\to }{\\mathbf{B}}& =0\\end{array}", "math_type": "mathml", "by": "mathjax"}}, {"type": "title", "raw_content": "

                                                          In-line Mathematics

                                                          ", "content": {"title_content": "In-line Mathematics", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression \\sqrt{3x-1}+\\left(1+x{\\right)}^{2}is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

                                                          ", "content": [{"c": "Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression", "t": "text"}, {"c": "\\sqrt{3x-1}+\\left(1+x{\\right)}^{2}", "t": "equation-inline"}, {"c": "is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.", "t": "text"}]}]], "main_html": "

                                                          MathJax v3 beta: MathML input, HTML output test

                                                          \n When\n a\\ne 0, there are two solutions toa{x}^{2}+bx+c=0and they are

                                                          x=\u2212b\u00b1b2\u22124ac2a.

                                                          The Lorenz Equations

                                                          x\u02d9=\u03c3(y\u2212x)y\u02d9=\u03c1x\u2212y\u2212xzz\u02d9=\u2212\u03b2z+xy

                                                          The Cauchy-Schwarz Inequality

                                                          (\u2211k=1nakbk)2\u2264(\u2211k=1nak2)(\u2211k=1nbk2)

                                                          A Cross Product Formula

                                                          V1\u00d7V2=|ijk\u2202X\u2202u\u2202Y\u2202u0\u2202X\u2202v\u2202Y\u2202v0|

                                                          The probability of getting kheads when flippingncoins is:

                                                          P(E)=(nk)pk(1\u2212p)n\u2212k

                                                          An Identity of Ramanujan

                                                          1(\u03d55\u2212\u03d5)e25\u03c0=1+e\u22122\u03c01+e\u22124\u03c01+e\u22126\u03c01+e\u22128\u03c01+\u2026

                                                          A Rogers-Ramanujan Identity

                                                          1+q2(1\u2212q)+q6(1\u2212q)(1\u2212q2)+\u22ef=\u220fj=0\u221e1(1\u2212q5j+2)(1\u2212q5j+3),for\u00a0|q|<1.

                                                          Maxwell's Equations

                                                          \u2207\u00d7B\u2192\u22121c\u2202E\u2192\u2202t=4\u03c0cj\u2192\u2207\u22c5E\u2192=4\u03c0\u03c1\u2207\u00d7E\u2192+1c\u2202B\u2192\u2202t=0\u2192\u2207\u22c5B\u2192=0

                                                          In-line Mathematics

                                                          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression \\sqrt{3x-1}+\\left(1+x{\\right)}^{2}is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

                                                          ", "statics": {"title": 9, "paragraph": 2, "paragraph.text": 5, "paragraph.equation-inline": 3, "equation-interline": 8}, "url": "https://mathjax.github.io/MathJax-demos-web/mml-chtml.html", "content": "# MathJax v3 beta: MathML input, HTML output test\n\nWhen $a\\ne 0$ , there are two solutions to $a{x}^{2}+bx+c=0$ and they are\n\n$$\nx=\\frac{-b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}\\text{.}\n$$\n\n## The Lorenz Equations\n\n$$\n\\begin{array}{rl}\\stackrel{\u02d9}{x}& =\\sigma \\left(y-x\\right)\\\\ \\stackrel{\u02d9}{y}& =\\rho x-y-xz\\\\ \\stackrel{\u02d9}{z}& =-\\beta z+xy\\end{array}\n$$\n\n## The Cauchy-Schwarz Inequality\n\n$$\n{\\left(\\sum _{k=1}^{n}{a}_{k}{b}_{k}\\right)}^{\\phantom{\\rule{negativethinmathspace}{0ex}}\\phantom{\\rule{negativethinmathspace}{0ex}}2}\\le \\left(\\sum _{k=1}^{n}{a}_{k}^{2}\\right)\\left(\\sum _{k=1}^{n}{b}_{k}^{2}\\right)\n$$\n\n## A Cross Product Formula\n\n$$\n{\\mathbf{V}}_{1}\u00d7{\\mathbf{V}}_{2}=|\\begin{array}{ccc}\\mathbf{i}& \\mathbf{j}& \\mathbf{k}\\\\ \\frac{\\mathrm{\\partial }X}{\\mathrm{\\partial }u}& \\frac{\\mathrm{\\partial }Y}{\\mathrm{\\partial }u}& 0\\\\ \\frac{\\mathrm{\\partial }X}{\\mathrm{\\partial }v}& \\frac{\\mathrm{\\partial }Y}{\\mathrm{\\partial }v}& 0\\end{array}|\n$$\n\n## The probability of getting k heads when flipping n coins is:\n\n$$\nP\\left(E\\right)=\\left(\\genfrac{}{}{0}{}{n}{k}\\right){p}^{k}\\left(1-p{\\right)}^{n-k}\n$$\n\n## An Identity of Ramanujan\n\n$$\n\\frac{1}{\\left(\\sqrt{\\varphi \\sqrt{5}}-\\varphi \\right){e}^{\\frac{2}{5}\\pi }}=1+\\frac{{e}^{-2\\pi }}{1+\\frac{{e}^{-4\\pi }}{1+\\frac{{e}^{-6\\pi }}{1+\\frac{{e}^{-8\\pi }}{1+\\dots }}}}\n$$\n\n## A Rogers-Ramanujan Identity\n\n$$\n1+\\frac{{q}^{2}}{\\left(1-q\\right)}+\\frac{{q}^{6}}{\\left(1-q\\right)\\left(1-{q}^{2}\\right)}+\\cdots =\\prod _{j=0}^{\\mathrm{\\infty }}\\frac{1}{\\left(1-{q}^{5j+2}\\right)\\left(1-{q}^{5j+3}\\right)},\\phantom{\\rule{1em}{0ex}}\\phantom{\\rule{1em}{0ex}}\\text{for\u00a0}|q|<1.\n$$\n\n## Maxwell's Equations\n\n$$\n\\begin{array}{rl}\\mathrm{\\nabla }\u00d7\\stackrel{\\to }{\\mathbf{B}}-\\phantom{\\rule{thinmathspace}{0ex}}\\frac{1}{c}\\phantom{\\rule{thinmathspace}{0ex}}\\frac{\\mathrm{\\partial }\\stackrel{\\to }{\\mathbf{E}}}{\\mathrm{\\partial }t}& =\\frac{4\\pi }{c}\\stackrel{\\to }{\\mathbf{j}}\\\\ \\mathrm{\\nabla }\\cdot \\stackrel{\\to }{\\mathbf{E}}& =4\\pi \\rho \\\\ \\mathrm{\\nabla }\u00d7\\stackrel{\\to }{\\mathbf{E}}\\phantom{\\rule{thinmathspace}{0ex}}+\\phantom{\\rule{thinmathspace}{0ex}}\\frac{1}{c}\\phantom{\\rule{thinmathspace}{0ex}}\\frac{\\mathrm{\\partial }\\stackrel{\\to }{\\mathbf{B}}}{\\mathrm{\\partial }t}& =\\stackrel{\\to }{\\mathbf{0}}\\\\ \\mathrm{\\nabla }\\cdot \\stackrel{\\to }{\\mathbf{B}}& =0\\end{array}\n$$\n\n## In-line Mathematics\n\nFinally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression $\\sqrt{3x-1}+\\left(1+x{\\right)}^{2}$ is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.\n", "html": "\n\n\n\n \n \n \n MathJax v3 with MathML input and HTML output\n \n\n\n

                                                          MathJax v3 beta: MathML input, HTML output test

                                                          \n

                                                          \n When\n \n a\n ≠\n 0\n , there are two solutions to\n \n a\n \n x\n 2\n \n +\n b\n x\n +\n c\n =\n 0\n \n and they are\n \n x\n =\n \n \n \n −\n b\n ±\n \n \n b\n 2\n \n −\n 4\n a\n c\n \n \n \n 2\n a\n \n \n \n .\n \n

                                                          \n\n

                                                          The Lorenz Equations

                                                          \n\n \n \n \n \n \n \n x\n ˙\n \n \n \n \n \n =\n σ\n (\n y\n −\n x\n )\n \n \n \n \n \n \n y\n ˙\n \n \n \n \n \n =\n ρ\n x\n −\n y\n −\n x\n z\n \n \n \n \n \n \n z\n ˙\n \n \n \n \n \n =\n −\n β\n z\n +\n x\n y\n \n \n \n \n\n

                                                          The Cauchy-Schwarz Inequality

                                                          \n\n \n \n \n (\n \n \n ∑\n \n k\n =\n 1\n \n n\n \n \n a\n k\n \n \n b\n k\n \n \n )\n \n \n \n \n 2\n \n \n ≤\n \n (\n \n \n ∑\n \n k\n =\n 1\n \n n\n \n \n a\n k\n 2\n \n \n )\n \n \n (\n \n \n ∑\n \n k\n =\n 1\n \n n\n \n \n b\n k\n 2\n \n \n )\n \n \n

                                                          A Cross Product Formula

                                                          \n\n \n \n \n V\n \n 1\n \n ×\n \n \n V\n \n 2\n \n =\n \n |\n \n \n \n \n i\n \n \n \n \n j\n \n \n \n \n k\n \n \n \n \n \n \n \n ∂\n X\n \n \n ∂\n u\n \n \n \n \n \n \n ∂\n Y\n \n \n ∂\n u\n \n \n \n \n 0\n \n \n \n \n \n \n ∂\n X\n \n \n ∂\n v\n \n \n \n \n \n \n ∂\n Y\n \n \n ∂\n v\n \n \n \n \n 0\n \n \n \n |\n \n \n\n

                                                          The probability of getting \n k\n heads when flipping \n n\n coins is:

                                                          \n\n \n P\n (\n E\n )\n =\n \n \n \n (\n \n \n n\n k\n \n \n )\n \n \n \n \n p\n k\n \n (\n 1\n −\n p\n \n )\n \n n\n −\n k\n \n \n \n\n

                                                          An Identity of Ramanujan

                                                          \n\n \n \n 1\n \n (\n \n ϕ\n \n 5\n \n \n −\n ϕ\n )\n \n e\n \n \n 2\n 5\n \n π\n \n \n \n \n =\n 1\n +\n \n \n e\n \n −\n 2\n π\n \n \n \n 1\n +\n \n \n e\n \n −\n 4\n π\n \n \n \n 1\n +\n \n \n e\n \n −\n 6\n π\n \n \n \n 1\n +\n \n \n e\n \n −\n 8\n π\n \n \n \n 1\n +\n …\n \n \n \n \n \n \n \n \n \n

                                                          A Rogers-Ramanujan Identity

                                                          \n\n \n 1\n +\n \n \n q\n 2\n \n \n (\n 1\n −\n q\n )\n \n \n +\n \n \n q\n 6\n \n \n (\n 1\n −\n q\n )\n (\n 1\n −\n \n q\n 2\n \n )\n \n \n +\n ⋯\n =\n \n ∏\n \n j\n =\n 0\n \n \n ∞\n \n \n \n 1\n \n (\n 1\n −\n \n q\n \n 5\n j\n +\n 2\n \n \n )\n (\n 1\n −\n \n q\n \n 5\n j\n +\n 3\n \n \n )\n \n \n ,\n \n \n \n for \n \n \n |\n \n q\n \n |\n \n <\n 1\n \n \n .\n \n

                                                          Maxwell's Equations

                                                          \n\n \n \n \n \n ∇\n ×\n \n \n \n B\n \n →\n \n \n −\n \n \n 1\n c\n \n \n \n \n ∂\n \n \n \n E\n \n →\n \n \n \n \n ∂\n t\n \n \n \n \n \n =\n \n \n 4\n π\n \n c\n \n \n \n \n j\n \n →\n \n \n \n \n \n \n ∇\n ⋅\n \n \n \n E\n \n →\n \n \n \n \n \n =\n 4\n π\n ρ\n \n \n \n \n ∇\n ×\n \n \n \n E\n \n →\n \n \n \n +\n \n \n 1\n c\n \n \n \n \n ∂\n \n \n \n B\n \n →\n \n \n \n \n ∂\n t\n \n \n \n \n \n =\n \n \n \n 0\n \n →\n \n \n \n \n \n \n ∇\n ⋅\n \n \n \n B\n \n →\n \n \n \n \n \n =\n 0\n \n \n \n \n\n

                                                          In-line Mathematics

                                                          \n\n

                                                          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression \n \n 3\n x\n −\n 1\n \n +\n (\n 1\n +\n x\n \n )\n 2\n \n is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

                                                          \n\n\n\n"} +{"content_list": [[{"type": "title", "raw_content": "

                                                          MathJax v3 beta: MathML input, HTML output test

                                                          ", "content": {"title_content": "MathJax v3 beta: MathML input, HTML output test", "level": "1"}}, {"type": "paragraph", "raw_content": "

                                                          \n When\n a\\ne 0, there are two solutions toa{x}^{2}+bx+c=0and they are

                                                          ", "content": [{"c": "When", "t": "text"}, {"c": "a\\ne 0", "t": "equation-inline"}, {"c": ", there are two solutions to", "t": "text"}, {"c": "a{x}^{2}+bx+c=0", "t": "equation-inline"}, {"c": "and they are", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x=\u2212b\u00b1b2\u22124ac2a.", "content": {"math_content": "x=\\frac{-b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}\\text{.}", "math_type": "mathml", "by": "mathjax"}}, {"type": "title", "raw_content": "

                                                          The Lorenz Equations

                                                          ", "content": {"title_content": "The Lorenz Equations", "level": "2"}}, {"type": "equation-interline", "raw_content": "x\u02d9=\u03c3(y\u2212x)y\u02d9=\u03c1x\u2212y\u2212xzz\u02d9=\u2212\u03b2z+xy", "content": {"math_content": "\\begin{array}{rl}\\stackrel{\u02d9}{x}& =\\sigma \\left(y-x\\right)\\\\ \\stackrel{\u02d9}{y}& =\\rho x-y-xz\\\\ \\stackrel{\u02d9}{z}& =-\\beta z+xy\\end{array}", "math_type": "mathml", "by": "mathjax"}}, {"type": "title", "raw_content": "

                                                          The Cauchy-Schwarz Inequality

                                                          ", "content": {"title_content": "The Cauchy-Schwarz Inequality", "level": "2"}}, {"type": "equation-interline", "raw_content": "(\u2211k=1nakbk)2\u2264(\u2211k=1nak2)(\u2211k=1nbk2)", "content": {"math_content": "{\\left(\\sum _{k=1}^{n}{a}_{k}{b}_{k}\\right)}^{2}\\le \\left(\\sum _{k=1}^{n}{a}_{k}^{2}\\right)\\left(\\sum _{k=1}^{n}{b}_{k}^{2}\\right)", "math_type": "mathml", "by": "mathjax"}}, {"type": "title", "raw_content": "

                                                          A Cross Product Formula

                                                          ", "content": {"title_content": "A Cross Product Formula", "level": "2"}}, {"type": "equation-interline", "raw_content": "V1\u00d7V2=|ijk\u2202X\u2202u\u2202Y\u2202u0\u2202X\u2202v\u2202Y\u2202v0|", "content": {"math_content": "{\\mathbf{V}}_{1}\u00d7{\\mathbf{V}}_{2}=|\\begin{array}{ccc}\\mathbf{i}& \\mathbf{j}& \\mathbf{k}\\\\ \\frac{\\mathrm{\\partial }X}{\\mathrm{\\partial }u}& \\frac{\\mathrm{\\partial }Y}{\\mathrm{\\partial }u}& 0\\\\ \\frac{\\mathrm{\\partial }X}{\\mathrm{\\partial }v}& \\frac{\\mathrm{\\partial }Y}{\\mathrm{\\partial }v}& 0\\end{array}|", "math_type": "mathml", "by": "mathjax"}}, {"type": "title", "raw_content": "

                                                          The probability of getting kheads when flippingncoins is:

                                                          ", "content": {"title_content": "The probability of getting k heads when flipping n coins is:", "level": "2"}}, {"type": "equation-interline", "raw_content": "P(E)=(nk)pk(1\u2212p)n\u2212k", "content": {"math_content": "P\\left(E\\right)=\\left(\\genfrac{}{}{0}{}{n}{k}\\right){p}^{k}{\\left(1-p\\right)}^{n-k}", "math_type": "mathml", "by": "mathjax"}}, {"type": "title", "raw_content": "

                                                          An Identity of Ramanujan

                                                          ", "content": {"title_content": "An Identity of Ramanujan", "level": "2"}}, {"type": "equation-interline", "raw_content": "1(\u03d55\u2212\u03d5)e25\u03c0=1+e\u22122\u03c01+e\u22124\u03c01+e\u22126\u03c01+e\u22128\u03c01+\u2026", "content": {"math_content": "\\frac{1}{\\left(\\sqrt{\\varphi \\sqrt{5}}-\\varphi \\right){e}^{\\frac{2}{5}\\pi }}=1+\\frac{{e}^{-2\\pi }}{1+\\frac{{e}^{-4\\pi }}{1+\\frac{{e}^{-6\\pi }}{1+\\frac{{e}^{-8\\pi }}{1+\\dots }}}}", "math_type": "mathml", "by": "mathjax"}}, {"type": "title", "raw_content": "

                                                          A Rogers-Ramanujan Identity

                                                          ", "content": {"title_content": "A Rogers-Ramanujan Identity", "level": "2"}}, {"type": "equation-interline", "raw_content": "1+q2(1\u2212q)+q6(1\u2212q)(1\u2212q2)+\u22ef=\u220fj=0\u221e1(1\u2212q5j+2)(1\u2212q5j+3),for\u00a0|q|<1.", "content": {"math_content": "1+\\frac{{q}^{2}}{\\left(1-q\\right)}+\\frac{{q}^{6}}{\\left(1-q\\right)\\left(1-{q}^{2}\\right)}+\\cdots =\\prod _{j=0}^{\\mathrm{\\infty }}\\frac{1}{\\left(1-{q}^{5j+2}\\right)\\left(1-{q}^{5j+3}\\right)},\\text{for\u00a0}|q|<1.", "math_type": "mathml", "by": "mathjax"}}, {"type": "title", "raw_content": "

                                                          Maxwell's Equations

                                                          ", "content": {"title_content": "Maxwell's Equations", "level": "2"}}, {"type": "equation-interline", "raw_content": "\u2207\u00d7B\u2192\u22121c\u2202E\u2192\u2202t=4\u03c0cj\u2192\u2207\u22c5E\u2192=4\u03c0\u03c1\u2207\u00d7E\u2192+1c\u2202B\u2192\u2202t=0\u2192\u2207\u22c5B\u2192=0", "content": {"math_content": "\\begin{array}{rl}\\mathrm{\\nabla }\u00d7\\stackrel{\\to }{\\mathbf{B}}-\\frac{1}{c}\\frac{\\mathrm{\\partial }\\stackrel{\\to }{\\mathbf{E}}}{\\mathrm{\\partial }t}& =\\frac{4\\pi }{c}\\stackrel{\\to }{\\mathbf{j}}\\\\ \\mathrm{\\nabla }\\cdot \\stackrel{\\to }{\\mathbf{E}}& =4\\pi \\rho \\\\ \\mathrm{\\nabla }\u00d7\\stackrel{\\to }{\\mathbf{E}}+\\frac{1}{c}\\frac{\\mathrm{\\partial }\\stackrel{\\to }{\\mathbf{B}}}{\\mathrm{\\partial }t}& =\\stackrel{\\to }{\\mathbf{0}}\\\\ \\mathrm{\\nabla }\\cdot \\stackrel{\\to }{\\mathbf{B}}& =0\\end{array}", "math_type": "mathml", "by": "mathjax"}}, {"type": "title", "raw_content": "

                                                          In-line Mathematics

                                                          ", "content": {"title_content": "In-line Mathematics", "level": "2"}}, {"type": "paragraph", "raw_content": "

                                                          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression \\sqrt{3x-1}+{\\left(1+x\\right)}^{2}is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

                                                          ", "content": [{"c": "Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression", "t": "text"}, {"c": "\\sqrt{3x-1}+{\\left(1+x\\right)}^{2}", "t": "equation-inline"}, {"c": "is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.", "t": "text"}]}]], "main_html": "

                                                          MathJax v3 beta: MathML input, HTML output test

                                                          \n When\n a\\ne 0, there are two solutions toa{x}^{2}+bx+c=0and they are

                                                          x=\u2212b\u00b1b2\u22124ac2a.

                                                          The Lorenz Equations

                                                          x\u02d9=\u03c3(y\u2212x)y\u02d9=\u03c1x\u2212y\u2212xzz\u02d9=\u2212\u03b2z+xy

                                                          The Cauchy-Schwarz Inequality

                                                          (\u2211k=1nakbk)2\u2264(\u2211k=1nak2)(\u2211k=1nbk2)

                                                          A Cross Product Formula

                                                          V1\u00d7V2=|ijk\u2202X\u2202u\u2202Y\u2202u0\u2202X\u2202v\u2202Y\u2202v0|

                                                          The probability of getting kheads when flippingncoins is:

                                                          P(E)=(nk)pk(1\u2212p)n\u2212k

                                                          An Identity of Ramanujan

                                                          1(\u03d55\u2212\u03d5)e25\u03c0=1+e\u22122\u03c01+e\u22124\u03c01+e\u22126\u03c01+e\u22128\u03c01+\u2026

                                                          A Rogers-Ramanujan Identity

                                                          1+q2(1\u2212q)+q6(1\u2212q)(1\u2212q2)+\u22ef=\u220fj=0\u221e1(1\u2212q5j+2)(1\u2212q5j+3),for\u00a0|q|<1.

                                                          Maxwell's Equations

                                                          \u2207\u00d7B\u2192\u22121c\u2202E\u2192\u2202t=4\u03c0cj\u2192\u2207\u22c5E\u2192=4\u03c0\u03c1\u2207\u00d7E\u2192+1c\u2202B\u2192\u2202t=0\u2192\u2207\u22c5B\u2192=0

                                                          In-line Mathematics

                                                          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression \\sqrt{3x-1}+{\\left(1+x\\right)}^{2}is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

                                                          ", "statics": {"title": 9, "paragraph": 2, "paragraph.text": 5, "paragraph.equation-inline": 3, "equation-interline": 8}, "url": "https://mathjax.github.io/MathJax-demos-web/mml-chtml.html", "content": "# MathJax v3 beta: MathML input, HTML output test\n\nWhen $a\\ne 0$ , there are two solutions to $a{x}^{2}+bx+c=0$ and they are\n\n$$\nx=\\frac{-b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}\\text{.}\n$$\n\n## The Lorenz Equations\n\n$$\n\\begin{array}{rl}\\stackrel{\u02d9}{x}& =\\sigma \\left(y-x\\right)\\\\ \\stackrel{\u02d9}{y}& =\\rho x-y-xz\\\\ \\stackrel{\u02d9}{z}& =-\\beta z+xy\\end{array}\n$$\n\n## The Cauchy-Schwarz Inequality\n\n$$\n{\\left(\\sum _{k=1}^{n}{a}_{k}{b}_{k}\\right)}^{2}\\le \\left(\\sum _{k=1}^{n}{a}_{k}^{2}\\right)\\left(\\sum _{k=1}^{n}{b}_{k}^{2}\\right)\n$$\n\n## A Cross Product Formula\n\n$$\n{\\mathbf{V}}_{1}\u00d7{\\mathbf{V}}_{2}=|\\begin{array}{ccc}\\mathbf{i}& \\mathbf{j}& \\mathbf{k}\\\\ \\frac{\\mathrm{\\partial }X}{\\mathrm{\\partial }u}& \\frac{\\mathrm{\\partial }Y}{\\mathrm{\\partial }u}& 0\\\\ \\frac{\\mathrm{\\partial }X}{\\mathrm{\\partial }v}& \\frac{\\mathrm{\\partial }Y}{\\mathrm{\\partial }v}& 0\\end{array}|\n$$\n\n## The probability of getting k heads when flipping n coins is:\n\n$$\nP\\left(E\\right)=\\left(\\genfrac{}{}{0}{}{n}{k}\\right){p}^{k}{\\left(1-p\\right)}^{n-k}\n$$\n\n## An Identity of Ramanujan\n\n$$\n\\frac{1}{\\left(\\sqrt{\\varphi \\sqrt{5}}-\\varphi \\right){e}^{\\frac{2}{5}\\pi }}=1+\\frac{{e}^{-2\\pi }}{1+\\frac{{e}^{-4\\pi }}{1+\\frac{{e}^{-6\\pi }}{1+\\frac{{e}^{-8\\pi }}{1+\\dots }}}}\n$$\n\n## A Rogers-Ramanujan Identity\n\n$$\n1+\\frac{{q}^{2}}{\\left(1-q\\right)}+\\frac{{q}^{6}}{\\left(1-q\\right)\\left(1-{q}^{2}\\right)}+\\cdots =\\prod _{j=0}^{\\mathrm{\\infty }}\\frac{1}{\\left(1-{q}^{5j+2}\\right)\\left(1-{q}^{5j+3}\\right)},\\text{for\u00a0}|q|<1.\n$$\n\n## Maxwell's Equations\n\n$$\n\\begin{array}{rl}\\mathrm{\\nabla }\u00d7\\stackrel{\\to }{\\mathbf{B}}-\\frac{1}{c}\\frac{\\mathrm{\\partial }\\stackrel{\\to }{\\mathbf{E}}}{\\mathrm{\\partial }t}& =\\frac{4\\pi }{c}\\stackrel{\\to }{\\mathbf{j}}\\\\ \\mathrm{\\nabla }\\cdot \\stackrel{\\to }{\\mathbf{E}}& =4\\pi \\rho \\\\ \\mathrm{\\nabla }\u00d7\\stackrel{\\to }{\\mathbf{E}}+\\frac{1}{c}\\frac{\\mathrm{\\partial }\\stackrel{\\to }{\\mathbf{B}}}{\\mathrm{\\partial }t}& =\\stackrel{\\to }{\\mathbf{0}}\\\\ \\mathrm{\\nabla }\\cdot \\stackrel{\\to }{\\mathbf{B}}& =0\\end{array}\n$$\n\n## In-line Mathematics\n\nFinally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression $\\sqrt{3x-1}+{\\left(1+x\\right)}^{2}$ is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.\n", "html": "\n\n\n\n \n \n \n MathJax v3 with MathML input and HTML output\n \n\n\n

                                                          MathJax v3 beta: MathML input, HTML output test

                                                          \n

                                                          \n When\n \n a\n ≠\n 0\n , there are two solutions to\n \n a\n \n x\n 2\n \n +\n b\n x\n +\n c\n =\n 0\n \n and they are\n \n x\n =\n \n \n \n −\n b\n ±\n \n \n b\n 2\n \n −\n 4\n a\n c\n \n \n \n 2\n a\n \n \n \n .\n \n

                                                          \n\n

                                                          The Lorenz Equations

                                                          \n\n \n \n \n \n \n \n x\n ˙\n \n \n \n \n \n =\n σ\n (\n y\n −\n x\n )\n \n \n \n \n \n \n y\n ˙\n \n \n \n \n \n =\n ρ\n x\n −\n y\n −\n x\n z\n \n \n \n \n \n \n z\n ˙\n \n \n \n \n \n =\n −\n β\n z\n +\n x\n y\n \n \n \n \n\n

                                                          The Cauchy-Schwarz Inequality

                                                          \n\n \n \n \n (\n \n \n ∑\n \n k\n =\n 1\n \n n\n \n \n a\n k\n \n \n b\n k\n \n \n )\n \n \n \n \n 2\n \n \n ≤\n \n (\n \n \n ∑\n \n k\n =\n 1\n \n n\n \n \n a\n k\n 2\n \n \n )\n \n \n (\n \n \n ∑\n \n k\n =\n 1\n \n n\n \n \n b\n k\n 2\n \n \n )\n \n \n

                                                          A Cross Product Formula

                                                          \n\n \n \n \n V\n \n 1\n \n ×\n \n \n V\n \n 2\n \n =\n \n |\n \n \n \n \n i\n \n \n \n \n j\n \n \n \n \n k\n \n \n \n \n \n \n \n ∂\n X\n \n \n ∂\n u\n \n \n \n \n \n \n ∂\n Y\n \n \n ∂\n u\n \n \n \n \n 0\n \n \n \n \n \n \n ∂\n X\n \n \n ∂\n v\n \n \n \n \n \n \n ∂\n Y\n \n \n ∂\n v\n \n \n \n \n 0\n \n \n \n |\n \n \n\n

                                                          The probability of getting \n k\n heads when flipping \n n\n coins is:

                                                          \n\n \n P\n (\n E\n )\n =\n \n \n \n (\n \n \n n\n k\n \n \n )\n \n \n \n \n p\n k\n \n (\n 1\n −\n p\n \n )\n \n n\n −\n k\n \n \n \n\n

                                                          An Identity of Ramanujan

                                                          \n\n \n \n 1\n \n (\n \n ϕ\n \n 5\n \n \n −\n ϕ\n )\n \n e\n \n \n 2\n 5\n \n π\n \n \n \n \n =\n 1\n +\n \n \n e\n \n −\n 2\n π\n \n \n \n 1\n +\n \n \n e\n \n −\n 4\n π\n \n \n \n 1\n +\n \n \n e\n \n −\n 6\n π\n \n \n \n 1\n +\n \n \n e\n \n −\n 8\n π\n \n \n \n 1\n +\n …\n \n \n \n \n \n \n \n \n \n

                                                          A Rogers-Ramanujan Identity

                                                          \n\n \n 1\n +\n \n \n q\n 2\n \n \n (\n 1\n −\n q\n )\n \n \n +\n \n \n q\n 6\n \n \n (\n 1\n −\n q\n )\n (\n 1\n −\n \n q\n 2\n \n )\n \n \n +\n ⋯\n =\n \n ∏\n \n j\n =\n 0\n \n \n ∞\n \n \n \n 1\n \n (\n 1\n −\n \n q\n \n 5\n j\n +\n 2\n \n \n )\n (\n 1\n −\n \n q\n \n 5\n j\n +\n 3\n \n \n )\n \n \n ,\n \n \n \n for \n \n \n |\n \n q\n \n |\n \n <\n 1\n \n \n .\n \n

                                                          Maxwell's Equations

                                                          \n\n \n \n \n \n ∇\n ×\n \n \n \n B\n \n →\n \n \n −\n \n \n 1\n c\n \n \n \n \n ∂\n \n \n \n E\n \n →\n \n \n \n \n ∂\n t\n \n \n \n \n \n =\n \n \n 4\n π\n \n c\n \n \n \n \n j\n \n →\n \n \n \n \n \n \n ∇\n ⋅\n \n \n \n E\n \n →\n \n \n \n \n \n =\n 4\n π\n ρ\n \n \n \n \n ∇\n ×\n \n \n \n E\n \n →\n \n \n \n +\n \n \n 1\n c\n \n \n \n \n ∂\n \n \n \n B\n \n →\n \n \n \n \n ∂\n t\n \n \n \n \n \n =\n \n \n \n 0\n \n →\n \n \n \n \n \n \n ∇\n ⋅\n \n \n \n B\n \n →\n \n \n \n \n \n =\n 0\n \n \n \n \n\n

                                                          In-line Mathematics

                                                          \n\n

                                                          Finally, while display equations look good for a page of samples, the\n ability to mix math and text in a paragraph is also important. This\n expression \n \n 3\n x\n −\n 1\n \n +\n (\n 1\n +\n x\n \n )\n 2\n \n is an example of an inline equation. As\n you see, MathJax equations can be used this way as well, without unduly\n disturbing the spacing between lines.

                                                          \n\n\n\n"} diff --git a/bench/data/groundtruth/math_none_mathml_1.jsonl b/bench/data/groundtruth/math_none_mathml_1.jsonl index e16dffa8..0501d3e2 100644 --- a/bench/data/groundtruth/math_none_mathml_1.jsonl +++ b/bench/data/groundtruth/math_none_mathml_1.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "paragraph", "raw_content": "

                                                          In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by {\\displaystyle \\sigma ^{2}}, {\\displaystyle s^{2}}, {\\displaystyle \\operatorname {Var} (X)}, {\\displaystyle V(X)}, or {\\displaystyle \\mathbb {V} (X)}.[1]

                                                          ", "content": [{"c": "In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation(SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by", "t": "text"}, {"c": "{\\displaystyle \\sigma ^{2}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "{\\displaystyle s^{2}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Var} (X)}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "{\\displaystyle V(X)}", "t": "equation-inline"}, {"c": ", or", "t": "text"}, {"c": "{\\displaystyle \\mathbb {V} (X)}", "t": "equation-inline"}, {"c": ".[ 1]", "t": "text"}]}, {"type": "image", "raw_content": "
                                                          Example of samples from two populations with the same mean but different variances. The red population has mean 100 and variance 100 (SD=10) while the blue population has mean 100 and variance 2500 (SD=50) where SD stands for Standard Deviation.
                                                          ", "content": {"url": "https://upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Comparison_standard_deviations.svg/400px-Comparison_standard_deviations.svg.png", "data": null, "alt": null, "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

                                                          An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.\n

                                                          ", "content": [{"c": "An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          There are two distinct concepts that are both called \"variance\". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.\n

                                                          ", "content": [{"c": "There are two distinct concepts that are both called \"variance\". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling.\n

                                                          ", "content": [{"c": "The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling.", "t": "text"}]}, {"type": "image", "raw_content": "
                                                          Geometric visualisation of the variance of an arbitrary distribution (2, 4, 4, 4, 5, 5, 7, 9):
                                                          ", "content": {"url": "https://upload.wikimedia.org/wikipedia/commons/thumb/6/64/Variance_visualisation.svg/220px-Variance_visualisation.svg.png", "data": null, "alt": null, "title": null, "caption": null}}, {"type": "list", "raw_content": "
                                                          1. A frequency distribution is constructed.
                                                          2. The centroid of the distribution gives its mean.
                                                          3. A square with sides equal to the difference of each value from the mean is formed for each value.
                                                          4. Arranging the squares into a rectangle with one side equal to the number of values, n, results in the other side being the distribution's variance, \u03c32.
                                                          ", "content": {"items": [[[{"c": "A frequency distribution is constructed.", "t": "text"}]], [[{"c": "The centroid of the distribution gives its mean.", "t": "text"}]], [[{"c": "A square with sides equal to the difference of each value from the mean is formed for each value.", "t": "text"}]], [[{"c": "Arranging the squares into a rectangle with one side equal to the number of values, ", "t": "text"}, {"c": "n", "t": "text"}, {"c": ", results in the other side being the distribution's variance, ", "t": "text"}, {"c": "\u03c3", "t": "text"}, {"c": "2", "t": "text"}, {"c": ".", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Definition

                                                          ", "content": {"title_content": "Definition", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The variance of a random variable {\\displaystyle X} is the expected value of the squared deviation from the mean of {\\displaystyle X}, {\\displaystyle \\mu =\\operatorname {E} [X]}:\n

                                                          ", "content": [{"c": "The variance of a random variable", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "is the expected value of the squared deviation from the mean of", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "{\\displaystyle \\mu =\\operatorname {E} [X]}", "t": "equation-inline"}, {"c": ":", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right].}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right].}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:\n

                                                          ", "content": [{"c": "This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X)=\\operatorname {Cov} (X,X).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\operatorname {Cov} (X,X).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          The variance is also equivalent to the second cumulant of a probability distribution that generates {\\displaystyle X}. The variance is typically designated as {\\displaystyle \\operatorname {Var} (X)}, or sometimes as {\\displaystyle V(X)} or {\\displaystyle \\mathbb {V} (X)}, or symbolically as {\\displaystyle \\sigma _{X}^{2}} or simply {\\displaystyle \\sigma ^{2}} (pronounced \"sigma squared\"). The expression for the variance can be expanded as follows:\n

                                                          ", "content": [{"c": "The variance is also equivalent to the second cumulant of a probability distribution that generates", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": ". The variance is typically designated as", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Var} (X)}", "t": "equation-inline"}, {"c": ", or sometimes as", "t": "text"}, {"c": "{\\displaystyle V(X)}", "t": "equation-inline"}, {"c": "or", "t": "text"}, {"c": "{\\displaystyle \\mathbb {V} (X)}", "t": "equation-inline"}, {"c": ", or symbolically as", "t": "text"}, {"c": "{\\displaystyle \\sigma _{X}^{2}}", "t": "equation-inline"}, {"c": "or simply", "t": "text"}, {"c": "{\\displaystyle \\sigma ^{2}}", "t": "equation-inline"}, {"c": "(pronounced \" sigma squared\"). The expression for the variance can be expanded as follows:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left[(X-\\operatorname {E} [X])^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}-2X\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]^{2}+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}\\end{aligned}}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left[(X-\\operatorname {E} [X])^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}-2X\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]^{2}+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          In other words, the variance of X is equal to the mean of the square of X minus the square of the mean of X. This equation should not be used for computations using floating point arithmetic, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see algorithms for calculating variance.\n

                                                          ", "content": [{"c": "In other words, the variance of X is equal to the mean of the square of X minus the square of the mean of X. This equation should not be used for computations using floating point arithmetic, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see algorithms for calculating variance.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Discrete random variable

                                                          ", "content": {"title_content": "Discrete random variable", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          If the generator of random variable {\\displaystyle X} is discrete with probability mass function{\\displaystyle x_{1}\\mapsto p_{1},x_{2}\\mapsto p_{2},\\ldots ,x_{n}\\mapsto p_{n}}, then\n

                                                          ", "content": [{"c": "If the generator of random variable", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "is discrete with probability mass function", "t": "text"}, {"c": "{\\displaystyle x_{1}\\mapsto p_{1},x_{2}\\mapsto p_{2},\\ldots ,x_{n}\\mapsto p_{n}}", "t": "equation-inline"}, {"c": ", then", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X)=\\sum _{i=1}^{n}p_{i}\\cdot (x_{i}-\\mu )^{2},}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\sum _{i=1}^{n}p_{i}\\cdot (x_{i}-\\mu )^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          where {\\displaystyle \\mu } is the expected value. That is,\n

                                                          ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\mu }", "t": "equation-inline"}, {"c": "is the expected value. That is,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\mu =\\sum _{i=1}^{n}p_{i}x_{i}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\mu =\\sum _{i=1}^{n}p_{i}x_{i}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          (When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)\n

                                                          ", "content": [{"c": "(When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The variance of a collection of {\\displaystyle n} equally likely values can be written as\n

                                                          ", "content": [{"c": "The variance of a collection of", "t": "text"}, {"c": "{\\displaystyle n}", "t": "equation-inline"}, {"c": "equally likely values can be written as", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n}}\\sum _{i=1}^{n}(x_{i}-\\mu )^{2}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n}}\\sum _{i=1}^{n}(x_{i}-\\mu )^{2}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          where {\\displaystyle \\mu } is the average value. That is,\n

                                                          ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\mu }", "t": "equation-inline"}, {"c": "is the average value. That is,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\mu ={\\frac {1}{n}}\\sum _{i=1}^{n}x_{i}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\mu ={\\frac {1}{n}}\\sum _{i=1}^{n}x_{i}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          The variance of a set of {\\displaystyle n} equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:[2]

                                                          ", "content": [{"c": "The variance of a set of", "t": "text"}, {"c": "{\\displaystyle n}", "t": "equation-inline"}, {"c": "equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:[ 2]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{\\frac {1}{2}}(x_{i}-x_{j})^{2}={\\frac {1}{n^{2}}}\\sum _{i}\\sum _{j>i}(x_{i}-x_{j})^{2}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{\\frac {1}{2}}(x_{i}-x_{j})^{2}={\\frac {1}{n^{2}}}\\sum _{i}\\sum _{j>i}(x_{i}-x_{j})^{2}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Absolutely continuous random variable

                                                          ", "content": {"title_content": "Absolutely continuous random variable", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          If the random variable {\\displaystyle X} has a probability density function{\\displaystyle f(x)}, and {\\displaystyle F(x)} is the corresponding cumulative distribution function, then\n

                                                          ", "content": [{"c": "If the random variable", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "has a probability density function", "t": "text"}, {"c": "{\\displaystyle f(x)}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "{\\displaystyle F(x)}", "t": "equation-inline"}, {"c": "is the corresponding cumulative distribution function, then", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)=\\sigma ^{2}&=\\int _{\\mathbb {R} }(x-\\mu )^{2}f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-2\\mu \\int _{\\mathbb {R} }xf(x)\\,dx+\\mu ^{2}\\int _{\\mathbb {R} }f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\int _{\\mathbb {R} }x\\,dF(x)+\\mu ^{2}\\int _{\\mathbb {R} }\\,dF(x)\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\cdot \\mu +\\mu ^{2}\\cdot 1\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-\\mu ^{2},\\end{aligned}}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)=\\sigma ^{2}&=\\int _{\\mathbb {R} }(x-\\mu )^{2}f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-2\\mu \\int _{\\mathbb {R} }xf(x)\\,dx+\\mu ^{2}\\int _{\\mathbb {R} }f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\int _{\\mathbb {R} }x\\,dF(x)+\\mu ^{2}\\int _{\\mathbb {R} }\\,dF(x)\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\cdot \\mu +\\mu ^{2}\\cdot 1\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-\\mu ^{2},\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          or equivalently,\n

                                                          ", "content": [{"c": "or equivalently,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X)=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-\\mu ^{2},}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-\\mu ^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          where {\\displaystyle \\mu } is the expected value of {\\displaystyle X} given by\n

                                                          ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\mu }", "t": "equation-inline"}, {"c": "is the expected value of", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\mu =\\int _{\\mathbb {R} }xf(x)\\,dx=\\int _{\\mathbb {R} }x\\,dF(x).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\mu =\\int _{\\mathbb {R} }xf(x)\\,dx=\\int _{\\mathbb {R} }x\\,dF(x).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          In these formulas, the integrals with respect to {\\displaystyle dx} and {\\displaystyle dF(x)}\nare Lebesgue and Lebesgue\u2013Stieltjes integrals, respectively.\n

                                                          ", "content": [{"c": "In these formulas, the integrals with respect to", "t": "text"}, {"c": "{\\displaystyle dx}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle dF(x)}", "t": "equation-inline"}, {"c": "are Lebesgue and Lebesgue\u2013Stieltjes integrals, respectively.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          If the function {\\displaystyle x^{2}f(x)} is Riemann-integrable on every finite interval {\\displaystyle [a,b]\\subset \\mathbb {R} ,} then\n

                                                          ", "content": [{"c": "If the function", "t": "text"}, {"c": "{\\displaystyle x^{2}f(x)}", "t": "equation-inline"}, {"c": "is Riemann-integrable on every finite interval", "t": "text"}, {"c": "{\\displaystyle [a,b]\\subset \\mathbb {R} ,}", "t": "equation-inline"}, {"c": "then", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X)=\\int _{-\\infty }^{+\\infty }x^{2}f(x)\\,dx-\\mu ^{2},}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\int _{-\\infty }^{+\\infty }x^{2}f(x)\\,dx-\\mu ^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          where the integral is an improper Riemann integral.\n

                                                          ", "content": [{"c": "where the integral is an improper Riemann integral.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Examples

                                                          ", "content": {"title_content": "Examples", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Exponential distribution

                                                          ", "content": {"title_content": "Exponential distribution", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The exponential distribution with parameter \u03bb is a continuous distribution whose probability density function is given by\n

                                                          ", "content": [{"c": "The exponential distribution with parameter \u03bb is a continuous distribution whose probability density function is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle f(x)=\\lambda e^{-\\lambda x}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle f(x)=\\lambda e^{-\\lambda x}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          on the interval [0, \u221e). Its mean can be shown to be\n

                                                          ", "content": [{"c": "on the interval[0, \u221e). Its mean can be shown to be", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {E} [X]=\\int _{0}^{\\infty }x\\lambda e^{-\\lambda x}\\,dx={\\frac {1}{\\lambda }}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {E} [X]=\\int _{0}^{\\infty }x\\lambda e^{-\\lambda x}\\,dx={\\frac {1}{\\lambda }}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Using integration by parts and making use of the expected value already calculated, we have:\n

                                                          ", "content": [{"c": "Using integration by parts and making use of the expected value already calculated, we have:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\begin{aligned}\\operatorname {E} \\left[X^{2}\\right]&=\\int _{0}^{\\infty }x^{2}\\lambda e^{-\\lambda x}\\,dx\\\\&=\\left[-x^{2}e^{-\\lambda x}\\right]_{0}^{\\infty }+\\int _{0}^{\\infty }2xe^{-\\lambda x}\\,dx\\\\&=0+{\\frac {2}{\\lambda }}\\operatorname {E} [X]\\\\&={\\frac {2}{\\lambda ^{2}}}.\\end{aligned}}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {E} \\left[X^{2}\\right]&=\\int _{0}^{\\infty }x^{2}\\lambda e^{-\\lambda x}\\,dx\\\\&=\\left[-x^{2}e^{-\\lambda x}\\right]_{0}^{\\infty }+\\int _{0}^{\\infty }2xe^{-\\lambda x}\\,dx\\\\&=0+{\\frac {2}{\\lambda }}\\operatorname {E} [X]\\\\&={\\frac {2}{\\lambda ^{2}}}.\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Thus, the variance of X is given by\n

                                                          ", "content": [{"c": "Thus, the variance of X is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}={\\frac {2}{\\lambda ^{2}}}-\\left({\\frac {1}{\\lambda }}\\right)^{2}={\\frac {1}{\\lambda ^{2}}}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}={\\frac {2}{\\lambda ^{2}}}-\\left({\\frac {1}{\\lambda }}\\right)^{2}={\\frac {1}{\\lambda ^{2}}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Fair dice

                                                          ", "content": {"title_content": "Fair dice", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          A fair six-sided dice can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is {\\displaystyle (1+2+3+4+5+6)/6=7/2.} Therefore, the variance of X is\n

                                                          ", "content": [{"c": "A fair six-sided dice can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is", "t": "text"}, {"c": "{\\displaystyle (1+2+3+4+5+6)/6=7/2.}", "t": "equation-inline"}, {"c": "Therefore, the variance of X is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\sum _{i=1}^{6}{\\frac {1}{6}}\\left(i-{\\frac {7}{2}}\\right)^{2}\\\\[5pt]&={\\frac {1}{6}}\\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\\right)\\\\[5pt]&={\\frac {35}{12}}\\approx 2.92.\\end{aligned}}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\sum _{i=1}^{6}{\\frac {1}{6}}\\left(i-{\\frac {7}{2}}\\right)^{2}\\\\[5pt]&={\\frac {1}{6}}\\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\\right)\\\\[5pt]&={\\frac {35}{12}}\\approx 2.92.\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          The general formula for the variance of the outcome, X, of an n-sided die is\n

                                                          ", "content": [{"c": "The general formula for the variance of the outcome, X, of an n-sided die is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left(X^{2}\\right)-(\\operatorname {E} (X))^{2}\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}i^{2}-\\left({\\frac {1}{n}}\\sum _{i=1}^{n}i\\right)^{2}\\\\[5pt]&={\\frac {(n+1)(2n+1)}{6}}-\\left({\\frac {n+1}{2}}\\right)^{2}\\\\[4pt]&={\\frac {n^{2}-1}{12}}.\\end{aligned}}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left(X^{2}\\right)-(\\operatorname {E} (X))^{2}\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}i^{2}-\\left({\\frac {1}{n}}\\sum _{i=1}^{n}i\\right)^{2}\\\\[5pt]&={\\frac {(n+1)(2n+1)}{6}}-\\left({\\frac {n+1}{2}}\\right)^{2}\\\\[4pt]&={\\frac {n^{2}-1}{12}}.\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Commonly used probability distributions

                                                          ", "content": {"title_content": "Commonly used probability distributions", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The following table lists the variance for some commonly used probability distributions.\n

                                                          ", "content": [{"c": "The following table lists the variance for some commonly used probability distributions.", "t": "text"}]}, {"type": "table", "raw_content": "
                                                          Name of the probability distribution\nProbability distribution function\nMean\nVariance\n
                                                          Binomial distributionPr(X=k)=(nk)pk(1\u2212p)n\u2212k{\\displaystyle \\Pr \\,(X=k)={\\binom {n}{k}}p^{k}(1-p)^{n-k}}np{\\displaystyle np}np(1\u2212p){\\displaystyle np(1-p)}
                                                          Geometric distributionPr(X=k)=(1\u2212p)k\u22121p{\\displaystyle \\Pr \\,(X=k)=(1-p)^{k-1}p}1p{\\displaystyle {\\frac {1}{p}}}(1\u2212p)p2{\\displaystyle {\\frac {(1-p)}{p^{2}}}}
                                                          Normal distributionf(x\u2223\u03bc,\u03c32)=12\u03c0\u03c32e\u2212(x\u2212\u03bc)22\u03c32{\\displaystyle f\\left(x\\mid \\mu ,\\sigma ^{2}\\right)={\\frac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}e^{-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}}\u03bc{\\displaystyle \\mu }\u03c32{\\displaystyle \\sigma ^{2}}
                                                          Uniform distribution (continuous)f(x\u2223a,b)={1b\u2212afor a\u2264x\u2264b,0for x<a or x>b{\\displaystyle f(x\\mid a,b)={\\begin{cases}{\\frac {1}{b-a}}&{\\text{for }}a\\leq x\\leq b,\\\\[3pt]0&{\\text{for }}x<a{\\text{ or }}x>b\\end{cases}}}a+b2{\\displaystyle {\\frac {a+b}{2}}}(b\u2212a)212{\\displaystyle {\\frac {(b-a)^{2}}{12}}}
                                                          Exponential distributionf(x\u2223\u03bb)=\u03bbe\u2212\u03bbx{\\displaystyle f(x\\mid \\lambda )=\\lambda e^{-\\lambda x}}1\u03bb{\\displaystyle {\\frac {1}{\\lambda }}}1\u03bb2{\\displaystyle {\\frac {1}{\\lambda ^{2}}}}
                                                          Poisson distributionf(k\u2223\u03bb)=e\u2212\u03bb\u03bbkk!{\\displaystyle f(k\\mid \\lambda )={\\frac {e^{-\\lambda }\\lambda ^{k}}{k!}}}\u03bb{\\displaystyle \\lambda }\u03bb{\\displaystyle \\lambda }
                                                          ", "content": {"html": "
                                                          Name of the probability distributionProbability distribution functionMeanVariance
                                                          Binomial distributionPr<br>(<br>X<br>=<br>k<br>)<br>=<br>(<br>n<br>k<br>)<br>p<br>k<br>(<br>1<br>\u2212<br>p<br>)<br>n<br>\u2212<br>k<br>{\\displaystyle \\Pr \\,(X=k)={\\binom {n}{k}}p^{k}(1-p)^{n-k}}n<br>p<br>{\\displaystyle np}n<br>p<br>(<br>1<br>\u2212<br>p<br>)<br>{\\displaystyle np(1-p)}
                                                          Geometric distributionPr<br>(<br>X<br>=<br>k<br>)<br>=<br>(<br>1<br>\u2212<br>p<br>)<br>k<br>\u2212<br>1<br>p<br>{\\displaystyle \\Pr \\,(X=k)=(1-p)^{k-1}p}1<br>p<br>{\\displaystyle {\\frac {1}{p}}}(<br>1<br>\u2212<br>p<br>)<br>p<br>2<br>{\\displaystyle {\\frac {(1-p)}{p^{2}}}}
                                                          Normal distributionf<br>(<br>x<br>\u2223<br>\u03bc<br>,<br>\u03c3<br>2<br>)<br>=<br>1<br>2<br>\u03c0<br>\u03c3<br>2<br>e<br>\u2212<br>(<br>x<br>\u2212<br>\u03bc<br>)<br>2<br>2<br>\u03c3<br>2<br>{\\displaystyle f\\left(x\\mid \\mu ,\\sigma ^{2}\\right)={\\frac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}e^{-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}}\u03bc<br>{\\displaystyle \\mu }\u03c3<br>2<br>{\\displaystyle \\sigma ^{2}}
                                                          Uniform distribution (continuous)f<br>(<br>x<br>\u2223<br>a<br>,<br>b<br>)<br>=<br>{<br>1<br>b<br>\u2212<br>a<br>for<br>a<br>\u2264<br>x<br>\u2264<br>b<br>,<br>0<br>for<br>x<br><<br>a<br>or<br>x<br>><br>b<br>{\\displaystyle f(x\\mid a,b)={\\begin{cases}{\\frac {1}{b-a}}&{\\text{for }}a\\leq x\\leq b,\\\\[3pt]0&{\\text{for }}x<a{\\text{ or }}x>b\\end{cases}}}a<br>+<br>b<br>2<br>{\\displaystyle {\\frac {a+b}{2}}}(<br>b<br>\u2212<br>a<br>)<br>2<br>12<br>{\\displaystyle {\\frac {(b-a)^{2}}{12}}}
                                                          Exponential distributionf<br>(<br>x<br>\u2223<br>\u03bb<br>)<br>=<br>\u03bb<br>e<br>\u2212<br>\u03bb<br>x<br>{\\displaystyle f(x\\mid \\lambda )=\\lambda e^{-\\lambda x}}1<br>\u03bb<br>{\\displaystyle {\\frac {1}{\\lambda }}}1<br>\u03bb<br>2<br>{\\displaystyle {\\frac {1}{\\lambda ^{2}}}}
                                                          Poisson distributionf<br>(<br>k<br>\u2223<br>\u03bb<br>)<br>=<br>e<br>\u2212<br>\u03bb<br>\u03bb<br>k<br>k<br>!<br>{\\displaystyle f(k\\mid \\lambda )={\\frac {e^{-\\lambda }\\lambda ^{k}}{k!}}}\u03bb<br>{\\displaystyle \\lambda }\u03bb<br>{\\displaystyle \\lambda }
                                                          ", "is_complex": false}}, {"type": "title", "raw_content": "

                                                          Properties

                                                          ", "content": {"title_content": "Properties", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Basic properties

                                                          ", "content": {"title_content": "Basic properties", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Variance is non-negative because the squares are positive or zero:\n

                                                          ", "content": [{"c": "Variance is non-negative because the squares are positive or zero:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X)\\geq 0.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)\\geq 0.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          The variance of a constant is zero.\n

                                                          ", "content": [{"c": "The variance of a constant is zero.", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (a)=0.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (a)=0.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Conversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value:\n

                                                          ", "content": [{"c": "Conversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X)=0\\iff \\exists a:P(X=a)=1.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=0\\iff \\exists a:P(X=a)=1.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Issues of finiteness

                                                          ", "content": {"title_content": "Issues of finiteness", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index{\\displaystyle k} satisfies {\\displaystyle 1<k\\leq 2.}

                                                          ", "content": [{"c": "If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index", "t": "text"}, {"c": "{\\displaystyle k}", "t": "equation-inline"}, {"c": "satisfies", "t": "text"}, {"c": "{\\displaystyle 1Decomposition", "content": {"title_content": "Decomposition", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The general formula for variance decomposition or the law of total variance is: If {\\displaystyle X} and {\\displaystyle Y} are two random variables, and the variance of {\\displaystyle X} exists, then\n

                                                          ", "content": [{"c": "The general formula for variance decomposition or the law of total variance is: If", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle Y}", "t": "equation-inline"}, {"c": "are two random variables, and the variance of", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "exists, then", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} [X]=\\operatorname {E} (\\operatorname {Var} [X\\mid Y])+\\operatorname {Var} (\\operatorname {E} [X\\mid Y]).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} [X]=\\operatorname {E} (\\operatorname {Var} [X\\mid Y])+\\operatorname {Var} (\\operatorname {E} [X\\mid Y]).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          The conditional expectation{\\displaystyle \\operatorname {E} (X\\mid Y)} of {\\displaystyle X} given {\\displaystyle Y}, and the conditional variance{\\displaystyle \\operatorname {Var} (X\\mid Y)} may be understood as follows. Given any particular value y of the random variable Y, there is a conditional expectation {\\displaystyle \\operatorname {E} (X\\mid Y=y)} given the event Y = y. This quantity depends on the particular value y; it is a function {\\displaystyle g(y)=\\operatorname {E} (X\\mid Y=y)}. That same function evaluated at the random variable Y is the conditional expectation {\\displaystyle \\operatorname {E} (X\\mid Y)=g(Y).}

                                                          ", "content": [{"c": "The conditional expectation", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} (X\\mid Y)}", "t": "equation-inline"}, {"c": "of", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "given", "t": "text"}, {"c": "{\\displaystyle Y}", "t": "equation-inline"}, {"c": ", and the conditional variance", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Var} (X\\mid Y)}", "t": "equation-inline"}, {"c": "may be understood as follows. Given any particular value y of the random variable Y, there is a conditional expectation", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} (X\\mid Y=y)}", "t": "equation-inline"}, {"c": "given the event Y= y. This quantity depends on the particular value y; it is a function", "t": "text"}, {"c": "{\\displaystyle g(y)=\\operatorname {E} (X\\mid Y=y)}", "t": "equation-inline"}, {"c": ". That same function evaluated at the random variable Y is the conditional expectation", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} (X\\mid Y)=g(Y).}", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

                                                          In particular, if {\\displaystyle Y} is a discrete random variable assuming possible values {\\displaystyle y_{1},y_{2},y_{3}\\ldots } with corresponding probabilities {\\displaystyle p_{1},p_{2},p_{3}\\ldots ,}, then in the formula for total variance, the first term on the right-hand side becomes\n

                                                          ", "content": [{"c": "In particular, if", "t": "text"}, {"c": "{\\displaystyle Y}", "t": "equation-inline"}, {"c": "is a discrete random variable assuming possible values", "t": "text"}, {"c": "{\\displaystyle y_{1},y_{2},y_{3}\\ldots }", "t": "equation-inline"}, {"c": "with corresponding probabilities", "t": "text"}, {"c": "{\\displaystyle p_{1},p_{2},p_{3}\\ldots ,}", "t": "equation-inline"}, {"c": ", then in the formula for total variance, the first term on the right-hand side becomes", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {E} (\\operatorname {Var} [X\\mid Y])=\\sum _{i}p_{i}\\sigma _{i}^{2},}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {E} (\\operatorname {Var} [X\\mid Y])=\\sum _{i}p_{i}\\sigma _{i}^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          where {\\displaystyle \\sigma _{i}^{2}=\\operatorname {Var} [X\\mid Y=y_{i}]}. Similarly, the second term on the right-hand side becomes\n

                                                          ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\sigma _{i}^{2}=\\operatorname {Var} [X\\mid Y=y_{i}]}", "t": "equation-inline"}, {"c": ". Similarly, the second term on the right-hand side becomes", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (\\operatorname {E} [X\\mid Y])=\\sum _{i}p_{i}\\mu _{i}^{2}-\\left(\\sum _{i}p_{i}\\mu _{i}\\right)^{2}=\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2},}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (\\operatorname {E} [X\\mid Y])=\\sum _{i}p_{i}\\mu _{i}^{2}-\\left(\\sum _{i}p_{i}\\mu _{i}\\right)^{2}=\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          where {\\displaystyle \\mu _{i}=\\operatorname {E} [X\\mid Y=y_{i}]} and {\\displaystyle \\mu =\\sum _{i}p_{i}\\mu _{i}}. Thus the total variance is given by\n

                                                          ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\mu _{i}=\\operatorname {E} [X\\mid Y=y_{i}]}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle \\mu =\\sum _{i}p_{i}\\mu _{i}}", "t": "equation-inline"}, {"c": ". Thus the total variance is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} [X]=\\sum _{i}p_{i}\\sigma _{i}^{2}+\\left(\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2}\\right).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} [X]=\\sum _{i}p_{i}\\sigma _{i}^{2}+\\left(\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2}\\right).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          A similar formula is applied in analysis of variance, where the corresponding formula is\n

                                                          ", "content": [{"c": "A similar formula is applied in analysis of variance, where the corresponding formula is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{between}}+{\\mathit {MS}}_{\\text{within}};}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{between}}+{\\mathit {MS}}_{\\text{within}};}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          here {\\displaystyle {\\mathit {MS}}} refers to the Mean of the Squares. In linear regression analysis the corresponding formula is\n

                                                          ", "content": [{"c": "here", "t": "text"}, {"c": "{\\displaystyle {\\mathit {MS}}}", "t": "equation-inline"}, {"c": "refers to the Mean of the Squares. In linear regression analysis the corresponding formula is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{regression}}+{\\mathit {MS}}_{\\text{residual}}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{regression}}+{\\mathit {MS}}_{\\text{residual}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.\n

                                                          ", "content": [{"c": "This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Similar decompositions are possible for the sum of squared deviations (sum of squares, {\\displaystyle {\\mathit {SS}}}):\n

                                                          ", "content": [{"c": "Similar decompositions are possible for the sum of squared deviations (sum of squares,", "t": "text"}, {"c": "{\\displaystyle {\\mathit {SS}}}", "t": "equation-inline"}, {"c": "):", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{between}}+{\\mathit {SS}}_{\\text{within}},}
                                                          {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{regression}}+{\\mathit {SS}}_{\\text{residual}}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{between}}+{\\mathit {SS}}_{\\text{within}},}", "t": "equation-inline"}]], [[{"c": "{\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{regression}}+{\\mathit {SS}}_{\\text{residual}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Calculation from the CDF

                                                          ", "content": {"title_content": "Calculation from the CDF", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution functionF using\n

                                                          ", "content": [{"c": "The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution function F using", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle 2\\int _{0}^{\\infty }u(1-F(u))\\,du-\\left(\\int _{0}^{\\infty }(1-F(u))\\,du\\right)^{2}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle 2\\int _{0}^{\\infty }u(1-F(u))\\,du-\\left(\\int _{0}^{\\infty }(1-F(u))\\,du\\right)^{2}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          This expression can be used to calculate the variance in situations where the CDF, but not the density, can be conveniently expressed.\n

                                                          ", "content": [{"c": "This expression can be used to calculate the variance in situations where the CDF, but not the density, can be conveniently expressed.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Characteristic property

                                                          ", "content": {"title_content": "Characteristic property", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} \\left(\\left(X-m\\right)^{2}\\right)=\\mathrm {E} (X)}. Conversely, if a continuous function {\\displaystyle \\varphi } satisfies {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} (\\varphi (X-m))=\\mathrm {E} (X)} for all random variables X, then it is necessarily of the form {\\displaystyle \\varphi (x)=ax^{2}+b}, where a > 0. This also holds in the multidimensional case.[3]

                                                          ", "content": [{"c": "The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e.", "t": "text"}, {"c": "{\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} \\left(\\left(X-m\\right)^{2}\\right)=\\mathrm {E} (X)}", "t": "equation-inline"}, {"c": ". Conversely, if a continuous function", "t": "text"}, {"c": "{\\displaystyle \\varphi }", "t": "equation-inline"}, {"c": "satisfies", "t": "text"}, {"c": "{\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} (\\varphi (X-m))=\\mathrm {E} (X)}", "t": "equation-inline"}, {"c": "for all random variables X, then it is necessarily of the form", "t": "text"}, {"c": "{\\displaystyle \\varphi (x)=ax^{2}+b}", "t": "equation-inline"}, {"c": ", where a> 0. This also holds in the multidimensional case.[ 3]", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Units of measurement

                                                          ", "content": {"title_content": "Units of measurement", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is \u221a2.9 \u2248 1.7, slightly larger than the expected absolute deviation of 1.5.\n

                                                          ", "content": [{"c": "Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is\u221a 2.9 \u2248 1.7, slightly larger than the expected absolute deviation of 1.5.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The standard deviation and the expected absolute deviation can both be used as an indicator of the \"spread\" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution.\n

                                                          ", "content": [{"c": "The standard deviation and the expected absolute deviation can both be used as an indicator of the \"spread\" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Propagation

                                                          ", "content": {"title_content": "Propagation", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Addition and multiplication by a constant

                                                          ", "content": {"title_content": "Addition and multiplication by a constant", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged:\n

                                                          ", "content": [{"c": "Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X+a)=\\operatorname {Var} (X).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X+a)=\\operatorname {Var} (X).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          If all values are scaled by a constant, the variance is scaled by the square of that constant:\n

                                                          ", "content": [{"c": "If all values are scaled by a constant, the variance is scaled by the square of that constant:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (aX)=a^{2}\\operatorname {Var} (X).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (aX)=a^{2}\\operatorname {Var} (X).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          The variance of a sum of two random variables is given by\n

                                                          ", "content": [{"c": "The variance of a sum of two random variables is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (aX+bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)+2ab\\,\\operatorname {Cov} (X,Y)}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (aX+bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)+2ab\\,\\operatorname {Cov} (X,Y)}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (aX-bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)-2ab\\,\\operatorname {Cov} (X,Y)}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (aX-bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)-2ab\\,\\operatorname {Cov} (X,Y)}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          where {\\displaystyle \\operatorname {Cov} (X,Y)} is the covariance.\n

                                                          ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Cov} (X,Y)}", "t": "equation-inline"}, {"c": "is the covariance.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Linear combinations

                                                          ", "content": {"title_content": "Linear combinations", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          In general, for the sum of {\\displaystyle N} random variables {\\displaystyle \\{X_{1},\\dots ,X_{N}\\}}, the variance becomes:\n

                                                          ", "content": [{"c": "In general, for the sum of", "t": "text"}, {"c": "{\\displaystyle N}", "t": "equation-inline"}, {"c": "random variables", "t": "text"}, {"c": "{\\displaystyle \\{X_{1},\\dots ,X_{N}\\}}", "t": "equation-inline"}, {"c": ", the variance becomes:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i,j=1}^{N}\\operatorname {Cov} (X_{i},X_{j})=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i})+\\sum _{i\\neq j}\\operatorname {Cov} (X_{i},X_{j}),}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i,j=1}^{N}\\operatorname {Cov} (X_{i},X_{j})=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i})+\\sum _{i\\neq j}\\operatorname {Cov} (X_{i},X_{j}),}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          see also general Bienaym\u00e9's identity.\n

                                                          ", "content": [{"c": "see also general Bienaym\u00e9's identity.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          These results lead to the variance of a linear combination as:\n

                                                          ", "content": [{"c": "These results lead to the variance of a linear combination as:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\begin{aligned}\\operatorname {Var} \\left(\\sum _{i=1}^{N}a_{i}X_{i}\\right)&=\\sum _{i,j=1}^{N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+\\sum _{i\\not =j}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i<j\\leq N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j}).\\end{aligned}}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} \\left(\\sum _{i=1}^{N}a_{i}X_{i}\\right)&=\\sum _{i,j=1}^{N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+\\sum _{i\\not =j}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i

                                                          If the random variables {\\displaystyle X_{1},\\dots ,X_{N}} are such that\n

                                                          ", "content": [{"c": "If the random variables", "t": "text"}, {"c": "{\\displaystyle X_{1},\\dots ,X_{N}}", "t": "equation-inline"}, {"c": "are such that", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Cov} (X_{i},X_{j})=0\\ ,\\ \\forall \\ (i\\neq j),}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Cov} (X_{i},X_{j})=0\\ ,\\ \\forall \\ (i\\neq j),}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          then they are said to be uncorrelated. It follows immediately from the expression given earlier that if the random variables {\\displaystyle X_{1},\\dots ,X_{N}} are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:\n

                                                          ", "content": [{"c": "then they are said to be uncorrelated. It follows immediately from the expression given earlier that if the random variables", "t": "text"}, {"c": "{\\displaystyle X_{1},\\dots ,X_{N}}", "t": "equation-inline"}, {"c": "are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i}).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i}).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Since independent random variables are always uncorrelated (see Covariance \u00a7 Uncorrelatedness and independence), the equation above holds in particular when the random variables {\\displaystyle X_{1},\\dots ,X_{n}} are independent. Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.\n

                                                          ", "content": [{"c": "Since independent random variables are always uncorrelated (see Covariance \u00a7 Uncorrelatedness and independence), the equation above holds in particular when the random variables", "t": "text"}, {"c": "{\\displaystyle X_{1},\\dots ,X_{n}}", "t": "equation-inline"}, {"c": "are independent. Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Matrix notation for the variance of a linear combination

                                                          ", "content": {"title_content": "Matrix notation for the variance of a linear combination", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Define {\\displaystyle X} as a column vector of {\\displaystyle n} random variables {\\displaystyle X_{1},\\ldots ,X_{n}}, and {\\displaystyle c} as a column vector of {\\displaystyle n} scalars {\\displaystyle c_{1},\\ldots ,c_{n}}. Therefore, {\\displaystyle c^{\\mathsf {T}}X} is a linear combination of these random variables, where {\\displaystyle c^{\\mathsf {T}}} denotes the transpose of {\\displaystyle c}. Also let {\\displaystyle \\Sigma } be the covariance matrix of {\\displaystyle X}. The variance of {\\displaystyle c^{\\mathsf {T}}X} is then given by:[4]

                                                          ", "content": [{"c": "Define", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "as a column vector of", "t": "text"}, {"c": "{\\displaystyle n}", "t": "equation-inline"}, {"c": "random variables", "t": "text"}, {"c": "{\\displaystyle X_{1},\\ldots ,X_{n}}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "{\\displaystyle c}", "t": "equation-inline"}, {"c": "as a column vector of", "t": "text"}, {"c": "{\\displaystyle n}", "t": "equation-inline"}, {"c": "scalars", "t": "text"}, {"c": "{\\displaystyle c_{1},\\ldots ,c_{n}}", "t": "equation-inline"}, {"c": ". Therefore,", "t": "text"}, {"c": "{\\displaystyle c^{\\mathsf {T}}X}", "t": "equation-inline"}, {"c": "is a linear combination of these random variables, where", "t": "text"}, {"c": "{\\displaystyle c^{\\mathsf {T}}}", "t": "equation-inline"}, {"c": "denotes the transpose of", "t": "text"}, {"c": "{\\displaystyle c}", "t": "equation-inline"}, {"c": ". Also let", "t": "text"}, {"c": "{\\displaystyle \\Sigma }", "t": "equation-inline"}, {"c": "be the covariance matrix of", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": ". The variance of", "t": "text"}, {"c": "{\\displaystyle c^{\\mathsf {T}}X}", "t": "equation-inline"}, {"c": "is then given by:[ 4]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left(c^{\\mathsf {T}}X\\right)=c^{\\mathsf {T}}\\Sigma c.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(c^{\\mathsf {T}}X\\right)=c^{\\mathsf {T}}\\Sigma c.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          This implies that the variance of the mean can be written as (with a column vector of ones)\n

                                                          ", "content": [{"c": "This implies that the variance of the mean can be written as (with a column vector of ones)", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left({\\bar {x}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}1'X\\right)={\\frac {1}{n^{2}}}1'\\Sigma 1.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left({\\bar {x}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}1'X\\right)={\\frac {1}{n^{2}}}1'\\Sigma 1.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Sum of variables

                                                          ", "content": {"title_content": "Sum of variables", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Sum of uncorrelated variables

                                                          ", "content": {"title_content": "Sum of uncorrelated variables", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances:\n

                                                          ", "content": [{"c": "One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} (X_{i}).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} (X_{i}).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          This statement is called the Bienaym\u00e9 formula[5] and was discovered in 1853.[6][7] It is often made with the stronger condition that the variables are independent, but being uncorrelated suffices. So if all the variables have the same variance \u03c32, then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is\n

                                                          ", "content": [{"c": "This statement is called the Bienaym\u00e9 formula[ 5] and was discovered in 1853.[ 6][ 7] It is often made with the stronger condition that the variables are independent, but being uncorrelated suffices. So if all the variables have the same variance \u03c3 2, then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}\\sum _{i=1}^{n}X_{i}\\right)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)={\\frac {1}{n^{2}}}n\\sigma ^{2}={\\frac {\\sigma ^{2}}{n}}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}\\sum _{i=1}^{n}X_{i}\\right)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)={\\frac {1}{n^{2}}}n\\sigma ^{2}={\\frac {\\sigma ^{2}}{n}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          That is, the variance of the mean decreases when n increases. This formula for the variance of the mean is used in the definition of the standard error of the sample mean, which is used in the central limit theorem.\n

                                                          ", "content": [{"c": "That is, the variance of the mean decreases when n increases. This formula for the variance of the mean is used in the definition of the standard error of the sample mean, which is used in the central limit theorem.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          To prove the initial statement, it suffices to show that\n

                                                          ", "content": [{"c": "To prove the initial statement, it suffices to show that", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (X+Y)=\\operatorname {Var} (X)+\\operatorname {Var} (Y).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X+Y)=\\operatorname {Var} (X)+\\operatorname {Var} (Y).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          The general result then follows by induction. Starting with the definition,\n

                                                          ", "content": [{"c": "The general result then follows by induction. Starting with the definition,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[(X+Y)^{2}\\right]-(\\operatorname {E} [X+Y])^{2}\\\\[5pt]&=\\operatorname {E} \\left[X^{2}+2XY+Y^{2}\\right]-(\\operatorname {E} [X]+\\operatorname {E} [Y])^{2}.\\end{aligned}}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[(X+Y)^{2}\\right]-(\\operatorname {E} [X+Y])^{2}\\\\[5pt]&=\\operatorname {E} \\left[X^{2}+2XY+Y^{2}\\right]-(\\operatorname {E} [X]+\\operatorname {E} [Y])^{2}.\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y, this further simplifies as follows:\n

                                                          ", "content": [{"c": "Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y, this further simplifies as follows:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[X^{2}\\right]+2\\operatorname {E} [XY]+\\operatorname {E} \\left[Y^{2}\\right]-\\left(\\operatorname {E} [X]^{2}+2\\operatorname {E} [X]\\operatorname {E} [Y]+\\operatorname {E} [Y]^{2}\\right)\\\\[5pt]&=\\operatorname {E} \\left[X^{2}\\right]+\\operatorname {E} \\left[Y^{2}\\right]-\\operatorname {E} [X]^{2}-\\operatorname {E} [Y]^{2}\\\\[5pt]&=\\operatorname {Var} (X)+\\operatorname {Var} (Y).\\end{aligned}}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[X^{2}\\right]+2\\operatorname {E} [XY]+\\operatorname {E} \\left[Y^{2}\\right]-\\left(\\operatorname {E} [X]^{2}+2\\operatorname {E} [X]\\operatorname {E} [Y]+\\operatorname {E} [Y]^{2}\\right)\\\\[5pt]&=\\operatorname {E} \\left[X^{2}\\right]+\\operatorname {E} \\left[Y^{2}\\right]-\\operatorname {E} [X]^{2}-\\operatorname {E} [Y]^{2}\\\\[5pt]&=\\operatorname {Var} (X)+\\operatorname {Var} (Y).\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Sum of correlated variables

                                                          ", "content": {"title_content": "Sum of correlated variables", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          Sum of correlated variables with fixed sample size
                                                          ", "content": {"title_content": "Sum of correlated variables with fixed sample size", "level": "5"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          In general, the variance of the sum of n variables is the sum of their covariances:\n

                                                          ", "content": [{"c": "In general, the variance of the sum of n variables is the sum of their covariances:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\sum _{j=1}^{n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)+2\\sum _{1\\leq i<j\\leq n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\sum _{j=1}^{n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)+2\\sum _{1\\leq i

                                                          (Note: The second equality comes from the fact that Cov(Xi,Xi) = Var(Xi).)\n

                                                          ", "content": [{"c": "(Note: The second equality comes from the fact that Cov( X i, X i) = Var( X i).)", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Here, {\\displaystyle \\operatorname {Cov} (\\cdot ,\\cdot )} is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. The next expression states equivalently that the variance of the sum is the sum of the diagonal of covariance matrix plus two times the sum of its upper triangular elements (or its lower triangular elements); this emphasizes that the covariance matrix is symmetric. This formula is used in the theory of Cronbach's alpha in classical test theory.\n

                                                          ", "content": [{"c": "Here,", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Cov} (\\cdot ,\\cdot )}", "t": "equation-inline"}, {"c": "is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. The next expression states equivalently that the variance of the sum is the sum of the diagonal of covariance matrix plus two times the sum of its upper triangular elements (or its lower triangular elements); this emphasizes that the covariance matrix is symmetric. This formula is used in the theory of Cronbach's alpha in classical test theory.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          So, if the variables have equal variance \u03c32 and the average correlation of distinct variables is \u03c1, then the variance of their mean is\n

                                                          ", "content": [{"c": "So, if the variables have equal variance \u03c3 2 and the average correlation of distinct variables is \u03c1, then the variance of their mean is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {\\sigma ^{2}}{n}}+{\\frac {n-1}{n}}\\rho \\sigma ^{2}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {\\sigma ^{2}}{n}}+{\\frac {n-1}{n}}\\rho \\sigma ^{2}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          This implies that the variance of the mean increases with the average of the correlations. In other words, additional correlated observations are not as effective as additional independent observations at reducing the uncertainty of the mean. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to\n

                                                          ", "content": [{"c": "This implies that the variance of the mean increases with the average of the correlations. In other words, additional correlated observations are not as effective as additional independent observations at reducing the uncertainty of the mean. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {1}{n}}+{\\frac {n-1}{n}}\\rho .}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {1}{n}}+{\\frac {n-1}{n}}\\rho .}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          This formula is used in the Spearman\u2013Brown prediction formula of classical test theory. This converges to \u03c1 if n goes to infinity, provided that the average correlation remains constant or converges too. So for the variance of the mean of standardized variables with equal correlations or converging average correlation we have\n

                                                          ", "content": [{"c": "This formula is used in the Spearman\u2013Brown prediction formula of classical test theory. This converges to \u03c1 if n goes to infinity, provided that the average correlation remains constant or converges too. So for the variance of the mean of standardized variables with equal correlations or converging average correlation we have", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\lim _{n\\to \\infty }\\operatorname {Var} \\left({\\overline {X}}\\right)=\\rho .}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\lim _{n\\to \\infty }\\operatorname {Var} \\left({\\overline {X}}\\right)=\\rho .}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. This makes clear that the sample mean of correlated variables does not generally converge to the population mean, even though the law of large numbers states that the sample mean will converge for independent variables.\n

                                                          ", "content": [{"c": "Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. This makes clear that the sample mean of correlated variables does not generally converge to the population mean, even though the law of large numbers states that the sample mean will converge for independent variables.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          Sum of uncorrelated variables with random sample size
                                                          ", "content": {"title_content": "Sum of uncorrelated variables with random sample size", "level": "5"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size N is a random variable whose variation adds to the variation of X, such that,\n

                                                          ", "content": [{"c": "There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size N is a random variable whose variation adds to the variation of X, such that,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\operatorname {E} \\left[N\\right]\\operatorname {Var} (X)+\\operatorname {Var} (N)(\\operatorname {E} \\left[X\\right])^{2}}[8]
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\operatorname {E} \\left[N\\right]\\operatorname {Var} (X)+\\operatorname {Var} (N)(\\operatorname {E} \\left[X\\right])^{2}}", "t": "equation-inline"}, {"c": "[", "t": "text"}, {"c": "8", "t": "text"}, {"c": "]", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          which follows from the law of total variance.\n

                                                          ", "content": [{"c": "which follows from the law of total variance.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          If N has a Poisson distribution, then {\\displaystyle \\operatorname {E} [N]=\\operatorname {Var} (N)} with estimator n = N. So, the estimator of {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)} becomes {\\displaystyle n{S_{x}}^{2}+n{\\bar {X}}^{2}}, giving {\\displaystyle \\operatorname {SE} ({\\bar {X}})={\\sqrt {\\frac {{S_{x}}^{2}+{\\bar {X}}^{2}}{n}}}}\n(see standard error of the sample mean).\n

                                                          ", "content": [{"c": "If N has a Poisson distribution, then", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} [N]=\\operatorname {Var} (N)}", "t": "equation-inline"}, {"c": "with estimator n= N. So, the estimator of", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)}", "t": "equation-inline"}, {"c": "becomes", "t": "text"}, {"c": "{\\displaystyle n{S_{x}}^{2}+n{\\bar {X}}^{2}}", "t": "equation-inline"}, {"c": ", giving", "t": "text"}, {"c": "{\\displaystyle \\operatorname {SE} ({\\bar {X}})={\\sqrt {\\frac {{S_{x}}^{2}+{\\bar {X}}^{2}}{n}}}}", "t": "equation-inline"}, {"c": "(see standard error of the sample mean).", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Weighted sum of variables

                                                          ", "content": {"title_content": "Weighted sum of variables", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The scaling property and the Bienaym\u00e9 formula, along with the property of the covarianceCov(aX, bY) = ab Cov(X, Y) jointly imply that\n

                                                          ", "content": [{"c": "The scaling property and the Bienaym\u00e9 formula, along with the property of the covariance Cov( aX, bY) = ab Cov( X, Y) jointly imply that", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (aX\\pm bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)\\pm 2ab\\,\\operatorname {Cov} (X,Y).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (aX\\pm bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)\\pm 2ab\\,\\operatorname {Cov} (X,Y).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y, then the weight of the variance of X will be four times the weight of the variance of Y.\n

                                                          ", "content": [{"c": "This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y, then the weight of the variance of X will be four times the weight of the variance of Y.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The expression above can be extended to a weighted sum of multiple variables:\n

                                                          ", "content": [{"c": "The expression above can be extended to a weighted sum of multiple variables:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left(\\sum _{i}^{n}a_{i}X_{i}\\right)=\\sum _{i=1}^{n}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i}\\sum _{<j\\leq n}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i}^{n}a_{i}X_{i}\\right)=\\sum _{i=1}^{n}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i}\\sum _{Product of variables", "content": {"title_content": "Product of variables", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Product of independent variables

                                                          ", "content": {"title_content": "Product of independent variables", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          If two variables X and Y are independent, the variance of their product is given by[9]

                                                          ", "content": [{"c": "If two variables X and Y are independent, the variance of their product is given by[ 9]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (XY)=[\\operatorname {E} (X)]^{2}\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\operatorname {Var} (X)+\\operatorname {Var} (X)\\operatorname {Var} (Y).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (XY)=[\\operatorname {E} (X)]^{2}\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\operatorname {Var} (X)+\\operatorname {Var} (X)\\operatorname {Var} (Y).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Equivalently, using the basic properties of expectation, it is given by\n

                                                          ", "content": [{"c": "Equivalently, using the basic properties of expectation, it is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} (XY)=\\operatorname {E} \\left(X^{2}\\right)\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (X)]^{2}[\\operatorname {E} (Y)]^{2}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (XY)=\\operatorname {E} \\left(X^{2}\\right)\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (X)]^{2}[\\operatorname {E} (Y)]^{2}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Product of statistically dependent variables

                                                          ", "content": {"title_content": "Product of statistically dependent variables", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          In general, if two variables are statistically dependent, then the variance of their product is given by:\n

                                                          ", "content": [{"c": "In general, if two variables are statistically dependent, then the variance of their product is given by:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\begin{aligned}\\operatorname {Var} (XY)={}&\\operatorname {E} \\left[X^{2}Y^{2}\\right]-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\operatorname {E} (X^{2})\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\left(\\operatorname {Var} (X)+[\\operatorname {E} (X)]^{2}\\right)\\left(\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\right)\\\\[5pt]&-[\\operatorname {Cov} (X,Y)+\\operatorname {E} (X)\\operatorname {E} (Y)]^{2}\\end{aligned}}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (XY)={}&\\operatorname {E} \\left[X^{2}Y^{2}\\right]-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\operatorname {E} (X^{2})\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\left(\\operatorname {Var} (X)+[\\operatorname {E} (X)]^{2}\\right)\\left(\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\right)\\\\[5pt]&-[\\operatorname {Cov} (X,Y)+\\operatorname {E} (X)\\operatorname {E} (Y)]^{2}\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Arbitrary functions

                                                          ", "content": {"title_content": "Arbitrary functions", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables: see Taylor expansions for the moments of functions of random variables. For example, the approximate variance of a function of one variable is given by\n

                                                          ", "content": [{"c": "The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables: see Taylor expansions for the moments of functions of random variables. For example, the approximate variance of a function of one variable is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left[f(X)\\right]\\approx \\left(f'(\\operatorname {E} \\left[X\\right])\\right)^{2}\\operatorname {Var} \\left[X\\right]}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left[f(X)\\right]\\approx \\left(f'(\\operatorname {E} \\left[X\\right])\\right)^{2}\\operatorname {Var} \\left[X\\right]}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          provided that f is twice differentiable and that the mean and variance of X are finite.\n

                                                          ", "content": [{"c": "provided that f is twice differentiable and that the mean and variance of X are finite.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Population variance and sample variance

                                                          ", "content": {"title_content": "Population variance and sample variance", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one estimates the mean and variance from a limited set of observations by using an estimator equation. The estimator is a function of the sample of nobservations drawn without observational bias from the whole population of potential observations. In this example, the sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.\n

                                                          ", "content": [{"c": "Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one estimates the mean and variance from a limited set of observations by using an estimator equation. The estimator is a function of the sample of n observations drawn without observational bias from the whole population of potential observations. In this example, the sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance \u2013 these are consistent estimators (they converge to the value of the whole population as the number of samples increases) but can be improved. Most simply, the sample variance is computed as the sum of squared deviations about the (sample) mean, divided by n as the number of samples. However, using values other than n improves the estimator in various ways. Four common values for the denominator are n,n \u2212 1, n + 1, and n \u2212 1.5: n is the simplest (the variance of the sample), n \u2212 1 eliminates bias,[10]n + 1 minimizes mean squared error for the normal distribution,[11] and n \u2212 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.[12]

                                                          ", "content": [{"c": "The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and(uncorrected) sample variance\u2013 these are consistent estimators(they converge to the value of the whole population as the number of samples increases) but can be improved. Most simply, the sample variance is computed as the sum of squared deviations about the (sample) mean, divided by n as the number of samples. However, using values other than n improves the estimator in various ways. Four common values for the denominator are n, n \u2212 1, n+ 1, and n \u2212 1.5: n is the simplest (the variance of the sample), n \u2212 1 eliminates bias,[ 10] n+ 1 minimizes mean squared error for the normal distribution,[ 11] and n \u2212 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.[ 12]", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (n \u2212 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n -1 instead of n, is called Bessel's correction.[10] The resulting estimator is unbiased and is called the (corrected) sample variance or unbiased sample variance. If the mean is determined in some other way than from the same samples used to estimate the variance, then this bias does not arise, and the variance can safely be estimated as that of the samples about the (independently known) mean.\n

                                                          ", "content": [{"c": "Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of ( n \u2212 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n-1 instead of n, is called Bessel's correction.[ 10] The resulting estimator is unbiased and is called the(corrected) sample variance or unbiased sample variance. If the mean is determined in some other way than from the same samples used to estimate the variance, then this bias does not arise, and the variance can safely be estimated as that of the samples about the (independently known) mean.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Secondly, the sample variance does not generally minimize mean squared error between sample variance and population variance. Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance) and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than n \u2212 1) and is a simple example of a shrinkage estimator: one \"shrinks\" the unbiased estimator towards zero. For the normal distribution, dividing by n + 1 (instead of n \u2212 1 or n) minimizes mean squared error.[11] The resulting estimator is biased, however, and is known as the biased sample variation.\n

                                                          ", "content": [{"c": "Secondly, the sample variance does not generally minimize mean squared error between sample variance and population variance. Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance) and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than n \u2212 1) and is a simple example of a shrinkage estimator: one \"shrinks\" the unbiased estimator towards zero. For the normal distribution, dividing by n+ 1 (instead of n \u2212 1 or n) minimizes mean squared error.[ 11] The resulting estimator is biased, however, and is known as the biased sample variation.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Population variance

                                                          ", "content": {"title_content": "Population variance", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          In general, the population variance of a finitepopulation of size N with values xi is given by

                                                          ", "content": [{"c": "In general, the population variance of a finite population of size N with values x i is given by", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\u03c32=1N\u2211i=1N(xi\u2212\u03bc)2=1N\u2211i=1N(xi2\u22122\u03bcxi+\u03bc2)=(1N\u2211i=1Nxi2)\u22122\u03bc(1N\u2211i=1Nxi)+\u03bc2=E\u2061[xi2]\u2212\u03bc2{\\displaystyle {\\begin{aligned}\\sigma ^{2}&={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}-\\mu \\right)^{2}={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}^{2}-2\\mu x_{i}+\\mu ^{2}\\right)\\\\[5pt]&=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-2\\mu \\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)+\\mu ^{2}\\\\[5pt]&=\\operatorname {E} [x_{i}^{2}]-\\mu ^{2}\\end{aligned}}}", "content": {"math_content": "{\\displaystyle {\\begin{aligned}\\sigma ^{2}&={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}-\\mu \\right)^{2}={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}^{2}-2\\mu x_{i}+\\mu ^{2}\\right)\\\\[5pt]&=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-2\\mu \\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)+\\mu ^{2}\\\\[5pt]&=\\operatorname {E} [x_{i}^{2}]-\\mu ^{2}\\end{aligned}}}", "math_type": "mathml", "by": "None"}}, {"type": "paragraph", "raw_content": "

                                                          where the population mean is {\\textstyle \\mu =\\operatorname {E} [x_{i}]={\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}} and {\\textstyle \\operatorname {E} [x_{i}^{2}]=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)}, where {\\textstyle \\operatorname {E} } is the expectation value operator.\n

                                                          ", "content": [{"c": "where the population mean is", "t": "text"}, {"c": "{\\textstyle \\mu =\\operatorname {E} [x_{i}]={\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\textstyle \\operatorname {E} [x_{i}^{2}]=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)}", "t": "equation-inline"}, {"c": ", where", "t": "text"}, {"c": "{\\textstyle \\operatorname {E} }", "t": "equation-inline"}, {"c": "is the expectation value operator.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The population variance can also be computed using[13]

                                                          ", "content": [{"c": "The population variance can also be computed using[ 13]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\sigma ^{2}={\\frac {1}{N^{2}}}\\sum _{i<j}\\left(x_{i}-x_{j}\\right)^{2}={\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\sigma ^{2}={\\frac {1}{N^{2}}}\\sum _{i

                                                          (The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because

                                                          ", "content": [{"c": "(The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because", "t": "text"}]}, {"type": "equation-interline", "raw_content": "12N2\u2211i,j=1N(xi\u2212xj)2=12N2\u2211i,j=1N(xi2\u22122xixj+xj2)=12N\u2211j=1N(1N\u2211i=1Nxi2)\u2212(1N\u2211i=1Nxi)(1N\u2211j=1Nxj)+12N\u2211i=1N(1N\u2211j=1Nxj2)=12(\u03c32+\u03bc2)\u2212\u03bc2+12(\u03c32+\u03bc2)=\u03c32.{\\displaystyle {\\begin{aligned}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}\\\\[5pt]={}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2N}}\\sum _{j=1}^{N}\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}\\right)+{\\frac {1}{2N}}\\sum _{i=1}^{N}\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-\\mu ^{2}+{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\\\[5pt]={}&\\sigma ^{2}.\\end{aligned}}}", "content": {"math_content": "{\\displaystyle {\\begin{aligned}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}\\\\[5pt]={}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2N}}\\sum _{j=1}^{N}\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}\\right)+{\\frac {1}{2N}}\\sum _{i=1}^{N}\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-\\mu ^{2}+{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\\\[5pt]={}&\\sigma ^{2}.\\end{aligned}}}", "math_type": "mathml", "by": "None"}}, {"type": "paragraph", "raw_content": "

                                                          The population variance matches the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.\n

                                                          ", "content": [{"c": "The population variance matches the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Sample variance

                                                          ", "content": {"title_content": "Sample variance", "level": "3"}}, {"type": "paragraph", "raw_content": "
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                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Biased sample variance

                                                          ", "content": {"title_content": "Biased sample variance", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population.[14] This is generally referred to as sample variance or empirical variance. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution.\n

                                                          ", "content": [{"c": "In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population.[ 14] This is generally referred to as sample variance or empirical variance. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          We take a sample with replacement of n values Y1, ..., Yn from the population of size {\\textstyle N}, where n < N, and estimate the variance on the basis of this sample.[15] Directly taking the variance of the sample data gives the average of the squared deviations:\n

                                                          ", "content": [{"c": "We take a sample with replacement of n values Y 1, ..., Y n from the population of size", "t": "text"}, {"c": "{\\textstyle N}", "t": "equation-inline"}, {"c": ", where n< N, and estimate the variance on the basis of this sample.[ 15] Directly taking the variance of the sample data gives the average of the squared deviations:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\tilde {S}}_{Y}^{2}={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}=\\left({\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}^{2}\\right)-{\\overline {Y}}^{2}={\\frac {1}{n^{2}}}\\sum _{i,j\\,:\\,i<j}\\left(Y_{i}-Y_{j}\\right)^{2}.}[16]
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\tilde {S}}_{Y}^{2}={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}=\\left({\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}^{2}\\right)-{\\overline {Y}}^{2}={\\frac {1}{n^{2}}}\\sum _{i,j\\,:\\,i

                                                          (See the section Population variance for the derivation of this formula.) Here, {\\displaystyle {\\overline {Y}}} denotes the sample mean:\n

                                                          ", "content": [{"c": "(See the section Population variance for the derivation of this formula.) Here,", "t": "text"}, {"c": "{\\displaystyle {\\overline {Y}}}", "t": "equation-inline"}, {"c": "denotes the sample mean:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\overline {Y}}={\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\overline {Y}}={\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Since the Yi are selected randomly, both {\\displaystyle {\\overline {Y}}} and {\\displaystyle {\\tilde {S}}_{Y}^{2}} are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples {Yi} of size n from the population. For {\\displaystyle {\\tilde {S}}_{Y}^{2}} this gives:\n

                                                          ", "content": [{"c": "Since the Y i are selected randomly, both", "t": "text"}, {"c": "{\\displaystyle {\\overline {Y}}}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle {\\tilde {S}}_{Y}^{2}}", "t": "equation-inline"}, {"c": "are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples { Y i} of size n from the population. For", "t": "text"}, {"c": "{\\displaystyle {\\tilde {S}}_{Y}^{2}}", "t": "equation-inline"}, {"c": "this gives:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle {\\begin{aligned}\\operatorname {E} [{\\tilde {S}}_{Y}^{2}]&=\\operatorname {E} \\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\frac {1}{n}}\\sum _{j=1}^{n}Y_{j}\\right)^{2}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\operatorname {E} \\left[Y_{i}^{2}-{\\frac {2}{n}}Y_{i}\\sum _{j=1}^{n}Y_{j}+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}Y_{j}\\sum _{k=1}^{n}Y_{k}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\left(\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+\\operatorname {E} \\left[Y_{i}^{2}\\right]\\right)+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left({\\frac {n-2}{n}}\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left[{\\frac {n-2}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-{\\frac {2}{n}}(n-1)\\mu ^{2}+{\\frac {1}{n^{2}}}n(n-1)\\mu ^{2}+{\\frac {1}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\right]\\\\[5pt]&={\\frac {n-1}{n}}\\sigma ^{2}.\\end{aligned}}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {E} [{\\tilde {S}}_{Y}^{2}]&=\\operatorname {E} \\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\frac {1}{n}}\\sum _{j=1}^{n}Y_{j}\\right)^{2}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\operatorname {E} \\left[Y_{i}^{2}-{\\frac {2}{n}}Y_{i}\\sum _{j=1}^{n}Y_{j}+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}Y_{j}\\sum _{k=1}^{n}Y_{k}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\left(\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+\\operatorname {E} \\left[Y_{i}^{2}\\right]\\right)+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left({\\frac {n-2}{n}}\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left[{\\frac {n-2}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-{\\frac {2}{n}}(n-1)\\mu ^{2}+{\\frac {1}{n^{2}}}n(n-1)\\mu ^{2}+{\\frac {1}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\right]\\\\[5pt]&={\\frac {n-1}{n}}\\sigma ^{2}.\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Here {\\textstyle \\sigma ^{2}=\\operatorname {E} [Y_{i}^{2}]-\\mu ^{2}} derived in the section Population variance and {\\textstyle \\operatorname {E} [Y_{i}Y_{j}]=\\operatorname {E} [Y_{i}]\\operatorname {E} [Y_{j}]=\\mu ^{2}} due to independency of {\\textstyle Y_{i}} and {\\textstyle Y_{j}} are used.\n

                                                          ", "content": [{"c": "Here", "t": "text"}, {"c": "{\\textstyle \\sigma ^{2}=\\operatorname {E} [Y_{i}^{2}]-\\mu ^{2}}", "t": "equation-inline"}, {"c": "derived in the section Population variance and", "t": "text"}, {"c": "{\\textstyle \\operatorname {E} [Y_{i}Y_{j}]=\\operatorname {E} [Y_{i}]\\operatorname {E} [Y_{j}]=\\mu ^{2}}", "t": "equation-inline"}, {"c": "due to independency of", "t": "text"}, {"c": "{\\textstyle Y_{i}}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\textstyle Y_{j}}", "t": "equation-inline"}, {"c": "are used.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Hence {\\textstyle {\\tilde {S}}_{Y}^{2}} gives an estimate of the population variance that is biased by a factor of {\\textstyle {\\frac {n-1}{n}}} as the expectation value of {\\textstyle {\\tilde {S}}_{Y}^{2}} is smaller than the population variance (true variance) by that factor. For this reason, {\\textstyle {\\tilde {S}}_{Y}^{2}} is referred to as the biased sample variance.\n

                                                          ", "content": [{"c": "Hence", "t": "text"}, {"c": "{\\textstyle {\\tilde {S}}_{Y}^{2}}", "t": "equation-inline"}, {"c": "gives an estimate of the population variance that is biased by a factor of", "t": "text"}, {"c": "{\\textstyle {\\frac {n-1}{n}}}", "t": "equation-inline"}, {"c": "as the expectation value of", "t": "text"}, {"c": "{\\textstyle {\\tilde {S}}_{Y}^{2}}", "t": "equation-inline"}, {"c": "is smaller than the population variance (true variance) by that factor. For this reason,", "t": "text"}, {"c": "{\\textstyle {\\tilde {S}}_{Y}^{2}}", "t": "equation-inline"}, {"c": "is referred to as the biased sample variance.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Unbiased sample variance

                                                          ", "content": {"title_content": "Unbiased sample variance", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Correcting for this bias yields the unbiased sample variance, denoted {\\displaystyle S^{2}}:\n

                                                          ", "content": [{"c": "Correcting for this bias yields the unbiased sample variance, denoted", "t": "text"}, {"c": "{\\displaystyle S^{2}}", "t": "equation-inline"}, {"c": ":", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle S^{2}={\\frac {n}{n-1}}{\\tilde {S}}_{Y}^{2}={\\frac {n}{n-1}}\\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}\\right]={\\frac {1}{n-1}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle S^{2}={\\frac {n}{n-1}}{\\tilde {S}}_{Y}^{2}={\\frac {n}{n-1}}\\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}\\right]={\\frac {1}{n-1}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          Either estimator may be simply referred to as the sample variance when the version can be determined by context. The same proof is also applicable for samples taken from a continuous probability distribution.\n

                                                          ", "content": [{"c": "Either estimator may be simply referred to as the sample variance when the version can be determined by context. The same proof is also applicable for samples taken from a continuous probability distribution.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The use of the term n \u2212 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation (the square root of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n \u2212 1.5 yields an almost unbiased estimator.\n

                                                          ", "content": [{"c": "The use of the term n \u2212 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation(the square root of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n \u2212 1.5 yields an almost unbiased estimator.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The unbiased sample variance is a U-statistic for the function \u0192(y1, y2) = (y1 \u2212 y2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.\n

                                                          ", "content": [{"c": "The unbiased sample variance is a U-statistic for the function \u0192( y 1, y 2) = ( y 1 \u2212 y 2) 2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                          Example
                                                          ", "content": {"title_content": "Example", "level": "5"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          For a set of numbers {10, 15, 30, 45, 57, 52 63, 72, 81, 93, 102, 105}, if this set is the whole data population for some measurement, then variance is the population variance 932.743 as the sum of the squared deviations about the mean of this set, divided by 12 as the number of the set members. If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.\n

                                                          ", "content": [{"c": "For a set of numbers {10, 15, 30, 45, 57, 52 63, 72, 81, 93, 102, 105}, if this set is the whole data population for some measurement, then variance is the population variance 932.743 as the sum of the squared deviations about the mean of this set, divided by 12 as the number of the set members. If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Distribution of the sample variance

                                                          ", "content": {"title_content": "Distribution of the sample variance", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Yi are independent observations from a normal distribution, Cochran's theorem shows that the unbiased sample varianceS2 follows a scaled chi-squared distribution (see also: asymptotic properties and an elementary proof):[17]

                                                          ", "content": [{"c": "Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Y i are independent observations from a normal distribution, Cochran's theorem shows that the unbiased sample variance S 2 follows a scaled chi-squared distribution(see also: asymptotic properties and an elementary proof):[ 17]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle (n-1){\\frac {S^{2}}{\\sigma ^{2}}}\\sim \\chi _{n-1}^{2}}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle (n-1){\\frac {S^{2}}{\\sigma ^{2}}}\\sim \\chi _{n-1}^{2}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          where \u03c32 is the population variance. As a direct consequence, it follows that\n

                                                          ", "content": [{"c": "where \u03c3 2 is the population variance. As a direct consequence, it follows that", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {E} \\left(S^{2}\\right)=\\operatorname {E} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)=\\sigma ^{2},}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {E} \\left(S^{2}\\right)=\\operatorname {E} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)=\\sigma ^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          and[18]

                                                          ", "content": [{"c": "and[ 18]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {Var} \\left[S^{2}\\right]=\\operatorname {Var} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)={\\frac {\\sigma ^{4}}{(n-1)^{2}}}\\operatorname {Var} \\left(\\chi _{n-1}^{2}\\right)={\\frac {2\\sigma ^{4}}{n-1}}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left[S^{2}\\right]=\\operatorname {Var} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)={\\frac {\\sigma ^{4}}{(n-1)^{2}}}\\operatorname {Var} \\left(\\chi _{n-1}^{2}\\right)={\\frac {2\\sigma ^{4}}{n-1}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          If Yi are independent and identically distributed, but not necessarily normally distributed, then[19]

                                                          ", "content": [{"c": "If Y i are independent and identically distributed, but not necessarily normally distributed, then[ 19]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\operatorname {E} \\left[S^{2}\\right]=\\sigma ^{2},\\quad \\operatorname {Var} \\left[S^{2}\\right]={\\frac {\\sigma ^{4}}{n}}\\left(\\kappa -1+{\\frac {2}{n-1}}\\right)={\\frac {1}{n}}\\left(\\mu _{4}-{\\frac {n-3}{n-1}}\\sigma ^{4}\\right),}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {E} \\left[S^{2}\\right]=\\sigma ^{2},\\quad \\operatorname {Var} \\left[S^{2}\\right]={\\frac {\\sigma ^{4}}{n}}\\left(\\kappa -1+{\\frac {2}{n-1}}\\right)={\\frac {1}{n}}\\left(\\mu _{4}-{\\frac {n-3}{n-1}}\\sigma ^{4}\\right),}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          where \u03ba is the kurtosis of the distribution and \u03bc4 is the fourth central moment.\n

                                                          ", "content": [{"c": "where \u03ba is the kurtosis of the distribution and \u03bc 4 is the fourth central moment.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          If the conditions of the law of large numbers hold for the squared observations, S2 is a consistent estimator of \u03c32. One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.).[20][21][22]

                                                          ", "content": [{"c": "If the conditions of the law of large numbers hold for the squared observations, S 2 is a consistent estimator of \u03c3 2. One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.).[ 20][ 21][ 22]", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Samuelson's inequality

                                                          ", "content": {"title_content": "Samuelson's inequality", "level": "4"}}, {"type": "paragraph", "raw_content": "
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                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Samuelson's inequality is a result that states bounds on the values that individual observations in a sample can take, given that the sample mean and (biased) variance have been calculated.[23] Values must lie within the limits {\\displaystyle {\\bar {y}}\\pm \\sigma _{Y}(n-1)^{1/2}.}

                                                          ", "content": [{"c": "Samuelson's inequality is a result that states bounds on the values that individual observations in a sample can take, given that the sample mean and (biased) variance have been calculated.[ 23] Values must lie within the limits", "t": "text"}, {"c": "{\\displaystyle {\\bar {y}}\\pm \\sigma _{Y}(n-1)^{1/2}.}", "t": "equation-inline"}]}, {"type": "title", "raw_content": "

                                                          Relations with the harmonic and arithmetic means

                                                          ", "content": {"title_content": "Relations with the harmonic and arithmetic means", "level": "3"}}, {"type": "paragraph", "raw_content": "
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                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          It has been shown[24] that for a sample {yi} of positive real numbers,\n

                                                          ", "content": [{"c": "It has been shown[ 24] that for a sample { y i} of positive real numbers,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\sigma _{y}^{2}\\leq 2y_{\\max }(A-H),}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\sigma _{y}^{2}\\leq 2y_{\\max }(A-H),}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          where ymax is the maximum of the sample, A is the arithmetic mean, H is the harmonic mean of the sample and {\\displaystyle \\sigma _{y}^{2}} is the (biased) variance of the sample.\n

                                                          ", "content": [{"c": "where y max is the maximum of the sample, A is the arithmetic mean, H is the harmonic mean of the sample and", "t": "text"}, {"c": "{\\displaystyle \\sigma _{y}^{2}}", "t": "equation-inline"}, {"c": "is the (biased) variance of the sample.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          This bound has been improved, and it is known that variance is bounded by\n

                                                          ", "content": [{"c": "This bound has been improved, and it is known that variance is bounded by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\sigma _{y}^{2}\\leq {\\frac {y_{\\max }(A-H)(y_{\\max }-A)}{y_{\\max }-H}},}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\sigma _{y}^{2}\\leq {\\frac {y_{\\max }(A-H)(y_{\\max }-A)}{y_{\\max }-H}},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\sigma _{y}^{2}\\geq {\\frac {y_{\\min }(A-H)(A-y_{\\min })}{H-y_{\\min }}},}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\sigma _{y}^{2}\\geq {\\frac {y_{\\min }(A-H)(A-y_{\\min })}{H-y_{\\min }}},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          where ymin is the minimum of the sample.[25]

                                                          ", "content": [{"c": "where y min is the minimum of the sample.[ 25]", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Tests of equality of variances

                                                          ", "content": {"title_content": "Tests of equality of variances", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult.\n

                                                          ", "content": [{"c": "The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Several non parametric tests have been proposed: these include the Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey test, the Capon test, Mood test, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that both medians be known and equal to zero. The Mood, Klotz, Capon and Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey tests also apply to two variances. They allow the median to be unknown but do require that the two medians are equal.\n

                                                          ", "content": [{"c": "Several non parametric tests have been proposed: these include the Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey test, the Capon test, Mood test, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that both medians be known and equal to zero. The Mood, Klotz, Capon and Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey tests also apply to two variances. They allow the median to be unknown but do require that the two medians are equal.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The Lehmann test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variances include the Box test, the Box\u2013Anderson test and the Moses test.\n

                                                          ", "content": [{"c": "The Lehmann test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variances include the Box test, the Box\u2013Anderson test and the Moses test.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Resampling methods, which include the bootstrap and the jackknife, may be used to test the equality of variances.\n

                                                          ", "content": [{"c": "Resampling methods, which include the bootstrap and the jackknife, may be used to test the equality of variances.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Moment of inertia

                                                          ", "content": {"title_content": "Moment of inertia", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass.[26] It is because of this analogy that such things as the variance are called moments of probability distributions.[26] The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of n points with a covariance matrix of {\\displaystyle \\Sigma } is given by\n

                                                          ", "content": [{"c": "The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass.[ 26] It is because of this analogy that such things as the variance are called moments of probability distributions.[ 26] The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of n points with a covariance matrix of", "t": "text"}, {"c": "{\\displaystyle \\Sigma }", "t": "equation-inline"}, {"c": "is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle I=n\\left(\\mathbf {1} _{3\\times 3}\\operatorname {tr} (\\Sigma )-\\Sigma \\right).}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle I=n\\left(\\mathbf {1} _{3\\times 3}\\operatorname {tr} (\\Sigma )-\\Sigma \\right).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the x axis and distributed along it. The covariance matrix might look like\n

                                                          ", "content": [{"c": "This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the x axis and distributed along it. The covariance matrix might look like", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle \\Sigma ={\\begin{bmatrix}10&0&0\\\\0&0.1&0\\\\0&0&0.1\\end{bmatrix}}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle \\Sigma ={\\begin{bmatrix}10&0&0\\\\0&0.1&0\\\\0&0&0.1\\end{bmatrix}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                          That is, there is the most variance in the x direction. Physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is\n

                                                          ", "content": [{"c": "That is, there is the most variance in the x direction. Physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                          {\\displaystyle I=n{\\begin{bmatrix}0.2&0&0\\\\0&10.1&0\\\\0&0&10.1\\end{bmatrix}}.}
                                                          ", "content": {"items": [[[{"c": "{\\displaystyle I=n{\\begin{bmatrix}0.2&0&0\\\\0&10.1&0\\\\0&0&10.1\\end{bmatrix}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                          Semivariance

                                                          ", "content": {"title_content": "Semivariance", "level": "2"}}, {"type": "paragraph", "raw_content": "
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                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation:

                                                          ", "content": [{"c": "The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "Semivariance=1n\u2211i:xi<\u03bc(xi\u2212\u03bc)2{\\displaystyle {\\text{Semivariance}}={1 \\over {n}}\\sum _{i:x_{i}<\\mu }(x_{i}-\\mu )^{2}}", "content": {"math_content": "{\\displaystyle {\\text{Semivariance}}={1 \\over {n}}\\sum _{i:x_{i}<\\mu }(x_{i}-\\mu )^{2}}", "math_type": "mathml", "by": "None"}}, {"type": "paragraph", "raw_content": "

                                                          It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not.[27]

                                                          ", "content": [{"c": "It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not.[ 27]", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          For inequalities associated with the semivariance, see Chebyshev's inequality \u00a7 Semivariances.\n

                                                          ", "content": [{"c": "For inequalities associated with the semivariance, see Chebyshev's inequality \u00a7 Semivariances.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Etymology

                                                          ", "content": {"title_content": "Etymology", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance:[28]

                                                          ", "content": [{"c": "The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance:[ 28]", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and, therefore, that the variability may be uniformly measured by the standard deviation corresponding to the square root of the mean square error. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations {\\displaystyle \\sigma _{1}} and {\\displaystyle \\sigma _{2}}, it is found that the distribution, when both causes act together, has a standard deviation {\\displaystyle {\\sqrt {\\sigma _{1}^{2}+\\sigma _{2}^{2}}}}. It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance...

                                                          ", "content": [{"c": "The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and, therefore, that the variability may be uniformly measured by the standard deviation corresponding to the square root of the mean square error. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations", "t": "text"}, {"c": "{\\displaystyle \\sigma _{1}}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle \\sigma _{2}}", "t": "equation-inline"}, {"c": ", it is found that the distribution, when both causes act together, has a standard deviation", "t": "text"}, {"c": "{\\displaystyle {\\sqrt {\\sigma _{1}^{2}+\\sigma _{2}^{2}}}}", "t": "equation-inline"}, {"c": ". It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance...", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          Generalizations

                                                          ", "content": {"title_content": "Generalizations", "level": "2"}}, {"type": "paragraph", "raw_content": "
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                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          For complex variables

                                                          ", "content": {"title_content": "For complex variables", "level": "3"}}, {"type": "paragraph", "raw_content": "
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                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          If {\\displaystyle x} is a scalar complex-valued random variable, with values in {\\displaystyle \\mathbb {C} ,} then its variance is {\\displaystyle \\operatorname {E} \\left[(x-\\mu )(x-\\mu )^{*}\\right],} where {\\displaystyle x^{*}} is the complex conjugate of {\\displaystyle x.} This variance is a real scalar.\n

                                                          ", "content": [{"c": "If", "t": "text"}, {"c": "{\\displaystyle x}", "t": "equation-inline"}, {"c": "is a scalar complex-valued random variable, with values in", "t": "text"}, {"c": "{\\displaystyle \\mathbb {C} ,}", "t": "equation-inline"}, {"c": "then its variance is", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} \\left[(x-\\mu )(x-\\mu )^{*}\\right],}", "t": "equation-inline"}, {"c": "where", "t": "text"}, {"c": "{\\displaystyle x^{*}}", "t": "equation-inline"}, {"c": "is the complex conjugate of", "t": "text"}, {"c": "{\\displaystyle x.}", "t": "equation-inline"}, {"c": "This variance is a real scalar.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          For vector-valued random variables

                                                          ", "content": {"title_content": "For vector-valued random variables", "level": "3"}}, {"type": "paragraph", "raw_content": "
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                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          As a matrix

                                                          ", "content": {"title_content": "As a matrix", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          If {\\displaystyle X} is a vector-valued random variable, with values in {\\displaystyle \\mathbb {R} ^{n},} and thought of as a column vector, then a natural generalization of variance is {\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\operatorname {T} }\\right],} where {\\displaystyle \\mu =\\operatorname {E} (X)} and {\\displaystyle X^{\\operatorname {T} }} is the transpose of {\\displaystyle X,} and so is a row vector. The result is a positive semi-definite square matrix, commonly referred to as the variance-covariance matrix (or simply as the covariance matrix).\n

                                                          ", "content": [{"c": "If", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "is a vector-valued random variable, with values in", "t": "text"}, {"c": "{\\displaystyle \\mathbb {R} ^{n},}", "t": "equation-inline"}, {"c": "and thought of as a column vector, then a natural generalization of variance is", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\operatorname {T} }\\right],}", "t": "equation-inline"}, {"c": "where", "t": "text"}, {"c": "{\\displaystyle \\mu =\\operatorname {E} (X)}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle X^{\\operatorname {T} }}", "t": "equation-inline"}, {"c": "is the transpose of", "t": "text"}, {"c": "{\\displaystyle X,}", "t": "equation-inline"}, {"c": "and so is a row vector. The result is a positive semi-definite square matrix, commonly referred to as the variance-covariance matrix(or simply as the covariance matrix).", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          If {\\displaystyle X} is a vector- and complex-valued random variable, with values in {\\displaystyle \\mathbb {C} ^{n},} then the covariance matrix is{\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\dagger }\\right],} where {\\displaystyle X^{\\dagger }} is the conjugate transpose of {\\displaystyle X.} This matrix is also positive semi-definite and square.\n

                                                          ", "content": [{"c": "If", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "is a vector- and complex-valued random variable, with values in", "t": "text"}, {"c": "{\\displaystyle \\mathbb {C} ^{n},}", "t": "equation-inline"}, {"c": "then the covariance matrix is", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\dagger }\\right],}", "t": "equation-inline"}, {"c": "where", "t": "text"}, {"c": "{\\displaystyle X^{\\dagger }}", "t": "equation-inline"}, {"c": "is the conjugate transpose of", "t": "text"}, {"c": "{\\displaystyle X.}", "t": "equation-inline"}, {"c": "This matrix is also positive semi-definite and square.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          As a scalar

                                                          ", "content": {"title_content": "As a scalar", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                          edit
                                                          ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          Another generalization of variance for vector-valued random variables {\\displaystyle X}, which results in a scalar value rather than in a matrix, is the generalized variance{\\displaystyle \\det(C)}, the determinant of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.[29]

                                                          ", "content": [{"c": "Another generalization of variance for vector-valued random variables", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": ", which results in a scalar value rather than in a matrix, is the generalized variance", "t": "text"}, {"c": "{\\displaystyle \\det(C)}", "t": "equation-inline"}, {"c": ", the determinant of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.[ 29]", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                          A different generalization is obtained by considering the equation for the scalar variance, {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right]}, and reinterpreting {\\displaystyle (X-\\mu )^{2}} as the squared Euclidean distance between the random variable and its mean, or, simply as the scalar product of the vector {\\displaystyle X-\\mu } with itself. This results in {\\displaystyle \\operatorname {E} \\left[(X-\\mu )^{\\operatorname {T} }(X-\\mu )\\right]=\\operatorname {tr} (C),} which is the trace of the covariance matrix.\n

                                                          ", "content": [{"c": "A different generalization is obtained by considering the equation for the scalar variance,", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right]}", "t": "equation-inline"}, {"c": ", and reinterpreting", "t": "text"}, {"c": "{\\displaystyle (X-\\mu )^{2}}", "t": "equation-inline"}, {"c": "as the squared Euclidean distance between the random variable and its mean, or, simply as the scalar product of the vector", "t": "text"}, {"c": "{\\displaystyle X-\\mu }", "t": "equation-inline"}, {"c": "with itself. This results in", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} \\left[(X-\\mu )^{\\operatorname {T} }(X-\\mu )\\right]=\\operatorname {tr} (C),}", "t": "equation-inline"}, {"c": "which is the trace of the covariance matrix.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                          See also

                                                          ", "content": {"title_content": "See also", "level": "2"}}, {"type": "paragraph", "raw_content": "
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                                                            Types of variance

                                                            ", "content": {"title_content": "Types of variance", "level": "3"}}, {"type": "paragraph", "raw_content": "
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                                                              References

                                                              ", "content": {"title_content": "References", "level": "2"}}, {"type": "paragraph", "raw_content": "
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                                                                In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by {\\displaystyle \\sigma ^{2}}, {\\displaystyle s^{2}}, {\\displaystyle \\operatorname {Var} (X)}, {\\displaystyle V(X)}, or {\\displaystyle \\mathbb {V} (X)}.[1]

                                                                Example of samples from two populations with the same mean but different variances. The red population has mean 100 and variance 100 (SD=10) while the blue population has mean 100 and variance 2500 (SD=50) where SD stands for Standard Deviation.

                                                                An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.\n

                                                                There are two distinct concepts that are both called \"variance\". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.\n

                                                                The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling.\n

                                                                Geometric visualisation of the variance of an arbitrary distribution (2, 4, 4, 4, 5, 5, 7, 9):
                                                                1. A frequency distribution is constructed.
                                                                2. The centroid of the distribution gives its mean.
                                                                3. A square with sides equal to the difference of each value from the mean is formed for each value.
                                                                4. Arranging the squares into a rectangle with one side equal to the number of values, n, results in the other side being the distribution's variance, \u03c32.

                                                                Definition

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                                                                The variance of a random variable {\\displaystyle X} is the expected value of the squared deviation from the mean of {\\displaystyle X}, {\\displaystyle \\mu =\\operatorname {E} [X]}:\n

                                                                {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right].}

                                                                This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:\n

                                                                {\\displaystyle \\operatorname {Var} (X)=\\operatorname {Cov} (X,X).}

                                                                The variance is also equivalent to the second cumulant of a probability distribution that generates {\\displaystyle X}. The variance is typically designated as {\\displaystyle \\operatorname {Var} (X)}, or sometimes as {\\displaystyle V(X)} or {\\displaystyle \\mathbb {V} (X)}, or symbolically as {\\displaystyle \\sigma _{X}^{2}} or simply {\\displaystyle \\sigma ^{2}} (pronounced \"sigma squared\"). The expression for the variance can be expanded as follows:\n

                                                                {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left[(X-\\operatorname {E} [X])^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}-2X\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]^{2}+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}\\end{aligned}}}

                                                                In other words, the variance of X is equal to the mean of the square of X minus the square of the mean of X. This equation should not be used for computations using floating point arithmetic, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see algorithms for calculating variance.\n

                                                                Discrete random variable

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                                                                If the generator of random variable {\\displaystyle X} is discrete with probability mass function{\\displaystyle x_{1}\\mapsto p_{1},x_{2}\\mapsto p_{2},\\ldots ,x_{n}\\mapsto p_{n}}, then\n

                                                                {\\displaystyle \\operatorname {Var} (X)=\\sum _{i=1}^{n}p_{i}\\cdot (x_{i}-\\mu )^{2},}

                                                                where {\\displaystyle \\mu } is the expected value. That is,\n

                                                                {\\displaystyle \\mu =\\sum _{i=1}^{n}p_{i}x_{i}.}

                                                                (When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)\n

                                                                The variance of a collection of {\\displaystyle n} equally likely values can be written as\n

                                                                {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n}}\\sum _{i=1}^{n}(x_{i}-\\mu )^{2}}

                                                                where {\\displaystyle \\mu } is the average value. That is,\n

                                                                {\\displaystyle \\mu ={\\frac {1}{n}}\\sum _{i=1}^{n}x_{i}.}

                                                                The variance of a set of {\\displaystyle n} equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:[2]

                                                                {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{\\frac {1}{2}}(x_{i}-x_{j})^{2}={\\frac {1}{n^{2}}}\\sum _{i}\\sum _{j>i}(x_{i}-x_{j})^{2}.}

                                                                Absolutely continuous random variable

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                                                                If the random variable {\\displaystyle X} has a probability density function{\\displaystyle f(x)}, and {\\displaystyle F(x)} is the corresponding cumulative distribution function, then\n

                                                                {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)=\\sigma ^{2}&=\\int _{\\mathbb {R} }(x-\\mu )^{2}f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-2\\mu \\int _{\\mathbb {R} }xf(x)\\,dx+\\mu ^{2}\\int _{\\mathbb {R} }f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\int _{\\mathbb {R} }x\\,dF(x)+\\mu ^{2}\\int _{\\mathbb {R} }\\,dF(x)\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\cdot \\mu +\\mu ^{2}\\cdot 1\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-\\mu ^{2},\\end{aligned}}}

                                                                or equivalently,\n

                                                                {\\displaystyle \\operatorname {Var} (X)=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-\\mu ^{2},}

                                                                where {\\displaystyle \\mu } is the expected value of {\\displaystyle X} given by\n

                                                                {\\displaystyle \\mu =\\int _{\\mathbb {R} }xf(x)\\,dx=\\int _{\\mathbb {R} }x\\,dF(x).}

                                                                In these formulas, the integrals with respect to {\\displaystyle dx} and {\\displaystyle dF(x)}\nare Lebesgue and Lebesgue\u2013Stieltjes integrals, respectively.\n

                                                                If the function {\\displaystyle x^{2}f(x)} is Riemann-integrable on every finite interval {\\displaystyle [a,b]\\subset \\mathbb {R} ,} then\n

                                                                {\\displaystyle \\operatorname {Var} (X)=\\int _{-\\infty }^{+\\infty }x^{2}f(x)\\,dx-\\mu ^{2},}

                                                                where the integral is an improper Riemann integral.\n

                                                                Examples

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                                                                Exponential distribution

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                                                                The exponential distribution with parameter \u03bb is a continuous distribution whose probability density function is given by\n

                                                                {\\displaystyle f(x)=\\lambda e^{-\\lambda x}}

                                                                on the interval [0, \u221e). Its mean can be shown to be\n

                                                                {\\displaystyle \\operatorname {E} [X]=\\int _{0}^{\\infty }x\\lambda e^{-\\lambda x}\\,dx={\\frac {1}{\\lambda }}.}

                                                                Using integration by parts and making use of the expected value already calculated, we have:\n

                                                                {\\displaystyle {\\begin{aligned}\\operatorname {E} \\left[X^{2}\\right]&=\\int _{0}^{\\infty }x^{2}\\lambda e^{-\\lambda x}\\,dx\\\\&=\\left[-x^{2}e^{-\\lambda x}\\right]_{0}^{\\infty }+\\int _{0}^{\\infty }2xe^{-\\lambda x}\\,dx\\\\&=0+{\\frac {2}{\\lambda }}\\operatorname {E} [X]\\\\&={\\frac {2}{\\lambda ^{2}}}.\\end{aligned}}}

                                                                Thus, the variance of X is given by\n

                                                                {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}={\\frac {2}{\\lambda ^{2}}}-\\left({\\frac {1}{\\lambda }}\\right)^{2}={\\frac {1}{\\lambda ^{2}}}.}

                                                                Fair dice

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                                                                A fair six-sided dice can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is {\\displaystyle (1+2+3+4+5+6)/6=7/2.} Therefore, the variance of X is\n

                                                                {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\sum _{i=1}^{6}{\\frac {1}{6}}\\left(i-{\\frac {7}{2}}\\right)^{2}\\\\[5pt]&={\\frac {1}{6}}\\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\\right)\\\\[5pt]&={\\frac {35}{12}}\\approx 2.92.\\end{aligned}}}

                                                                The general formula for the variance of the outcome, X, of an n-sided die is\n

                                                                {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left(X^{2}\\right)-(\\operatorname {E} (X))^{2}\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}i^{2}-\\left({\\frac {1}{n}}\\sum _{i=1}^{n}i\\right)^{2}\\\\[5pt]&={\\frac {(n+1)(2n+1)}{6}}-\\left({\\frac {n+1}{2}}\\right)^{2}\\\\[4pt]&={\\frac {n^{2}-1}{12}}.\\end{aligned}}}

                                                                Commonly used probability distributions

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                                                                The following table lists the variance for some commonly used probability distributions.\n

                                                                Name of the probability distribution\nProbability distribution function\nMean\nVariance\n
                                                                Binomial distributionPr(X=k)=(nk)pk(1\u2212p)n\u2212k{\\displaystyle \\Pr \\,(X=k)={\\binom {n}{k}}p^{k}(1-p)^{n-k}}np{\\displaystyle np}np(1\u2212p){\\displaystyle np(1-p)}
                                                                Geometric distributionPr(X=k)=(1\u2212p)k\u22121p{\\displaystyle \\Pr \\,(X=k)=(1-p)^{k-1}p}1p{\\displaystyle {\\frac {1}{p}}}(1\u2212p)p2{\\displaystyle {\\frac {(1-p)}{p^{2}}}}
                                                                Normal distributionf(x\u2223\u03bc,\u03c32)=12\u03c0\u03c32e\u2212(x\u2212\u03bc)22\u03c32{\\displaystyle f\\left(x\\mid \\mu ,\\sigma ^{2}\\right)={\\frac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}e^{-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}}\u03bc{\\displaystyle \\mu }\u03c32{\\displaystyle \\sigma ^{2}}
                                                                Uniform distribution (continuous)f(x\u2223a,b)={1b\u2212afor a\u2264x\u2264b,0for x<a or x>b{\\displaystyle f(x\\mid a,b)={\\begin{cases}{\\frac {1}{b-a}}&{\\text{for }}a\\leq x\\leq b,\\\\[3pt]0&{\\text{for }}x<a{\\text{ or }}x>b\\end{cases}}}a+b2{\\displaystyle {\\frac {a+b}{2}}}(b\u2212a)212{\\displaystyle {\\frac {(b-a)^{2}}{12}}}
                                                                Exponential distributionf(x\u2223\u03bb)=\u03bbe\u2212\u03bbx{\\displaystyle f(x\\mid \\lambda )=\\lambda e^{-\\lambda x}}1\u03bb{\\displaystyle {\\frac {1}{\\lambda }}}1\u03bb2{\\displaystyle {\\frac {1}{\\lambda ^{2}}}}
                                                                Poisson distributionf(k\u2223\u03bb)=e\u2212\u03bb\u03bbkk!{\\displaystyle f(k\\mid \\lambda )={\\frac {e^{-\\lambda }\\lambda ^{k}}{k!}}}\u03bb{\\displaystyle \\lambda }\u03bb{\\displaystyle \\lambda }

                                                                Properties

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                                                                Basic properties

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                                                                Variance is non-negative because the squares are positive or zero:\n

                                                                {\\displaystyle \\operatorname {Var} (X)\\geq 0.}

                                                                The variance of a constant is zero.\n

                                                                {\\displaystyle \\operatorname {Var} (a)=0.}

                                                                Conversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value:\n

                                                                {\\displaystyle \\operatorname {Var} (X)=0\\iff \\exists a:P(X=a)=1.}

                                                                Issues of finiteness

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                                                                If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index{\\displaystyle k} satisfies {\\displaystyle 1<k\\leq 2.}

                                                                Decomposition

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                                                                The general formula for variance decomposition or the law of total variance is: If {\\displaystyle X} and {\\displaystyle Y} are two random variables, and the variance of {\\displaystyle X} exists, then\n

                                                                {\\displaystyle \\operatorname {Var} [X]=\\operatorname {E} (\\operatorname {Var} [X\\mid Y])+\\operatorname {Var} (\\operatorname {E} [X\\mid Y]).}

                                                                The conditional expectation{\\displaystyle \\operatorname {E} (X\\mid Y)} of {\\displaystyle X} given {\\displaystyle Y}, and the conditional variance{\\displaystyle \\operatorname {Var} (X\\mid Y)} may be understood as follows. Given any particular value y of the random variable Y, there is a conditional expectation {\\displaystyle \\operatorname {E} (X\\mid Y=y)} given the event Y = y. This quantity depends on the particular value y; it is a function {\\displaystyle g(y)=\\operatorname {E} (X\\mid Y=y)}. That same function evaluated at the random variable Y is the conditional expectation {\\displaystyle \\operatorname {E} (X\\mid Y)=g(Y).}

                                                                In particular, if {\\displaystyle Y} is a discrete random variable assuming possible values {\\displaystyle y_{1},y_{2},y_{3}\\ldots } with corresponding probabilities {\\displaystyle p_{1},p_{2},p_{3}\\ldots ,}, then in the formula for total variance, the first term on the right-hand side becomes\n

                                                                {\\displaystyle \\operatorname {E} (\\operatorname {Var} [X\\mid Y])=\\sum _{i}p_{i}\\sigma _{i}^{2},}

                                                                where {\\displaystyle \\sigma _{i}^{2}=\\operatorname {Var} [X\\mid Y=y_{i}]}. Similarly, the second term on the right-hand side becomes\n

                                                                {\\displaystyle \\operatorname {Var} (\\operatorname {E} [X\\mid Y])=\\sum _{i}p_{i}\\mu _{i}^{2}-\\left(\\sum _{i}p_{i}\\mu _{i}\\right)^{2}=\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2},}

                                                                where {\\displaystyle \\mu _{i}=\\operatorname {E} [X\\mid Y=y_{i}]} and {\\displaystyle \\mu =\\sum _{i}p_{i}\\mu _{i}}. Thus the total variance is given by\n

                                                                {\\displaystyle \\operatorname {Var} [X]=\\sum _{i}p_{i}\\sigma _{i}^{2}+\\left(\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2}\\right).}

                                                                A similar formula is applied in analysis of variance, where the corresponding formula is\n

                                                                {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{between}}+{\\mathit {MS}}_{\\text{within}};}

                                                                here {\\displaystyle {\\mathit {MS}}} refers to the Mean of the Squares. In linear regression analysis the corresponding formula is\n

                                                                {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{regression}}+{\\mathit {MS}}_{\\text{residual}}.}

                                                                This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.\n

                                                                Similar decompositions are possible for the sum of squared deviations (sum of squares, {\\displaystyle {\\mathit {SS}}}):\n

                                                                {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{between}}+{\\mathit {SS}}_{\\text{within}},}
                                                                {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{regression}}+{\\mathit {SS}}_{\\text{residual}}.}

                                                                Calculation from the CDF

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                                                                The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution functionF using\n

                                                                {\\displaystyle 2\\int _{0}^{\\infty }u(1-F(u))\\,du-\\left(\\int _{0}^{\\infty }(1-F(u))\\,du\\right)^{2}.}

                                                                This expression can be used to calculate the variance in situations where the CDF, but not the density, can be conveniently expressed.\n

                                                                Characteristic property

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                                                                The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} \\left(\\left(X-m\\right)^{2}\\right)=\\mathrm {E} (X)}. Conversely, if a continuous function {\\displaystyle \\varphi } satisfies {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} (\\varphi (X-m))=\\mathrm {E} (X)} for all random variables X, then it is necessarily of the form {\\displaystyle \\varphi (x)=ax^{2}+b}, where a > 0. This also holds in the multidimensional case.[3]

                                                                Units of measurement

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                                                                Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is \u221a2.9 \u2248 1.7, slightly larger than the expected absolute deviation of 1.5.\n

                                                                The standard deviation and the expected absolute deviation can both be used as an indicator of the \"spread\" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution.\n

                                                                Propagation

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                                                                Addition and multiplication by a constant

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                                                                Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged:\n

                                                                {\\displaystyle \\operatorname {Var} (X+a)=\\operatorname {Var} (X).}

                                                                If all values are scaled by a constant, the variance is scaled by the square of that constant:\n

                                                                {\\displaystyle \\operatorname {Var} (aX)=a^{2}\\operatorname {Var} (X).}

                                                                The variance of a sum of two random variables is given by\n

                                                                {\\displaystyle \\operatorname {Var} (aX+bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)+2ab\\,\\operatorname {Cov} (X,Y)}
                                                                {\\displaystyle \\operatorname {Var} (aX-bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)-2ab\\,\\operatorname {Cov} (X,Y)}

                                                                where {\\displaystyle \\operatorname {Cov} (X,Y)} is the covariance.\n

                                                                Linear combinations

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                                                                In general, for the sum of {\\displaystyle N} random variables {\\displaystyle \\{X_{1},\\dots ,X_{N}\\}}, the variance becomes:\n

                                                                {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i,j=1}^{N}\\operatorname {Cov} (X_{i},X_{j})=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i})+\\sum _{i\\neq j}\\operatorname {Cov} (X_{i},X_{j}),}

                                                                see also general Bienaym\u00e9's identity.\n

                                                                These results lead to the variance of a linear combination as:\n

                                                                {\\displaystyle {\\begin{aligned}\\operatorname {Var} \\left(\\sum _{i=1}^{N}a_{i}X_{i}\\right)&=\\sum _{i,j=1}^{N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+\\sum _{i\\not =j}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i<j\\leq N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j}).\\end{aligned}}}

                                                                If the random variables {\\displaystyle X_{1},\\dots ,X_{N}} are such that\n

                                                                {\\displaystyle \\operatorname {Cov} (X_{i},X_{j})=0\\ ,\\ \\forall \\ (i\\neq j),}

                                                                then they are said to be uncorrelated. It follows immediately from the expression given earlier that if the random variables {\\displaystyle X_{1},\\dots ,X_{N}} are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:\n

                                                                {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i}).}

                                                                Since independent random variables are always uncorrelated (see Covariance \u00a7 Uncorrelatedness and independence), the equation above holds in particular when the random variables {\\displaystyle X_{1},\\dots ,X_{n}} are independent. Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.\n

                                                                Matrix notation for the variance of a linear combination

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                                                                Define {\\displaystyle X} as a column vector of {\\displaystyle n} random variables {\\displaystyle X_{1},\\ldots ,X_{n}}, and {\\displaystyle c} as a column vector of {\\displaystyle n} scalars {\\displaystyle c_{1},\\ldots ,c_{n}}. Therefore, {\\displaystyle c^{\\mathsf {T}}X} is a linear combination of these random variables, where {\\displaystyle c^{\\mathsf {T}}} denotes the transpose of {\\displaystyle c}. Also let {\\displaystyle \\Sigma } be the covariance matrix of {\\displaystyle X}. The variance of {\\displaystyle c^{\\mathsf {T}}X} is then given by:[4]

                                                                {\\displaystyle \\operatorname {Var} \\left(c^{\\mathsf {T}}X\\right)=c^{\\mathsf {T}}\\Sigma c.}

                                                                This implies that the variance of the mean can be written as (with a column vector of ones)\n

                                                                {\\displaystyle \\operatorname {Var} \\left({\\bar {x}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}1'X\\right)={\\frac {1}{n^{2}}}1'\\Sigma 1.}

                                                                Sum of variables

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                                                                Sum of uncorrelated variables

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                                                                One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances:\n

                                                                {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} (X_{i}).}

                                                                This statement is called the Bienaym\u00e9 formula[5] and was discovered in 1853.[6][7] It is often made with the stronger condition that the variables are independent, but being uncorrelated suffices. So if all the variables have the same variance \u03c32, then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is\n

                                                                {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}\\sum _{i=1}^{n}X_{i}\\right)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)={\\frac {1}{n^{2}}}n\\sigma ^{2}={\\frac {\\sigma ^{2}}{n}}.}

                                                                That is, the variance of the mean decreases when n increases. This formula for the variance of the mean is used in the definition of the standard error of the sample mean, which is used in the central limit theorem.\n

                                                                To prove the initial statement, it suffices to show that\n

                                                                {\\displaystyle \\operatorname {Var} (X+Y)=\\operatorname {Var} (X)+\\operatorname {Var} (Y).}

                                                                The general result then follows by induction. Starting with the definition,\n

                                                                {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[(X+Y)^{2}\\right]-(\\operatorname {E} [X+Y])^{2}\\\\[5pt]&=\\operatorname {E} \\left[X^{2}+2XY+Y^{2}\\right]-(\\operatorname {E} [X]+\\operatorname {E} [Y])^{2}.\\end{aligned}}}

                                                                Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y, this further simplifies as follows:\n

                                                                {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[X^{2}\\right]+2\\operatorname {E} [XY]+\\operatorname {E} \\left[Y^{2}\\right]-\\left(\\operatorname {E} [X]^{2}+2\\operatorname {E} [X]\\operatorname {E} [Y]+\\operatorname {E} [Y]^{2}\\right)\\\\[5pt]&=\\operatorname {E} \\left[X^{2}\\right]+\\operatorname {E} \\left[Y^{2}\\right]-\\operatorname {E} [X]^{2}-\\operatorname {E} [Y]^{2}\\\\[5pt]&=\\operatorname {Var} (X)+\\operatorname {Var} (Y).\\end{aligned}}}

                                                                Sum of correlated variables

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                                                                Sum of correlated variables with fixed sample size
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                                                                In general, the variance of the sum of n variables is the sum of their covariances:\n

                                                                {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\sum _{j=1}^{n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)+2\\sum _{1\\leq i<j\\leq n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right).}

                                                                (Note: The second equality comes from the fact that Cov(Xi,Xi) = Var(Xi).)\n

                                                                Here, {\\displaystyle \\operatorname {Cov} (\\cdot ,\\cdot )} is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. The next expression states equivalently that the variance of the sum is the sum of the diagonal of covariance matrix plus two times the sum of its upper triangular elements (or its lower triangular elements); this emphasizes that the covariance matrix is symmetric. This formula is used in the theory of Cronbach's alpha in classical test theory.\n

                                                                So, if the variables have equal variance \u03c32 and the average correlation of distinct variables is \u03c1, then the variance of their mean is\n

                                                                {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {\\sigma ^{2}}{n}}+{\\frac {n-1}{n}}\\rho \\sigma ^{2}.}

                                                                This implies that the variance of the mean increases with the average of the correlations. In other words, additional correlated observations are not as effective as additional independent observations at reducing the uncertainty of the mean. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to\n

                                                                {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {1}{n}}+{\\frac {n-1}{n}}\\rho .}

                                                                This formula is used in the Spearman\u2013Brown prediction formula of classical test theory. This converges to \u03c1 if n goes to infinity, provided that the average correlation remains constant or converges too. So for the variance of the mean of standardized variables with equal correlations or converging average correlation we have\n

                                                                {\\displaystyle \\lim _{n\\to \\infty }\\operatorname {Var} \\left({\\overline {X}}\\right)=\\rho .}

                                                                Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. This makes clear that the sample mean of correlated variables does not generally converge to the population mean, even though the law of large numbers states that the sample mean will converge for independent variables.\n

                                                                Sum of uncorrelated variables with random sample size
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                                                                There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size N is a random variable whose variation adds to the variation of X, such that,\n

                                                                {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\operatorname {E} \\left[N\\right]\\operatorname {Var} (X)+\\operatorname {Var} (N)(\\operatorname {E} \\left[X\\right])^{2}}[8]

                                                                which follows from the law of total variance.\n

                                                                If N has a Poisson distribution, then {\\displaystyle \\operatorname {E} [N]=\\operatorname {Var} (N)} with estimator n = N. So, the estimator of {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)} becomes {\\displaystyle n{S_{x}}^{2}+n{\\bar {X}}^{2}}, giving {\\displaystyle \\operatorname {SE} ({\\bar {X}})={\\sqrt {\\frac {{S_{x}}^{2}+{\\bar {X}}^{2}}{n}}}}\n(see standard error of the sample mean).\n

                                                                Weighted sum of variables

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                                                                The scaling property and the Bienaym\u00e9 formula, along with the property of the covarianceCov(aX, bY) = ab Cov(X, Y) jointly imply that\n

                                                                {\\displaystyle \\operatorname {Var} (aX\\pm bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)\\pm 2ab\\,\\operatorname {Cov} (X,Y).}

                                                                This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y, then the weight of the variance of X will be four times the weight of the variance of Y.\n

                                                                The expression above can be extended to a weighted sum of multiple variables:\n

                                                                {\\displaystyle \\operatorname {Var} \\left(\\sum _{i}^{n}a_{i}X_{i}\\right)=\\sum _{i=1}^{n}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i}\\sum _{<j\\leq n}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})}

                                                                Product of variables

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                                                                Product of independent variables

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                                                                If two variables X and Y are independent, the variance of their product is given by[9]

                                                                {\\displaystyle \\operatorname {Var} (XY)=[\\operatorname {E} (X)]^{2}\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\operatorname {Var} (X)+\\operatorname {Var} (X)\\operatorname {Var} (Y).}

                                                                Equivalently, using the basic properties of expectation, it is given by\n

                                                                {\\displaystyle \\operatorname {Var} (XY)=\\operatorname {E} \\left(X^{2}\\right)\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (X)]^{2}[\\operatorname {E} (Y)]^{2}.}

                                                                Product of statistically dependent variables

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                                                                In general, if two variables are statistically dependent, then the variance of their product is given by:\n

                                                                {\\displaystyle {\\begin{aligned}\\operatorname {Var} (XY)={}&\\operatorname {E} \\left[X^{2}Y^{2}\\right]-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\operatorname {E} (X^{2})\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\left(\\operatorname {Var} (X)+[\\operatorname {E} (X)]^{2}\\right)\\left(\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\right)\\\\[5pt]&-[\\operatorname {Cov} (X,Y)+\\operatorname {E} (X)\\operatorname {E} (Y)]^{2}\\end{aligned}}}

                                                                Arbitrary functions

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                                                                The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables: see Taylor expansions for the moments of functions of random variables. For example, the approximate variance of a function of one variable is given by\n

                                                                {\\displaystyle \\operatorname {Var} \\left[f(X)\\right]\\approx \\left(f'(\\operatorname {E} \\left[X\\right])\\right)^{2}\\operatorname {Var} \\left[X\\right]}

                                                                provided that f is twice differentiable and that the mean and variance of X are finite.\n

                                                                Population variance and sample variance

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                                                                Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one estimates the mean and variance from a limited set of observations by using an estimator equation. The estimator is a function of the sample of nobservations drawn without observational bias from the whole population of potential observations. In this example, the sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.\n

                                                                The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance \u2013 these are consistent estimators (they converge to the value of the whole population as the number of samples increases) but can be improved. Most simply, the sample variance is computed as the sum of squared deviations about the (sample) mean, divided by n as the number of samples. However, using values other than n improves the estimator in various ways. Four common values for the denominator are n,n \u2212 1, n + 1, and n \u2212 1.5: n is the simplest (the variance of the sample), n \u2212 1 eliminates bias,[10]n + 1 minimizes mean squared error for the normal distribution,[11] and n \u2212 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.[12]

                                                                Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (n \u2212 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n -1 instead of n, is called Bessel's correction.[10] The resulting estimator is unbiased and is called the (corrected) sample variance or unbiased sample variance. If the mean is determined in some other way than from the same samples used to estimate the variance, then this bias does not arise, and the variance can safely be estimated as that of the samples about the (independently known) mean.\n

                                                                Secondly, the sample variance does not generally minimize mean squared error between sample variance and population variance. Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance) and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than n \u2212 1) and is a simple example of a shrinkage estimator: one \"shrinks\" the unbiased estimator towards zero. For the normal distribution, dividing by n + 1 (instead of n \u2212 1 or n) minimizes mean squared error.[11] The resulting estimator is biased, however, and is known as the biased sample variation.\n

                                                                Population variance

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                                                                In general, the population variance of a finitepopulation of size N with values xi is given by

                                                                \u03c32=1N\u2211i=1N(xi\u2212\u03bc)2=1N\u2211i=1N(xi2\u22122\u03bcxi+\u03bc2)=(1N\u2211i=1Nxi2)\u22122\u03bc(1N\u2211i=1Nxi)+\u03bc2=E\u2061[xi2]\u2212\u03bc2{\\displaystyle {\\begin{aligned}\\sigma ^{2}&={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}-\\mu \\right)^{2}={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}^{2}-2\\mu x_{i}+\\mu ^{2}\\right)\\\\[5pt]&=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-2\\mu \\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)+\\mu ^{2}\\\\[5pt]&=\\operatorname {E} [x_{i}^{2}]-\\mu ^{2}\\end{aligned}}}

                                                                where the population mean is {\\textstyle \\mu =\\operatorname {E} [x_{i}]={\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}} and {\\textstyle \\operatorname {E} [x_{i}^{2}]=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)}, where {\\textstyle \\operatorname {E} } is the expectation value operator.\n

                                                                The population variance can also be computed using[13]

                                                                {\\displaystyle \\sigma ^{2}={\\frac {1}{N^{2}}}\\sum _{i<j}\\left(x_{i}-x_{j}\\right)^{2}={\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}.}

                                                                (The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because

                                                                12N2\u2211i,j=1N(xi\u2212xj)2=12N2\u2211i,j=1N(xi2\u22122xixj+xj2)=12N\u2211j=1N(1N\u2211i=1Nxi2)\u2212(1N\u2211i=1Nxi)(1N\u2211j=1Nxj)+12N\u2211i=1N(1N\u2211j=1Nxj2)=12(\u03c32+\u03bc2)\u2212\u03bc2+12(\u03c32+\u03bc2)=\u03c32.{\\displaystyle {\\begin{aligned}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}\\\\[5pt]={}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2N}}\\sum _{j=1}^{N}\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}\\right)+{\\frac {1}{2N}}\\sum _{i=1}^{N}\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-\\mu ^{2}+{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\\\[5pt]={}&\\sigma ^{2}.\\end{aligned}}}

                                                                The population variance matches the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.\n

                                                                Sample variance

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                                                                Biased sample variance

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                                                                In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population.[14] This is generally referred to as sample variance or empirical variance. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution.\n

                                                                We take a sample with replacement of n values Y1, ..., Yn from the population of size {\\textstyle N}, where n < N, and estimate the variance on the basis of this sample.[15] Directly taking the variance of the sample data gives the average of the squared deviations:\n

                                                                {\\displaystyle {\\tilde {S}}_{Y}^{2}={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}=\\left({\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}^{2}\\right)-{\\overline {Y}}^{2}={\\frac {1}{n^{2}}}\\sum _{i,j\\,:\\,i<j}\\left(Y_{i}-Y_{j}\\right)^{2}.}[16]

                                                                (See the section Population variance for the derivation of this formula.) Here, {\\displaystyle {\\overline {Y}}} denotes the sample mean:\n

                                                                {\\displaystyle {\\overline {Y}}={\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}.}

                                                                Since the Yi are selected randomly, both {\\displaystyle {\\overline {Y}}} and {\\displaystyle {\\tilde {S}}_{Y}^{2}} are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples {Yi} of size n from the population. For {\\displaystyle {\\tilde {S}}_{Y}^{2}} this gives:\n

                                                                {\\displaystyle {\\begin{aligned}\\operatorname {E} [{\\tilde {S}}_{Y}^{2}]&=\\operatorname {E} \\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\frac {1}{n}}\\sum _{j=1}^{n}Y_{j}\\right)^{2}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\operatorname {E} \\left[Y_{i}^{2}-{\\frac {2}{n}}Y_{i}\\sum _{j=1}^{n}Y_{j}+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}Y_{j}\\sum _{k=1}^{n}Y_{k}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\left(\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+\\operatorname {E} \\left[Y_{i}^{2}\\right]\\right)+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left({\\frac {n-2}{n}}\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left[{\\frac {n-2}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-{\\frac {2}{n}}(n-1)\\mu ^{2}+{\\frac {1}{n^{2}}}n(n-1)\\mu ^{2}+{\\frac {1}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\right]\\\\[5pt]&={\\frac {n-1}{n}}\\sigma ^{2}.\\end{aligned}}}

                                                                Here {\\textstyle \\sigma ^{2}=\\operatorname {E} [Y_{i}^{2}]-\\mu ^{2}} derived in the section Population variance and {\\textstyle \\operatorname {E} [Y_{i}Y_{j}]=\\operatorname {E} [Y_{i}]\\operatorname {E} [Y_{j}]=\\mu ^{2}} due to independency of {\\textstyle Y_{i}} and {\\textstyle Y_{j}} are used.\n

                                                                Hence {\\textstyle {\\tilde {S}}_{Y}^{2}} gives an estimate of the population variance that is biased by a factor of {\\textstyle {\\frac {n-1}{n}}} as the expectation value of {\\textstyle {\\tilde {S}}_{Y}^{2}} is smaller than the population variance (true variance) by that factor. For this reason, {\\textstyle {\\tilde {S}}_{Y}^{2}} is referred to as the biased sample variance.\n

                                                                Unbiased sample variance

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                                                                Correcting for this bias yields the unbiased sample variance, denoted {\\displaystyle S^{2}}:\n

                                                                {\\displaystyle S^{2}={\\frac {n}{n-1}}{\\tilde {S}}_{Y}^{2}={\\frac {n}{n-1}}\\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}\\right]={\\frac {1}{n-1}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}}

                                                                Either estimator may be simply referred to as the sample variance when the version can be determined by context. The same proof is also applicable for samples taken from a continuous probability distribution.\n

                                                                The use of the term n \u2212 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation (the square root of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n \u2212 1.5 yields an almost unbiased estimator.\n

                                                                The unbiased sample variance is a U-statistic for the function \u0192(y1, y2) = (y1 \u2212 y2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.\n

                                                                Example
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                                                                For a set of numbers {10, 15, 30, 45, 57, 52 63, 72, 81, 93, 102, 105}, if this set is the whole data population for some measurement, then variance is the population variance 932.743 as the sum of the squared deviations about the mean of this set, divided by 12 as the number of the set members. If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.\n

                                                                Distribution of the sample variance

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                                                                Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Yi are independent observations from a normal distribution, Cochran's theorem shows that the unbiased sample varianceS2 follows a scaled chi-squared distribution (see also: asymptotic properties and an elementary proof):[17]

                                                                {\\displaystyle (n-1){\\frac {S^{2}}{\\sigma ^{2}}}\\sim \\chi _{n-1}^{2}}

                                                                where \u03c32 is the population variance. As a direct consequence, it follows that\n

                                                                {\\displaystyle \\operatorname {E} \\left(S^{2}\\right)=\\operatorname {E} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)=\\sigma ^{2},}

                                                                and[18]

                                                                {\\displaystyle \\operatorname {Var} \\left[S^{2}\\right]=\\operatorname {Var} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)={\\frac {\\sigma ^{4}}{(n-1)^{2}}}\\operatorname {Var} \\left(\\chi _{n-1}^{2}\\right)={\\frac {2\\sigma ^{4}}{n-1}}.}

                                                                If Yi are independent and identically distributed, but not necessarily normally distributed, then[19]

                                                                {\\displaystyle \\operatorname {E} \\left[S^{2}\\right]=\\sigma ^{2},\\quad \\operatorname {Var} \\left[S^{2}\\right]={\\frac {\\sigma ^{4}}{n}}\\left(\\kappa -1+{\\frac {2}{n-1}}\\right)={\\frac {1}{n}}\\left(\\mu _{4}-{\\frac {n-3}{n-1}}\\sigma ^{4}\\right),}

                                                                where \u03ba is the kurtosis of the distribution and \u03bc4 is the fourth central moment.\n

                                                                If the conditions of the law of large numbers hold for the squared observations, S2 is a consistent estimator of \u03c32. One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.).[20][21][22]

                                                                Samuelson's inequality

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                                                                Samuelson's inequality is a result that states bounds on the values that individual observations in a sample can take, given that the sample mean and (biased) variance have been calculated.[23] Values must lie within the limits {\\displaystyle {\\bar {y}}\\pm \\sigma _{Y}(n-1)^{1/2}.}

                                                                Relations with the harmonic and arithmetic means

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                                                                It has been shown[24] that for a sample {yi} of positive real numbers,\n

                                                                {\\displaystyle \\sigma _{y}^{2}\\leq 2y_{\\max }(A-H),}

                                                                where ymax is the maximum of the sample, A is the arithmetic mean, H is the harmonic mean of the sample and {\\displaystyle \\sigma _{y}^{2}} is the (biased) variance of the sample.\n

                                                                This bound has been improved, and it is known that variance is bounded by\n

                                                                {\\displaystyle \\sigma _{y}^{2}\\leq {\\frac {y_{\\max }(A-H)(y_{\\max }-A)}{y_{\\max }-H}},}
                                                                {\\displaystyle \\sigma _{y}^{2}\\geq {\\frac {y_{\\min }(A-H)(A-y_{\\min })}{H-y_{\\min }}},}

                                                                where ymin is the minimum of the sample.[25]

                                                                Tests of equality of variances

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                                                                The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult.\n

                                                                Several non parametric tests have been proposed: these include the Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey test, the Capon test, Mood test, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that both medians be known and equal to zero. The Mood, Klotz, Capon and Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey tests also apply to two variances. They allow the median to be unknown but do require that the two medians are equal.\n

                                                                The Lehmann test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variances include the Box test, the Box\u2013Anderson test and the Moses test.\n

                                                                Resampling methods, which include the bootstrap and the jackknife, may be used to test the equality of variances.\n

                                                                Moment of inertia

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                                                                The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass.[26] It is because of this analogy that such things as the variance are called moments of probability distributions.[26] The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of n points with a covariance matrix of {\\displaystyle \\Sigma } is given by\n

                                                                {\\displaystyle I=n\\left(\\mathbf {1} _{3\\times 3}\\operatorname {tr} (\\Sigma )-\\Sigma \\right).}

                                                                This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the x axis and distributed along it. The covariance matrix might look like\n

                                                                {\\displaystyle \\Sigma ={\\begin{bmatrix}10&0&0\\\\0&0.1&0\\\\0&0&0.1\\end{bmatrix}}.}

                                                                That is, there is the most variance in the x direction. Physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is\n

                                                                {\\displaystyle I=n{\\begin{bmatrix}0.2&0&0\\\\0&10.1&0\\\\0&0&10.1\\end{bmatrix}}.}

                                                                Semivariance

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                                                                The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation:

                                                                Semivariance=1n\u2211i:xi<\u03bc(xi\u2212\u03bc)2{\\displaystyle {\\text{Semivariance}}={1 \\over {n}}\\sum _{i:x_{i}<\\mu }(x_{i}-\\mu )^{2}}

                                                                It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not.[27]

                                                                For inequalities associated with the semivariance, see Chebyshev's inequality \u00a7 Semivariances.\n

                                                                Etymology

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                                                                The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance:[28]

                                                                The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and, therefore, that the variability may be uniformly measured by the standard deviation corresponding to the square root of the mean square error. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations {\\displaystyle \\sigma _{1}} and {\\displaystyle \\sigma _{2}}, it is found that the distribution, when both causes act together, has a standard deviation {\\displaystyle {\\sqrt {\\sigma _{1}^{2}+\\sigma _{2}^{2}}}}. It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance...

                                                                Generalizations

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                                                                For complex variables

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                                                                If {\\displaystyle x} is a scalar complex-valued random variable, with values in {\\displaystyle \\mathbb {C} ,} then its variance is {\\displaystyle \\operatorname {E} \\left[(x-\\mu )(x-\\mu )^{*}\\right],} where {\\displaystyle x^{*}} is the complex conjugate of {\\displaystyle x.} This variance is a real scalar.\n

                                                                For vector-valued random variables

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                                                                As a matrix

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                                                                If {\\displaystyle X} is a vector-valued random variable, with values in {\\displaystyle \\mathbb {R} ^{n},} and thought of as a column vector, then a natural generalization of variance is {\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\operatorname {T} }\\right],} where {\\displaystyle \\mu =\\operatorname {E} (X)} and {\\displaystyle X^{\\operatorname {T} }} is the transpose of {\\displaystyle X,} and so is a row vector. The result is a positive semi-definite square matrix, commonly referred to as the variance-covariance matrix (or simply as the covariance matrix).\n

                                                                If {\\displaystyle X} is a vector- and complex-valued random variable, with values in {\\displaystyle \\mathbb {C} ^{n},} then the covariance matrix is{\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\dagger }\\right],} where {\\displaystyle X^{\\dagger }} is the conjugate transpose of {\\displaystyle X.} This matrix is also positive semi-definite and square.\n

                                                                As a scalar

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                                                                Another generalization of variance for vector-valued random variables {\\displaystyle X}, which results in a scalar value rather than in a matrix, is the generalized variance{\\displaystyle \\det(C)}, the determinant of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.[29]

                                                                A different generalization is obtained by considering the equation for the scalar variance, {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right]}, and reinterpreting {\\displaystyle (X-\\mu )^{2}} as the squared Euclidean distance between the random variable and its mean, or, simply as the scalar product of the vector {\\displaystyle X-\\mu } with itself. This results in {\\displaystyle \\operatorname {E} \\left[(X-\\mu )^{\\operatorname {T} }(X-\\mu )\\right]=\\operatorname {tr} (C),} which is the trace of the covariance matrix.\n

                                                                See also

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                                                                  Types of variance

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                                                                    References

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                                                                      ", "statics": {"paragraph": 178, "paragraph.text": 297, "paragraph.equation-inline": 122, "image": 2, "list": 74, "list.text": 15, "title": 49, "list.equation-inline": 71, "table": 1, "equation-interline": 3}, "url": "https://en.m.wikipedia.org/wiki/Variance", "content": "In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation(SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by ${\\displaystyle \\sigma ^{2}}$ , ${\\displaystyle s^{2}}$ , ${\\displaystyle \\operatorname {Var} (X)}$ , ${\\displaystyle V(X)}$ , or ${\\displaystyle \\mathbb {V} (X)}$ .[ 1]\n\nAn advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.\n\nThere are two distinct concepts that are both called \"variance\". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.\n\nThe two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling.\n\n1. A frequency distribution is constructed.\n2. The centroid of the distribution gives its mean.\n3. A square with sides equal to the difference of each value from the mean is formed for each value.\n4. Arranging the squares into a rectangle with one side equal to the number of values, n , results in the other side being the distribution's variance, \u03c3 2 .\n\n## Definition\n\nedit\n\nThe variance of a random variable ${\\displaystyle X}$ is the expected value of the squared deviation from the mean of ${\\displaystyle X}$ , ${\\displaystyle \\mu =\\operatorname {E} [X]}$ :\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right].}$\n\nThis definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\operatorname {Cov} (X,X).}$\n\nThe variance is also equivalent to the second cumulant of a probability distribution that generates ${\\displaystyle X}$ . The variance is typically designated as ${\\displaystyle \\operatorname {Var} (X)}$ , or sometimes as ${\\displaystyle V(X)}$ or ${\\displaystyle \\mathbb {V} (X)}$ , or symbolically as ${\\displaystyle \\sigma _{X}^{2}}$ or simply ${\\displaystyle \\sigma ^{2}}$ (pronounced \" sigma squared\"). The expression for the variance can be expanded as follows:\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left[(X-\\operatorname {E} [X])^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}-2X\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]^{2}+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}\\end{aligned}}}$\n\nIn other words, the variance of X is equal to the mean of the square of X minus the square of the mean of X. This equation should not be used for computations using floating point arithmetic, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see algorithms for calculating variance.\n\n### Discrete random variable\n\nedit\n\nIf the generator of random variable ${\\displaystyle X}$ is discrete with probability mass function ${\\displaystyle x_{1}\\mapsto p_{1},x_{2}\\mapsto p_{2},\\ldots ,x_{n}\\mapsto p_{n}}$ , then\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\sum _{i=1}^{n}p_{i}\\cdot (x_{i}-\\mu )^{2},}$\n\nwhere ${\\displaystyle \\mu }$ is the expected value. That is,\n\n1. ${\\displaystyle \\mu =\\sum _{i=1}^{n}p_{i}x_{i}.}$\n\n(When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)\n\nThe variance of a collection of ${\\displaystyle n}$ equally likely values can be written as\n\n1. ${\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n}}\\sum _{i=1}^{n}(x_{i}-\\mu )^{2}}$\n\nwhere ${\\displaystyle \\mu }$ is the average value. That is,\n\n1. ${\\displaystyle \\mu ={\\frac {1}{n}}\\sum _{i=1}^{n}x_{i}.}$\n\nThe variance of a set of ${\\displaystyle n}$ equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:[ 2]\n\n1. ${\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{\\frac {1}{2}}(x_{i}-x_{j})^{2}={\\frac {1}{n^{2}}}\\sum _{i}\\sum _{j>i}(x_{i}-x_{j})^{2}.}$\n\n### Absolutely continuous random variable\n\nedit\n\nIf the random variable ${\\displaystyle X}$ has a probability density function ${\\displaystyle f(x)}$ , and ${\\displaystyle F(x)}$ is the corresponding cumulative distribution function, then\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)=\\sigma ^{2}&=\\int _{\\mathbb {R} }(x-\\mu )^{2}f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-2\\mu \\int _{\\mathbb {R} }xf(x)\\,dx+\\mu ^{2}\\int _{\\mathbb {R} }f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\int _{\\mathbb {R} }x\\,dF(x)+\\mu ^{2}\\int _{\\mathbb {R} }\\,dF(x)\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\cdot \\mu +\\mu ^{2}\\cdot 1\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-\\mu ^{2},\\end{aligned}}}$\n\nor equivalently,\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-\\mu ^{2},}$\n\nwhere ${\\displaystyle \\mu }$ is the expected value of ${\\displaystyle X}$ given by\n\n1. ${\\displaystyle \\mu =\\int _{\\mathbb {R} }xf(x)\\,dx=\\int _{\\mathbb {R} }x\\,dF(x).}$\n\nIn these formulas, the integrals with respect to ${\\displaystyle dx}$ and ${\\displaystyle dF(x)}$ are Lebesgue and Lebesgue\u2013Stieltjes integrals, respectively.\n\nIf the function ${\\displaystyle x^{2}f(x)}$ is Riemann-integrable on every finite interval ${\\displaystyle [a,b]\\subset \\mathbb {R} ,}$ then\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\int _{-\\infty }^{+\\infty }x^{2}f(x)\\,dx-\\mu ^{2},}$\n\nwhere the integral is an improper Riemann integral.\n\n## Examples\n\nedit\n\n### Exponential distribution\n\nedit\n\nThe exponential distribution with parameter \u03bb is a continuous distribution whose probability density function is given by\n\n1. ${\\displaystyle f(x)=\\lambda e^{-\\lambda x}}$\n\non the interval[0, \u221e). Its mean can be shown to be\n\n1. ${\\displaystyle \\operatorname {E} [X]=\\int _{0}^{\\infty }x\\lambda e^{-\\lambda x}\\,dx={\\frac {1}{\\lambda }}.}$\n\nUsing integration by parts and making use of the expected value already calculated, we have:\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {E} \\left[X^{2}\\right]&=\\int _{0}^{\\infty }x^{2}\\lambda e^{-\\lambda x}\\,dx\\\\&=\\left[-x^{2}e^{-\\lambda x}\\right]_{0}^{\\infty }+\\int _{0}^{\\infty }2xe^{-\\lambda x}\\,dx\\\\&=0+{\\frac {2}{\\lambda }}\\operatorname {E} [X]\\\\&={\\frac {2}{\\lambda ^{2}}}.\\end{aligned}}}$\n\nThus, the variance of X is given by\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}={\\frac {2}{\\lambda ^{2}}}-\\left({\\frac {1}{\\lambda }}\\right)^{2}={\\frac {1}{\\lambda ^{2}}}.}$\n\n### Fair dice\n\nedit\n\nA fair six-sided dice can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is ${\\displaystyle (1+2+3+4+5+6)/6=7/2.}$ Therefore, the variance of X is\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\sum _{i=1}^{6}{\\frac {1}{6}}\\left(i-{\\frac {7}{2}}\\right)^{2}\\\\[5pt]&={\\frac {1}{6}}\\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\\right)\\\\[5pt]&={\\frac {35}{12}}\\approx 2.92.\\end{aligned}}}$\n\nThe general formula for the variance of the outcome, X, of an n-sided die is\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left(X^{2}\\right)-(\\operatorname {E} (X))^{2}\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}i^{2}-\\left({\\frac {1}{n}}\\sum _{i=1}^{n}i\\right)^{2}\\\\[5pt]&={\\frac {(n+1)(2n+1)}{6}}-\\left({\\frac {n+1}{2}}\\right)^{2}\\\\[4pt]&={\\frac {n^{2}-1}{12}}.\\end{aligned}}}$\n\n### Commonly used probability distributions\n\nedit\n\nThe following table lists the variance for some commonly used probability distributions.\n\n| Name of the probability distribution | Probability distribution function | Mean | Variance |\n|---|---|---|---|\n| Binomial distribution | Pr
                                                                      (
                                                                      X
                                                                      =
                                                                      k
                                                                      )
                                                                      =
                                                                      (
                                                                      n
                                                                      k
                                                                      )
                                                                      p
                                                                      k
                                                                      (
                                                                      1
                                                                      \u2212
                                                                      p
                                                                      )
                                                                      n
                                                                      \u2212
                                                                      k
                                                                      {\\displaystyle \\Pr \\,(X=k)={\\binom {n}{k}}p^{k}(1-p)^{n-k}} | n
                                                                      p
                                                                      {\\displaystyle np} | n
                                                                      p
                                                                      (
                                                                      1
                                                                      \u2212
                                                                      p
                                                                      )
                                                                      {\\displaystyle np(1-p)} |\n| Geometric distribution | Pr
                                                                      (
                                                                      X
                                                                      =
                                                                      k
                                                                      )
                                                                      =
                                                                      (
                                                                      1
                                                                      \u2212
                                                                      p
                                                                      )
                                                                      k
                                                                      \u2212
                                                                      1
                                                                      p
                                                                      {\\displaystyle \\Pr \\,(X=k)=(1-p)^{k-1}p} | 1
                                                                      p
                                                                      {\\displaystyle {\\frac {1}{p}}} | (
                                                                      1
                                                                      \u2212
                                                                      p
                                                                      )
                                                                      p
                                                                      2
                                                                      {\\displaystyle {\\frac {(1-p)}{p^{2}}}} |\n| Normal distribution | f
                                                                      (
                                                                      x
                                                                      \u2223
                                                                      \u03bc
                                                                      ,
                                                                      \u03c3
                                                                      2
                                                                      )
                                                                      =
                                                                      1
                                                                      2
                                                                      \u03c0
                                                                      \u03c3
                                                                      2
                                                                      e
                                                                      \u2212
                                                                      (
                                                                      x
                                                                      \u2212
                                                                      \u03bc
                                                                      )
                                                                      2
                                                                      2
                                                                      \u03c3
                                                                      2
                                                                      {\\displaystyle f\\left(x\\mid \\mu ,\\sigma ^{2}\\right)={\\frac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}e^{-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}} | \u03bc
                                                                      {\\displaystyle \\mu } | \u03c3
                                                                      2
                                                                      {\\displaystyle \\sigma ^{2}} |\n| Uniform distribution (continuous) | f
                                                                      (
                                                                      x
                                                                      \u2223
                                                                      a
                                                                      ,
                                                                      b
                                                                      )
                                                                      =
                                                                      {
                                                                      1
                                                                      b
                                                                      \u2212
                                                                      a
                                                                      for
                                                                      a
                                                                      \u2264
                                                                      x
                                                                      \u2264
                                                                      b
                                                                      ,
                                                                      0
                                                                      for
                                                                      x
                                                                      <
                                                                      a
                                                                      or
                                                                      x
                                                                      >
                                                                      b
                                                                      {\\displaystyle f(x\\mid a,b)={\\begin{cases}{\\frac {1}{b-a}}&{\\text{for }}a\\leq x\\leq b,\\\\[3pt]0&{\\text{for }}xb\\end{cases}}} | a
                                                                      +
                                                                      b
                                                                      2
                                                                      {\\displaystyle {\\frac {a+b}{2}}} | (
                                                                      b
                                                                      \u2212
                                                                      a
                                                                      )
                                                                      2
                                                                      12
                                                                      {\\displaystyle {\\frac {(b-a)^{2}}{12}}} |\n| Exponential distribution | f
                                                                      (
                                                                      x
                                                                      \u2223
                                                                      \u03bb
                                                                      )
                                                                      =
                                                                      \u03bb
                                                                      e
                                                                      \u2212
                                                                      \u03bb
                                                                      x
                                                                      {\\displaystyle f(x\\mid \\lambda )=\\lambda e^{-\\lambda x}} | 1
                                                                      \u03bb
                                                                      {\\displaystyle {\\frac {1}{\\lambda }}} | 1
                                                                      \u03bb
                                                                      2
                                                                      {\\displaystyle {\\frac {1}{\\lambda ^{2}}}} |\n| Poisson distribution | f
                                                                      (
                                                                      k
                                                                      \u2223
                                                                      \u03bb
                                                                      )
                                                                      =
                                                                      e
                                                                      \u2212
                                                                      \u03bb
                                                                      \u03bb
                                                                      k
                                                                      k
                                                                      !
                                                                      {\\displaystyle f(k\\mid \\lambda )={\\frac {e^{-\\lambda }\\lambda ^{k}}{k!}}} | \u03bb
                                                                      {\\displaystyle \\lambda } | \u03bb
                                                                      {\\displaystyle \\lambda } |\n\n## Properties\n\nedit\n\n### Basic properties\n\nedit\n\nVariance is non-negative because the squares are positive or zero:\n\n1. ${\\displaystyle \\operatorname {Var} (X)\\geq 0.}$\n\nThe variance of a constant is zero.\n\n1. ${\\displaystyle \\operatorname {Var} (a)=0.}$\n\nConversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value:\n\n1. ${\\displaystyle \\operatorname {Var} (X)=0\\iff \\exists a:P(X=a)=1.}$\n\n### Issues of finiteness\n\nedit\n\nIf a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index ${\\displaystyle k}$ satisfies ${\\displaystyle 1 0. This also holds in the multidimensional case.[ 3]\n\n### Units of measurement\n\nedit\n\nUnlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is\u221a 2.9 \u2248 1.7, slightly larger than the expected absolute deviation of 1.5.\n\nThe standard deviation and the expected absolute deviation can both be used as an indicator of the \"spread\" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution.\n\n## Propagation\n\nedit\n\n### Addition and multiplication by a constant\n\nedit\n\nVariance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged:\n\n1. ${\\displaystyle \\operatorname {Var} (X+a)=\\operatorname {Var} (X).}$\n\nIf all values are scaled by a constant, the variance is scaled by the square of that constant:\n\n1. ${\\displaystyle \\operatorname {Var} (aX)=a^{2}\\operatorname {Var} (X).}$\n\nThe variance of a sum of two random variables is given by\n\n1. ${\\displaystyle \\operatorname {Var} (aX+bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)+2ab\\,\\operatorname {Cov} (X,Y)}$\n\n1. ${\\displaystyle \\operatorname {Var} (aX-bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)-2ab\\,\\operatorname {Cov} (X,Y)}$\n\nwhere ${\\displaystyle \\operatorname {Cov} (X,Y)}$ is the covariance.\n\n### Linear combinations\n\nedit\n\nIn general, for the sum of ${\\displaystyle N}$ random variables ${\\displaystyle \\{X_{1},\\dots ,X_{N}\\}}$ , the variance becomes:\n\n1. ${\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i,j=1}^{N}\\operatorname {Cov} (X_{i},X_{j})=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i})+\\sum _{i\\neq j}\\operatorname {Cov} (X_{i},X_{j}),}$\n\nsee also general Bienaym\u00e9's identity.\n\nThese results lead to the variance of a linear combination as:\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {Var} \\left(\\sum _{i=1}^{N}a_{i}X_{i}\\right)&=\\sum _{i,j=1}^{N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+\\sum _{i\\not =j}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i\n\n\n\nVariance - Wikipedia\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
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                                                                      This article is about the mathematical concept. For other uses, see Variance (disambiguation).
                                                                      \n

                                                                      In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \n \n \n \n \n \u03c3\n \n 2\n \n \n \n \n {\\displaystyle \\sigma ^{2}}\n \n\"{\\displaystyle, \n \n \n \n \n s\n \n 2\n \n \n \n \n {\\displaystyle s^{2}}\n \n\"{\\displaystyle, \n \n \n \n Var\n \u2061\n (\n X\n )\n \n \n {\\displaystyle \\operatorname {Var} (X)}\n \n\"{\\displaystyle, \n \n \n \n V\n (\n X\n )\n \n \n {\\displaystyle V(X)}\n \n\"{\\displaystyle, or \n \n \n \n \n V\n \n (\n X\n )\n \n \n {\\displaystyle \\mathbb {V} (X)}\n \n\"{\\displaystyle.[1]\n

                                                                      Example of samples from two populations with the same mean but different variances. The red population has mean 100 and variance 100 (SD=10) while the blue population has mean 100 and variance 2500 (SD=50) where SD stands for Standard Deviation.
                                                                      \n

                                                                      An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.\n

                                                                      There are two distinct concepts that are both called \"variance\". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.\n

                                                                      The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling.\n

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                                                                      Contents

                                                                      \n\n
                                                                      \n
                                                                      \n
                                                                      Geometric visualisation of the variance of an arbitrary distribution (2, 4, 4, 4, 5, 5, 7, 9):
                                                                      1. A frequency distribution is constructed.
                                                                      2. The centroid of the distribution gives its mean.
                                                                      3. A square with sides equal to the difference of each value from the mean is formed for each value.
                                                                      4. Arranging the squares into a rectangle with one side equal to the number of values, n, results in the other side being the distribution's variance, \u03c32.
                                                                      \n

                                                                      Definition

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      The variance of a random variable \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  is the expected value of the squared deviation from the mean of \n \n \n \n X\n \n \n {\\displaystyle X}\n \n , \n \n \n \n \u03bc\n =\n E\n \u2061\n [\n X\n ]\n \n \n {\\displaystyle \\mu =\\operatorname {E} [X]}\n \n :\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n E\n \u2061\n \n [\n \n (\n X\n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n ]\n \n .\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right].}\n \n 
                                                                      \n

                                                                      This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n Cov\n \u2061\n (\n X\n ,\n X\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\operatorname {Cov} (X,X).}\n \n 
                                                                      \n

                                                                      The variance is also equivalent to the second cumulant of a probability distribution that generates \n \n \n \n X\n \n \n {\\displaystyle X}\n \n . The variance is typically designated as \n \n \n \n Var\n \u2061\n (\n X\n )\n \n \n {\\displaystyle \\operatorname {Var} (X)}\n \n , or sometimes as \n \n \n \n V\n (\n X\n )\n \n \n {\\displaystyle V(X)}\n \n  or \n \n \n \n \n V\n \n (\n X\n )\n \n \n {\\displaystyle \\mathbb {V} (X)}\n \n , or symbolically as \n \n \n \n \n \u03c3\n \n X\n \n \n 2\n \n \n \n \n {\\displaystyle \\sigma _{X}^{2}}\n \n  or simply \n \n \n \n \n \u03c3\n \n 2\n \n \n \n \n {\\displaystyle \\sigma ^{2}}\n \n  (pronounced \"sigma squared\"). The expression for the variance can be expanded as follows:\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n )\n \n \n \n =\n E\n \u2061\n \n [\n \n (\n X\n \u2212\n E\n \u2061\n [\n X\n ]\n \n )\n \n 2\n \n \n \n ]\n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n \n X\n \n 2\n \n \n \u2212\n 2\n X\n E\n \u2061\n [\n X\n ]\n +\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n \n ]\n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n \u2212\n 2\n E\n \u2061\n [\n X\n ]\n E\n \u2061\n [\n X\n ]\n +\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n \u2212\n 2\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n +\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n \u2212\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left[(X-\\operatorname {E} [X])^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}-2X\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]^{2}+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}\\end{aligned}}}\n \n 
                                                                      \n

                                                                      In other words, the variance of X is equal to the mean of the square of X minus the square of the mean of X. This equation should not be used for computations using floating point arithmetic, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see algorithms for calculating variance.\n

                                                                      \n

                                                                      Discrete random variable

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      If the generator of random variable \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  is discrete with probability mass function \n \n \n \n \n x\n \n 1\n \n \n \u21a6\n \n p\n \n 1\n \n \n ,\n \n x\n \n 2\n \n \n \u21a6\n \n p\n \n 2\n \n \n ,\n \u2026\n ,\n \n x\n \n n\n \n \n \u21a6\n \n p\n \n n\n \n \n \n \n {\\displaystyle x_{1}\\mapsto p_{1},x_{2}\\mapsto p_{2},\\ldots ,x_{n}\\mapsto p_{n}}\n \n , then\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n p\n \n i\n \n \n \u22c5\n (\n \n x\n \n i\n \n \n \u2212\n \u03bc\n \n )\n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\sum _{i=1}^{n}p_{i}\\cdot (x_{i}-\\mu )^{2},}\n \n 
                                                                      \n

                                                                      where \n \n \n \n \u03bc\n \n \n {\\displaystyle \\mu }\n \n  is the expected value. That is,\n

                                                                      \n
                                                                      \n \n \n \n \u03bc\n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n p\n \n i\n \n \n \n x\n \n i\n \n \n .\n \n \n {\\displaystyle \\mu =\\sum _{i=1}^{n}p_{i}x_{i}.}\n \n 
                                                                      \n

                                                                      (When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)\n

                                                                      The variance of a collection of \n \n \n \n n\n \n \n {\\displaystyle n}\n \n  equally likely values can be written as\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n (\n \n x\n \n i\n \n \n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n \n {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n}}\\sum _{i=1}^{n}(x_{i}-\\mu )^{2}}\n \n 
                                                                      \n

                                                                      where \n \n \n \n \u03bc\n \n \n {\\displaystyle \\mu }\n \n  is the average value. That is,\n

                                                                      \n
                                                                      \n \n \n \n \u03bc\n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n x\n \n i\n \n \n .\n \n \n {\\displaystyle \\mu ={\\frac {1}{n}}\\sum _{i=1}^{n}x_{i}.}\n \n 
                                                                      \n

                                                                      The variance of a set of \n \n \n \n n\n \n \n {\\displaystyle n}\n \n  equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:[2]\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n \n 1\n 2\n \n \n (\n \n x\n \n i\n \n \n \u2212\n \n x\n \n j\n \n \n \n )\n \n 2\n \n \n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n i\n \n \n \n \u2211\n \n j\n >\n i\n \n \n (\n \n x\n \n i\n \n \n \u2212\n \n x\n \n j\n \n \n \n )\n \n 2\n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{\\frac {1}{2}}(x_{i}-x_{j})^{2}={\\frac {1}{n^{2}}}\\sum _{i}\\sum _{j>i}(x_{i}-x_{j})^{2}.}\n \n 
                                                                      \n

                                                                      Absolutely continuous random variable

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      If the random variable \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  has a probability density function \n \n \n \n f\n (\n x\n )\n \n \n {\\displaystyle f(x)}\n \n , and \n \n \n \n F\n (\n x\n )\n \n \n {\\displaystyle F(x)}\n \n  is the corresponding cumulative distribution function, then\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \u03c3\n \n 2\n \n \n \n \n \n =\n \n \u222b\n \n \n R\n \n \n \n (\n x\n \u2212\n \u03bc\n \n )\n \n 2\n \n \n f\n (\n x\n )\n \n d\n x\n \n \n \n \n \n \n =\n \n \u222b\n \n \n R\n \n \n \n \n x\n \n 2\n \n \n f\n (\n x\n )\n \n d\n x\n \u2212\n 2\n \u03bc\n \n \u222b\n \n \n R\n \n \n \n x\n f\n (\n x\n )\n \n d\n x\n +\n \n \u03bc\n \n 2\n \n \n \n \u222b\n \n \n R\n \n \n \n f\n (\n x\n )\n \n d\n x\n \n \n \n \n \n \n =\n \n \u222b\n \n \n R\n \n \n \n \n x\n \n 2\n \n \n \n d\n F\n (\n x\n )\n \u2212\n 2\n \u03bc\n \n \u222b\n \n \n R\n \n \n \n x\n \n d\n F\n (\n x\n )\n +\n \n \u03bc\n \n 2\n \n \n \n \u222b\n \n \n R\n \n \n \n \n d\n F\n (\n x\n )\n \n \n \n \n \n \n =\n \n \u222b\n \n \n R\n \n \n \n \n x\n \n 2\n \n \n \n d\n F\n (\n x\n )\n \u2212\n 2\n \u03bc\n \u22c5\n \u03bc\n +\n \n \u03bc\n \n 2\n \n \n \u22c5\n 1\n \n \n \n \n \n \n =\n \n \u222b\n \n \n R\n \n \n \n \n x\n \n 2\n \n \n \n d\n F\n (\n x\n )\n \u2212\n \n \u03bc\n \n 2\n \n \n ,\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)=\\sigma ^{2}&=\\int _{\\mathbb {R} }(x-\\mu )^{2}f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-2\\mu \\int _{\\mathbb {R} }xf(x)\\,dx+\\mu ^{2}\\int _{\\mathbb {R} }f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\int _{\\mathbb {R} }x\\,dF(x)+\\mu ^{2}\\int _{\\mathbb {R} }\\,dF(x)\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\cdot \\mu +\\mu ^{2}\\cdot 1\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-\\mu ^{2},\\end{aligned}}}\n \n 
                                                                      \n

                                                                      or equivalently,\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \u222b\n \n \n R\n \n \n \n \n x\n \n 2\n \n \n f\n (\n x\n )\n \n d\n x\n \u2212\n \n \u03bc\n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-\\mu ^{2},}\n \n 
                                                                      \n

                                                                      where \n \n \n \n \u03bc\n \n \n {\\displaystyle \\mu }\n \n  is the expected value of \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  given by\n

                                                                      \n
                                                                      \n \n \n \n \u03bc\n =\n \n \u222b\n \n \n R\n \n \n \n x\n f\n (\n x\n )\n \n d\n x\n =\n \n \u222b\n \n \n R\n \n \n \n x\n \n d\n F\n (\n x\n )\n .\n \n \n {\\displaystyle \\mu =\\int _{\\mathbb {R} }xf(x)\\,dx=\\int _{\\mathbb {R} }x\\,dF(x).}\n \n 
                                                                      \n

                                                                      In these formulas, the integrals with respect to \n \n \n \n d\n x\n \n \n {\\displaystyle dx}\n \n  and \n \n \n \n d\n F\n (\n x\n )\n \n \n {\\displaystyle dF(x)}\n \n \nare Lebesgue and Lebesgue\u2013Stieltjes integrals, respectively.\n

                                                                      If the function \n \n \n \n \n x\n \n 2\n \n \n f\n (\n x\n )\n \n \n {\\displaystyle x^{2}f(x)}\n \n  is Riemann-integrable on every finite interval \n \n \n \n [\n a\n ,\n b\n ]\n \u2282\n \n R\n \n ,\n \n \n {\\displaystyle [a,b]\\subset \\mathbb {R} ,}\n \n  then\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \u222b\n \n \u2212\n \u221e\n \n \n +\n \u221e\n \n \n \n x\n \n 2\n \n \n f\n (\n x\n )\n \n d\n x\n \u2212\n \n \u03bc\n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\int _{-\\infty }^{+\\infty }x^{2}f(x)\\,dx-\\mu ^{2},}\n \n 
                                                                      \n

                                                                      where the integral is an improper Riemann integral.\n

                                                                      \n

                                                                      Examples

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      Exponential distribution

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      The exponential distribution with parameter \u03bb is a continuous distribution whose probability density function is given by\n

                                                                      \n
                                                                      \n \n \n \n f\n (\n x\n )\n =\n \u03bb\n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n \n {\\displaystyle f(x)=\\lambda e^{-\\lambda x}}\n \n 
                                                                      \n

                                                                      on the interval [0, \u221e). Its mean can be shown to be\n

                                                                      \n
                                                                      \n \n \n \n E\n \u2061\n [\n X\n ]\n =\n \n \u222b\n \n 0\n \n \n \u221e\n \n \n x\n \u03bb\n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n d\n x\n =\n \n \n 1\n \u03bb\n \n \n .\n \n \n {\\displaystyle \\operatorname {E} [X]=\\int _{0}^{\\infty }x\\lambda e^{-\\lambda x}\\,dx={\\frac {1}{\\lambda }}.}\n \n 
                                                                      \n

                                                                      Using integration by parts and making use of the expected value already calculated, we have:\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n \n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n \n \n \n =\n \n \u222b\n \n 0\n \n \n \u221e\n \n \n \n x\n \n 2\n \n \n \u03bb\n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n d\n x\n \n \n \n \n \n \n =\n \n \n [\n \n \u2212\n \n x\n \n 2\n \n \n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n ]\n \n \n 0\n \n \n \u221e\n \n \n +\n \n \u222b\n \n 0\n \n \n \u221e\n \n \n 2\n x\n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n d\n x\n \n \n \n \n \n \n =\n 0\n +\n \n \n 2\n \u03bb\n \n \n E\n \u2061\n [\n X\n ]\n \n \n \n \n \n \n =\n \n \n 2\n \n \u03bb\n \n 2\n \n \n \n \n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {E} \\left[X^{2}\\right]&=\\int _{0}^{\\infty }x^{2}\\lambda e^{-\\lambda x}\\,dx\\\\&=\\left[-x^{2}e^{-\\lambda x}\\right]_{0}^{\\infty }+\\int _{0}^{\\infty }2xe^{-\\lambda x}\\,dx\\\\&=0+{\\frac {2}{\\lambda }}\\operatorname {E} [X]\\\\&={\\frac {2}{\\lambda ^{2}}}.\\end{aligned}}}\n \n 
                                                                      \n

                                                                      Thus, the variance of X is given by\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n \u2212\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n =\n \n \n 2\n \n \u03bb\n \n 2\n \n \n \n \n \u2212\n \n \n (\n \n \n 1\n \u03bb\n \n \n )\n \n \n 2\n \n \n =\n \n \n 1\n \n \u03bb\n \n 2\n \n \n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}={\\frac {2}{\\lambda ^{2}}}-\\left({\\frac {1}{\\lambda }}\\right)^{2}={\\frac {1}{\\lambda ^{2}}}.}\n \n 
                                                                      \n

                                                                      Fair dice

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      A fair six-sided dice can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is \n \n \n \n (\n 1\n +\n 2\n +\n 3\n +\n 4\n +\n 5\n +\n 6\n )\n \n /\n \n 6\n =\n 7\n \n /\n \n 2.\n \n \n {\\displaystyle (1+2+3+4+5+6)/6=7/2.}\n \n  Therefore, the variance of X is\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n )\n \n \n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n 6\n \n \n \n \n 1\n 6\n \n \n \n \n (\n \n i\n \u2212\n \n \n 7\n 2\n \n \n \n )\n \n \n 2\n \n \n \n \n \n \n \n \n =\n \n \n 1\n 6\n \n \n \n (\n \n (\n \u2212\n 5\n \n /\n \n 2\n \n )\n \n 2\n \n \n +\n (\n \u2212\n 3\n \n /\n \n 2\n \n )\n \n 2\n \n \n +\n (\n \u2212\n 1\n \n /\n \n 2\n \n )\n \n 2\n \n \n +\n (\n 1\n \n /\n \n 2\n \n )\n \n 2\n \n \n +\n (\n 3\n \n /\n \n 2\n \n )\n \n 2\n \n \n +\n (\n 5\n \n /\n \n 2\n \n )\n \n 2\n \n \n \n )\n \n \n \n \n \n \n \n =\n \n \n 35\n 12\n \n \n \u2248\n 2.92.\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\sum _{i=1}^{6}{\\frac {1}{6}}\\left(i-{\\frac {7}{2}}\\right)^{2}\\\\[5pt]&={\\frac {1}{6}}\\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\\right)\\\\[5pt]&={\\frac {35}{12}}\\approx 2.92.\\end{aligned}}}\n \n 
                                                                      \n

                                                                      The general formula for the variance of the outcome, X, of an n-sided die is\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n )\n \n \n \n =\n E\n \u2061\n \n (\n \n X\n \n 2\n \n \n )\n \n \u2212\n (\n E\n \u2061\n (\n X\n )\n \n )\n \n 2\n \n \n \n \n \n \n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n i\n \n 2\n \n \n \u2212\n \n \n (\n \n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n i\n \n )\n \n \n 2\n \n \n \n \n \n \n \n \n =\n \n \n \n (\n n\n +\n 1\n )\n (\n 2\n n\n +\n 1\n )\n \n 6\n \n \n \u2212\n \n \n (\n \n \n \n n\n +\n 1\n \n 2\n \n \n )\n \n \n 2\n \n \n \n \n \n \n \n \n =\n \n \n \n \n n\n \n 2\n \n \n \u2212\n 1\n \n 12\n \n \n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left(X^{2}\\right)-(\\operatorname {E} (X))^{2}\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}i^{2}-\\left({\\frac {1}{n}}\\sum _{i=1}^{n}i\\right)^{2}\\\\[5pt]&={\\frac {(n+1)(2n+1)}{6}}-\\left({\\frac {n+1}{2}}\\right)^{2}\\\\[4pt]&={\\frac {n^{2}-1}{12}}.\\end{aligned}}}\n \n 
                                                                      \n

                                                                      Commonly used probability distributions

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      The following table lists the variance for some commonly used probability distributions.\n

                                                                      \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                                                      Name of the probability distribution\nProbability distribution function\nMean\nVariance\n
                                                                      Binomial distribution\n\n \n \n \n Pr\n \n (\n X\n =\n k\n )\n =\n \n \n \n (\n \n \n n\n k\n \n \n )\n \n \n \n \n p\n \n k\n \n \n (\n 1\n \u2212\n p\n \n )\n \n n\n \u2212\n k\n \n \n \n \n {\\displaystyle \\Pr \\,(X=k)={\\binom {n}{k}}p^{k}(1-p)^{n-k}}\n \n \n\n \n \n \n n\n p\n \n \n {\\displaystyle np}\n \n \n\n \n \n \n n\n p\n (\n 1\n \u2212\n p\n )\n \n \n {\\displaystyle np(1-p)}\n \n \n
                                                                      Geometric distribution\n\n \n \n \n Pr\n \n (\n X\n =\n k\n )\n =\n (\n 1\n \u2212\n p\n \n )\n \n k\n \u2212\n 1\n \n \n p\n \n \n {\\displaystyle \\Pr \\,(X=k)=(1-p)^{k-1}p}\n \n \n\n \n \n \n \n \n 1\n p\n \n \n \n \n {\\displaystyle {\\frac {1}{p}}}\n \n \n\n \n \n \n \n \n \n (\n 1\n \u2212\n p\n )\n \n \n p\n \n 2\n \n \n \n \n \n \n {\\displaystyle {\\frac {(1-p)}{p^{2}}}}\n \n \n
                                                                      Normal distribution\n\n \n \n \n f\n \n (\n \n x\n \u2223\n \u03bc\n ,\n \n \u03c3\n \n 2\n \n \n \n )\n \n =\n \n \n 1\n \n 2\n \u03c0\n \n \u03c3\n \n 2\n \n \n \n \n \n \n e\n \n \u2212\n \n \n \n (\n x\n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n \n 2\n \n \u03c3\n \n 2\n \n \n \n \n \n \n \n \n \n {\\displaystyle f\\left(x\\mid \\mu ,\\sigma ^{2}\\right)={\\frac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}e^{-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}}\n \n \n\n \n \n \n \u03bc\n \n \n {\\displaystyle \\mu }\n \n \n\n \n \n \n \n \u03c3\n \n 2\n \n \n \n \n {\\displaystyle \\sigma ^{2}}\n \n \n
                                                                      Uniform distribution (continuous)\n\n \n \n \n f\n (\n x\n \u2223\n a\n ,\n b\n )\n =\n \n \n {\n \n \n \n \n \n 1\n \n b\n \u2212\n a\n \n \n \n \n \n \n for \n \n a\n \u2264\n x\n \u2264\n b\n ,\n \n \n \n \n 0\n \n \n \n for \n \n x\n <\n a\n \n or \n \n x\n >\n b\n \n \n \n \n \n \n \n \n {\\displaystyle f(x\\mid a,b)={\\begin{cases}{\\frac {1}{b-a}}&{\\text{for }}a\\leq x\\leq b,\\\\[3pt]0&{\\text{for }}x<a{\\text{ or }}x>b\\end{cases}}}\n \n \n\n \n \n \n \n \n \n a\n +\n b\n \n 2\n \n \n \n \n {\\displaystyle {\\frac {a+b}{2}}}\n \n \n\n \n \n \n \n \n \n (\n b\n \u2212\n a\n \n )\n \n 2\n \n \n \n 12\n \n \n \n \n {\\displaystyle {\\frac {(b-a)^{2}}{12}}}\n \n \n
                                                                      Exponential distribution\n\n \n \n \n f\n (\n x\n \u2223\n \u03bb\n )\n =\n \u03bb\n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n \n {\\displaystyle f(x\\mid \\lambda )=\\lambda e^{-\\lambda x}}\n \n \n\n \n \n \n \n \n 1\n \u03bb\n \n \n \n \n {\\displaystyle {\\frac {1}{\\lambda }}}\n \n \n\n \n \n \n \n \n 1\n \n \u03bb\n \n 2\n \n \n \n \n \n \n {\\displaystyle {\\frac {1}{\\lambda ^{2}}}}\n \n \n
                                                                      Poisson distribution\n\n \n \n \n f\n (\n k\n \u2223\n \u03bb\n )\n =\n \n \n \n \n e\n \n \u2212\n \u03bb\n \n \n \n \u03bb\n \n k\n \n \n \n \n k\n !\n \n \n \n \n \n {\\displaystyle f(k\\mid \\lambda )={\\frac {e^{-\\lambda }\\lambda ^{k}}{k!}}}\n \n \n\n \n \n \n \u03bb\n \n \n {\\displaystyle \\lambda }\n \n \n\n \n \n \n \u03bb\n \n \n {\\displaystyle \\lambda }\n \n \n
                                                                      \n

                                                                      Properties

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                                                                      \n

                                                                      Basic properties

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                                                                      Variance is non-negative because the squares are positive or zero:\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n )\n \u2265\n 0.\n \n \n {\\displaystyle \\operatorname {Var} (X)\\geq 0.}\n \n 
                                                                      \n

                                                                      The variance of a constant is zero.\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n a\n )\n =\n 0.\n \n \n {\\displaystyle \\operatorname {Var} (a)=0.}\n \n 
                                                                      \n

                                                                      Conversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value:\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n 0\n \n \u27fa\n \n \u2203\n a\n :\n P\n (\n X\n =\n a\n )\n =\n 1.\n \n \n {\\displaystyle \\operatorname {Var} (X)=0\\iff \\exists a:P(X=a)=1.}\n \n 
                                                                      \n

                                                                      Issues of finiteness

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                                                                      \n

                                                                      If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index \n \n \n \n k\n \n \n {\\displaystyle k}\n \n  satisfies \n \n \n \n 1\n <\n k\n \u2264\n 2.\n \n \n {\\displaystyle 1<k\\leq 2.}\n \n \n

                                                                      \n

                                                                      Decomposition

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                                                                      The general formula for variance decomposition or the law of total variance is: If \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  and \n \n \n \n Y\n \n \n {\\displaystyle Y}\n \n  are two random variables, and the variance of \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  exists, then\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n [\n X\n ]\n =\n E\n \u2061\n (\n Var\n \u2061\n [\n X\n \u2223\n Y\n ]\n )\n +\n Var\n \u2061\n (\n E\n \u2061\n [\n X\n \u2223\n Y\n ]\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} [X]=\\operatorname {E} (\\operatorname {Var} [X\\mid Y])+\\operatorname {Var} (\\operatorname {E} [X\\mid Y]).}\n \n 
                                                                      \n

                                                                      The conditional expectation \n \n \n \n E\n \u2061\n (\n X\n \u2223\n Y\n )\n \n \n {\\displaystyle \\operatorname {E} (X\\mid Y)}\n \n  of \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  given \n \n \n \n Y\n \n \n {\\displaystyle Y}\n \n , and the conditional variance \n \n \n \n Var\n \u2061\n (\n X\n \u2223\n Y\n )\n \n \n {\\displaystyle \\operatorname {Var} (X\\mid Y)}\n \n  may be understood as follows. Given any particular value y of the random variable Y, there is a conditional expectation \n \n \n \n E\n \u2061\n (\n X\n \u2223\n Y\n =\n y\n )\n \n \n {\\displaystyle \\operatorname {E} (X\\mid Y=y)}\n \n  given the event Y = y. This quantity depends on the particular value y; it is a function \n \n \n \n g\n (\n y\n )\n =\n E\n \u2061\n (\n X\n \u2223\n Y\n =\n y\n )\n \n \n {\\displaystyle g(y)=\\operatorname {E} (X\\mid Y=y)}\n \n . That same function evaluated at the random variable Y is the conditional expectation \n \n \n \n E\n \u2061\n (\n X\n \u2223\n Y\n )\n =\n g\n (\n Y\n )\n .\n \n \n {\\displaystyle \\operatorname {E} (X\\mid Y)=g(Y).}\n \n \n

                                                                      In particular, if \n \n \n \n Y\n \n \n {\\displaystyle Y}\n \n  is a discrete random variable assuming possible values \n \n \n \n \n y\n \n 1\n \n \n ,\n \n y\n \n 2\n \n \n ,\n \n y\n \n 3\n \n \n \u2026\n \n \n {\\displaystyle y_{1},y_{2},y_{3}\\ldots }\n \n  with corresponding probabilities \n \n \n \n \n p\n \n 1\n \n \n ,\n \n p\n \n 2\n \n \n ,\n \n p\n \n 3\n \n \n \u2026\n ,\n \n \n {\\displaystyle p_{1},p_{2},p_{3}\\ldots ,}\n \n , then in the formula for total variance, the first term on the right-hand side becomes\n

                                                                      \n
                                                                      \n \n \n \n E\n \u2061\n (\n Var\n \u2061\n [\n X\n \u2223\n Y\n ]\n )\n =\n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03c3\n \n i\n \n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {E} (\\operatorname {Var} [X\\mid Y])=\\sum _{i}p_{i}\\sigma _{i}^{2},}\n \n 
                                                                      \n

                                                                      where \n \n \n \n \n \u03c3\n \n i\n \n \n 2\n \n \n =\n Var\n \u2061\n [\n X\n \u2223\n Y\n =\n \n y\n \n i\n \n \n ]\n \n \n {\\displaystyle \\sigma _{i}^{2}=\\operatorname {Var} [X\\mid Y=y_{i}]}\n \n . Similarly, the second term on the right-hand side becomes\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n E\n \u2061\n [\n X\n \u2223\n Y\n ]\n )\n =\n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03bc\n \n i\n \n \n 2\n \n \n \u2212\n \n \n (\n \n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03bc\n \n i\n \n \n \n )\n \n \n 2\n \n \n =\n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03bc\n \n i\n \n \n 2\n \n \n \u2212\n \n \u03bc\n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {Var} (\\operatorname {E} [X\\mid Y])=\\sum _{i}p_{i}\\mu _{i}^{2}-\\left(\\sum _{i}p_{i}\\mu _{i}\\right)^{2}=\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2},}\n \n 
                                                                      \n

                                                                      where \n \n \n \n \n \u03bc\n \n i\n \n \n =\n E\n \u2061\n [\n X\n \u2223\n Y\n =\n \n y\n \n i\n \n \n ]\n \n \n {\\displaystyle \\mu _{i}=\\operatorname {E} [X\\mid Y=y_{i}]}\n \n  and \n \n \n \n \u03bc\n =\n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03bc\n \n i\n \n \n \n \n {\\displaystyle \\mu =\\sum _{i}p_{i}\\mu _{i}}\n \n . Thus the total variance is given by\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n [\n X\n ]\n =\n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03c3\n \n i\n \n \n 2\n \n \n +\n \n (\n \n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03bc\n \n i\n \n \n 2\n \n \n \u2212\n \n \u03bc\n \n 2\n \n \n \n )\n \n .\n \n \n {\\displaystyle \\operatorname {Var} [X]=\\sum _{i}p_{i}\\sigma _{i}^{2}+\\left(\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2}\\right).}\n \n 
                                                                      \n

                                                                      A similar formula is applied in analysis of variance, where the corresponding formula is\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n M\n S\n \n \n \n total\n \n \n =\n \n \n \n M\n S\n \n \n \n between\n \n \n +\n \n \n \n M\n S\n \n \n \n within\n \n \n ;\n \n \n {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{between}}+{\\mathit {MS}}_{\\text{within}};}\n \n 
                                                                      \n

                                                                      here \n \n \n \n \n \n M\n S\n \n \n \n \n {\\displaystyle {\\mathit {MS}}}\n \n  refers to the Mean of the Squares. In linear regression analysis the corresponding formula is\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n M\n S\n \n \n \n total\n \n \n =\n \n \n \n M\n S\n \n \n \n regression\n \n \n +\n \n \n \n M\n S\n \n \n \n residual\n \n \n .\n \n \n {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{regression}}+{\\mathit {MS}}_{\\text{residual}}.}\n \n 
                                                                      \n

                                                                      This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.\n

                                                                      Similar decompositions are possible for the sum of squared deviations (sum of squares, \n \n \n \n \n \n S\n S\n \n \n \n \n {\\displaystyle {\\mathit {SS}}}\n \n ):\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n S\n S\n \n \n \n total\n \n \n =\n \n \n \n S\n S\n \n \n \n between\n \n \n +\n \n \n \n S\n S\n \n \n \n within\n \n \n ,\n \n \n {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{between}}+{\\mathit {SS}}_{\\text{within}},}\n \n 
                                                                      \n
                                                                      \n \n \n \n \n \n \n S\n S\n \n \n \n total\n \n \n =\n \n \n \n S\n S\n \n \n \n regression\n \n \n +\n \n \n \n S\n S\n \n \n \n residual\n \n \n .\n \n \n {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{regression}}+{\\mathit {SS}}_{\\text{residual}}.}\n \n 
                                                                      \n

                                                                      Calculation from the CDF

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                                                                      The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution function F using\n

                                                                      \n
                                                                      \n \n \n \n 2\n \n \u222b\n \n 0\n \n \n \u221e\n \n \n u\n (\n 1\n \u2212\n F\n (\n u\n )\n )\n \n d\n u\n \u2212\n \n \n (\n \n \n \u222b\n \n 0\n \n \n \u221e\n \n \n (\n 1\n \u2212\n F\n (\n u\n )\n )\n \n d\n u\n \n )\n \n \n 2\n \n \n .\n \n \n {\\displaystyle 2\\int _{0}^{\\infty }u(1-F(u))\\,du-\\left(\\int _{0}^{\\infty }(1-F(u))\\,du\\right)^{2}.}\n \n 
                                                                      \n

                                                                      This expression can be used to calculate the variance in situations where the CDF, but not the density, can be conveniently expressed.\n

                                                                      \n

                                                                      Characteristic property

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                                                                      The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. \n \n \n \n \n \n a\n r\n g\n m\n i\n n\n \n \n m\n \n \n \n \n E\n \n \n (\n \n \n (\n \n X\n \u2212\n m\n \n )\n \n \n 2\n \n \n )\n \n =\n \n E\n \n (\n X\n )\n \n \n {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} \\left(\\left(X-m\\right)^{2}\\right)=\\mathrm {E} (X)}\n \n . Conversely, if a continuous function \n \n \n \n \u03c6\n \n \n {\\displaystyle \\varphi }\n \n  satisfies \n \n \n \n \n \n a\n r\n g\n m\n i\n n\n \n \n m\n \n \n \n \n E\n \n (\n \u03c6\n (\n X\n \u2212\n m\n )\n )\n =\n \n E\n \n (\n X\n )\n \n \n {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} (\\varphi (X-m))=\\mathrm {E} (X)}\n \n  for all random variables X, then it is necessarily of the form \n \n \n \n \u03c6\n (\n x\n )\n =\n a\n \n x\n \n 2\n \n \n +\n b\n \n \n {\\displaystyle \\varphi (x)=ax^{2}+b}\n \n , where a > 0. This also holds in the multidimensional case.[3]\n

                                                                      \n

                                                                      Units of measurement

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                                                                      Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is \u221a2.9 \u2248 1.7, slightly larger than the expected absolute deviation of 1.5.\n

                                                                      The standard deviation and the expected absolute deviation can both be used as an indicator of the \"spread\" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution.\n

                                                                      \n

                                                                      Propagation

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                                                                      Addition and multiplication by a constant

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                                                                      Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged:\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n +\n a\n )\n =\n Var\n \u2061\n (\n X\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (X+a)=\\operatorname {Var} (X).}\n \n 
                                                                      \n

                                                                      If all values are scaled by a constant, the variance is scaled by the square of that constant:\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n a\n X\n )\n =\n \n a\n \n 2\n \n \n Var\n \u2061\n (\n X\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (aX)=a^{2}\\operatorname {Var} (X).}\n \n 
                                                                      \n

                                                                      The variance of a sum of two random variables is given by\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n a\n X\n +\n b\n Y\n )\n =\n \n a\n \n 2\n \n \n Var\n \u2061\n (\n X\n )\n +\n \n b\n \n 2\n \n \n Var\n \u2061\n (\n Y\n )\n +\n 2\n a\n b\n \n Cov\n \u2061\n (\n X\n ,\n Y\n )\n \n \n {\\displaystyle \\operatorname {Var} (aX+bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)+2ab\\,\\operatorname {Cov} (X,Y)}\n \n 
                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n a\n X\n \u2212\n b\n Y\n )\n =\n \n a\n \n 2\n \n \n Var\n \u2061\n (\n X\n )\n +\n \n b\n \n 2\n \n \n Var\n \u2061\n (\n Y\n )\n \u2212\n 2\n a\n b\n \n Cov\n \u2061\n (\n X\n ,\n Y\n )\n \n \n {\\displaystyle \\operatorname {Var} (aX-bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)-2ab\\,\\operatorname {Cov} (X,Y)}\n \n 
                                                                      \n

                                                                      where \n \n \n \n Cov\n \u2061\n (\n X\n ,\n Y\n )\n \n \n {\\displaystyle \\operatorname {Cov} (X,Y)}\n \n  is the covariance.\n

                                                                      \n

                                                                      Linear combinations

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                                                                      In general, for the sum of \n \n \n \n N\n \n \n {\\displaystyle N}\n \n  random variables \n \n \n \n {\n \n X\n \n 1\n \n \n ,\n \u2026\n ,\n \n X\n \n N\n \n \n }\n \n \n {\\displaystyle \\{X_{1},\\dots ,X_{N}\\}}\n \n , the variance becomes:\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \u2211\n \n i\n ,\n j\n =\n 1\n \n \n N\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n =\n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n +\n \n \u2211\n \n i\n \u2260\n j\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n ,\n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i,j=1}^{N}\\operatorname {Cov} (X_{i},X_{j})=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i})+\\sum _{i\\neq j}\\operatorname {Cov} (X_{i},X_{j}),}\n \n 
                                                                      \n

                                                                      see also general Bienaym\u00e9's identity.\n

                                                                      These results lead to the variance of a linear combination as:\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n a\n \n i\n \n \n \n X\n \n i\n \n \n \n )\n \n \n \n \n =\n \n \u2211\n \n i\n ,\n j\n =\n 1\n \n \n N\n \n \n \n a\n \n i\n \n \n \n a\n \n j\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n \n \n \n \n \n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n a\n \n i\n \n \n 2\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n +\n \n \u2211\n \n i\n \u2260\n j\n \n \n \n a\n \n i\n \n \n \n a\n \n j\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n \n \n \n \n \n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n a\n \n i\n \n \n 2\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n +\n 2\n \n \u2211\n \n 1\n \u2264\n i\n <\n j\n \u2264\n N\n \n \n \n a\n \n i\n \n \n \n a\n \n j\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} \\left(\\sum _{i=1}^{N}a_{i}X_{i}\\right)&=\\sum _{i,j=1}^{N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+\\sum _{i\\not =j}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i<j\\leq N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j}).\\end{aligned}}}\n \n 
                                                                      \n

                                                                      If the random variables \n \n \n \n \n X\n \n 1\n \n \n ,\n \u2026\n ,\n \n X\n \n N\n \n \n \n \n {\\displaystyle X_{1},\\dots ,X_{N}}\n \n  are such that\n

                                                                      \n
                                                                      \n \n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n =\n 0\n \n ,\n \n \u2200\n \n (\n i\n \u2260\n j\n )\n ,\n \n \n {\\displaystyle \\operatorname {Cov} (X_{i},X_{j})=0\\ ,\\ \\forall \\ (i\\neq j),}\n \n 
                                                                      \n

                                                                      then they are said to be uncorrelated. It follows immediately from the expression given earlier that if the random variables \n \n \n \n \n X\n \n 1\n \n \n ,\n \u2026\n ,\n \n X\n \n N\n \n \n \n \n {\\displaystyle X_{1},\\dots ,X_{N}}\n \n  are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n .\n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i}).}\n \n 
                                                                      \n

                                                                      Since independent random variables are always uncorrelated (see Covariance \u00a7 Uncorrelatedness and independence), the equation above holds in particular when the random variables \n \n \n \n \n X\n \n 1\n \n \n ,\n \u2026\n ,\n \n X\n \n n\n \n \n \n \n {\\displaystyle X_{1},\\dots ,X_{n}}\n \n  are independent. Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.\n

                                                                      \n

                                                                      Matrix notation for the variance of a linear combination

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      Define \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  as a column vector of \n \n \n \n n\n \n \n {\\displaystyle n}\n \n  random variables \n \n \n \n \n X\n \n 1\n \n \n ,\n \u2026\n ,\n \n X\n \n n\n \n \n \n \n {\\displaystyle X_{1},\\ldots ,X_{n}}\n \n , and \n \n \n \n c\n \n \n {\\displaystyle c}\n \n  as a column vector of \n \n \n \n n\n \n \n {\\displaystyle n}\n \n  scalars \n \n \n \n \n c\n \n 1\n \n \n ,\n \u2026\n ,\n \n c\n \n n\n \n \n \n \n {\\displaystyle c_{1},\\ldots ,c_{n}}\n \n . Therefore, \n \n \n \n \n c\n \n \n T\n \n \n \n X\n \n \n {\\displaystyle c^{\\mathsf {T}}X}\n \n  is a linear combination of these random variables, where \n \n \n \n \n c\n \n \n T\n \n \n \n \n \n {\\displaystyle c^{\\mathsf {T}}}\n \n  denotes the transpose of \n \n \n \n c\n \n \n {\\displaystyle c}\n \n . Also let \n \n \n \n \u03a3\n \n \n {\\displaystyle \\Sigma }\n \n  be the covariance matrix of \n \n \n \n X\n \n \n {\\displaystyle X}\n \n . The variance of \n \n \n \n \n c\n \n \n T\n \n \n \n X\n \n \n {\\displaystyle c^{\\mathsf {T}}X}\n \n  is then given by:[4]\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n (\n \n \n c\n \n \n T\n \n \n \n X\n \n )\n \n =\n \n c\n \n \n T\n \n \n \n \u03a3\n c\n .\n \n \n {\\displaystyle \\operatorname {Var} \\left(c^{\\mathsf {T}}X\\right)=c^{\\mathsf {T}}\\Sigma c.}\n \n 
                                                                      \n

                                                                      This implies that the variance of the mean can be written as (with a column vector of ones)\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n (\n \n \n \n x\n \u00af\n \n \n \n )\n \n =\n Var\n \u2061\n \n (\n \n \n \n 1\n n\n \n \n \n 1\n \u2032\n \n X\n \n )\n \n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n 1\n \u2032\n \n \u03a3\n 1.\n \n \n {\\displaystyle \\operatorname {Var} \\left({\\bar {x}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}1'X\\right)={\\frac {1}{n^{2}}}1'\\Sigma 1.}\n \n 
                                                                      \n

                                                                      Sum of variables

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      Sum of uncorrelated variables

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n
                                                                      Main article: Bienaym\u00e9's identity
                                                                      \n
                                                                      See also: Sum of normally distributed random variables
                                                                      \n

                                                                      One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances:\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n .\n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} (X_{i}).}\n \n 
                                                                      \n

                                                                      This statement is called the Bienaym\u00e9 formula[5] and was discovered in 1853.[6][7] It is often made with the stronger condition that the variables are independent, but being uncorrelated suffices. So if all the variables have the same variance \u03c32, then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n (\n \n \n X\n \u00af\n \n \n )\n \n =\n Var\n \u2061\n \n (\n \n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n Var\n \u2061\n \n (\n \n X\n \n i\n \n \n )\n \n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n n\n \n \u03c3\n \n 2\n \n \n =\n \n \n \n \u03c3\n \n 2\n \n \n n\n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}\\sum _{i=1}^{n}X_{i}\\right)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)={\\frac {1}{n^{2}}}n\\sigma ^{2}={\\frac {\\sigma ^{2}}{n}}.}\n \n 
                                                                      \n

                                                                      That is, the variance of the mean decreases when n increases. This formula for the variance of the mean is used in the definition of the standard error of the sample mean, which is used in the central limit theorem.\n

                                                                      To prove the initial statement, it suffices to show that\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n +\n Y\n )\n =\n Var\n \u2061\n (\n X\n )\n +\n Var\n \u2061\n (\n Y\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (X+Y)=\\operatorname {Var} (X)+\\operatorname {Var} (Y).}\n \n 
                                                                      \n

                                                                      The general result then follows by induction. Starting with the definition,\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n +\n Y\n )\n \n \n \n =\n E\n \u2061\n \n [\n \n (\n X\n +\n Y\n \n )\n \n 2\n \n \n \n ]\n \n \u2212\n (\n E\n \u2061\n [\n X\n +\n Y\n ]\n \n )\n \n 2\n \n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n \n X\n \n 2\n \n \n +\n 2\n X\n Y\n +\n \n Y\n \n 2\n \n \n \n ]\n \n \u2212\n (\n E\n \u2061\n [\n X\n ]\n +\n E\n \u2061\n [\n Y\n ]\n \n )\n \n 2\n \n \n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[(X+Y)^{2}\\right]-(\\operatorname {E} [X+Y])^{2}\\\\[5pt]&=\\operatorname {E} \\left[X^{2}+2XY+Y^{2}\\right]-(\\operatorname {E} [X]+\\operatorname {E} [Y])^{2}.\\end{aligned}}}\n \n 
                                                                      \n

                                                                      Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y, this further simplifies as follows:\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n +\n Y\n )\n \n \n \n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n +\n 2\n E\n \u2061\n [\n X\n Y\n ]\n +\n E\n \u2061\n \n [\n \n Y\n \n 2\n \n \n ]\n \n \u2212\n \n (\n \n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n +\n 2\n E\n \u2061\n [\n X\n ]\n E\n \u2061\n [\n Y\n ]\n +\n E\n \u2061\n [\n Y\n \n ]\n \n 2\n \n \n \n )\n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n +\n E\n \u2061\n \n [\n \n Y\n \n 2\n \n \n ]\n \n \u2212\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n \u2212\n E\n \u2061\n [\n Y\n \n ]\n \n 2\n \n \n \n \n \n \n \n \n =\n Var\n \u2061\n (\n X\n )\n +\n Var\n \u2061\n (\n Y\n )\n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[X^{2}\\right]+2\\operatorname {E} [XY]+\\operatorname {E} \\left[Y^{2}\\right]-\\left(\\operatorname {E} [X]^{2}+2\\operatorname {E} [X]\\operatorname {E} [Y]+\\operatorname {E} [Y]^{2}\\right)\\\\[5pt]&=\\operatorname {E} \\left[X^{2}\\right]+\\operatorname {E} \\left[Y^{2}\\right]-\\operatorname {E} [X]^{2}-\\operatorname {E} [Y]^{2}\\\\[5pt]&=\\operatorname {Var} (X)+\\operatorname {Var} (Y).\\end{aligned}}}\n \n 
                                                                      \n

                                                                      Sum of correlated variables

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n
                                                                      Sum of correlated variables with fixed sample size
                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n
                                                                      Main article: Bienaym\u00e9's identity
                                                                      \n

                                                                      In general, the variance of the sum of n variables is the sum of their covariances:\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n Cov\n \u2061\n \n (\n \n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n \n )\n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n Var\n \u2061\n \n (\n \n X\n \n i\n \n \n )\n \n +\n 2\n \n \u2211\n \n 1\n \u2264\n i\n <\n j\n \u2264\n n\n \n \n Cov\n \u2061\n \n (\n \n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n \n )\n \n .\n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\sum _{j=1}^{n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)+2\\sum _{1\\leq i<j\\leq n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right).}\n \n 
                                                                      \n

                                                                      (Note: The second equality comes from the fact that Cov(Xi,Xi) = Var(Xi).)\n

                                                                      Here, \n \n \n \n Cov\n \u2061\n (\n \u22c5\n ,\n \u22c5\n )\n \n \n {\\displaystyle \\operatorname {Cov} (\\cdot ,\\cdot )}\n \n  is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. The next expression states equivalently that the variance of the sum is the sum of the diagonal of covariance matrix plus two times the sum of its upper triangular elements (or its lower triangular elements); this emphasizes that the covariance matrix is symmetric. This formula is used in the theory of Cronbach's alpha in classical test theory.\n

                                                                      So, if the variables have equal variance \u03c32 and the average correlation of distinct variables is \u03c1, then the variance of their mean is\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n (\n \n \n X\n \u00af\n \n \n )\n \n =\n \n \n \n \u03c3\n \n 2\n \n \n n\n \n \n +\n \n \n \n n\n \u2212\n 1\n \n n\n \n \n \u03c1\n \n \u03c3\n \n 2\n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {\\sigma ^{2}}{n}}+{\\frac {n-1}{n}}\\rho \\sigma ^{2}.}\n \n 
                                                                      \n

                                                                      This implies that the variance of the mean increases with the average of the correlations. In other words, additional correlated observations are not as effective as additional independent observations at reducing the uncertainty of the mean. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n (\n \n \n X\n \u00af\n \n \n )\n \n =\n \n \n 1\n n\n \n \n +\n \n \n \n n\n \u2212\n 1\n \n n\n \n \n \u03c1\n .\n \n \n {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {1}{n}}+{\\frac {n-1}{n}}\\rho .}\n \n 
                                                                      \n

                                                                      This formula is used in the Spearman\u2013Brown prediction formula of classical test theory. This converges to \u03c1 if n goes to infinity, provided that the average correlation remains constant or converges too. So for the variance of the mean of standardized variables with equal correlations or converging average correlation we have\n

                                                                      \n
                                                                      \n \n \n \n \n lim\n \n n\n \u2192\n \u221e\n \n \n Var\n \u2061\n \n (\n \n \n X\n \u00af\n \n \n )\n \n =\n \u03c1\n .\n \n \n {\\displaystyle \\lim _{n\\to \\infty }\\operatorname {Var} \\left({\\overline {X}}\\right)=\\rho .}\n \n 
                                                                      \n

                                                                      Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. This makes clear that the sample mean of correlated variables does not generally converge to the population mean, even though the law of large numbers states that the sample mean will converge for independent variables.\n

                                                                      \n
                                                                      Sum of uncorrelated variables with random sample size
                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size N is a random variable whose variation adds to the variation of X, such that,\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n E\n \u2061\n \n [\n N\n ]\n \n Var\n \u2061\n (\n X\n )\n +\n Var\n \u2061\n (\n N\n )\n (\n E\n \u2061\n \n [\n X\n ]\n \n \n )\n \n 2\n \n \n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\operatorname {E} \\left[N\\right]\\operatorname {Var} (X)+\\operatorname {Var} (N)(\\operatorname {E} \\left[X\\right])^{2}}\n \n [8]
                                                                      \n

                                                                      which follows from the law of total variance.\n

                                                                      If N has a Poisson distribution, then \n \n \n \n E\n \u2061\n [\n N\n ]\n =\n Var\n \u2061\n (\n N\n )\n \n \n {\\displaystyle \\operatorname {E} [N]=\\operatorname {Var} (N)}\n \n  with estimator n = N. So, the estimator of \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n X\n \n i\n \n \n \n )\n \n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)}\n \n  becomes \n \n \n \n n\n \n \n \n S\n \n x\n \n \n \n \n 2\n \n \n +\n n\n \n \n \n \n X\n \u00af\n \n \n \n \n 2\n \n \n \n \n {\\displaystyle n{S_{x}}^{2}+n{\\bar {X}}^{2}}\n \n , giving \n \n \n \n SE\n \u2061\n (\n \n \n \n X\n \u00af\n \n \n \n )\n =\n \n \n \n \n \n \n \n S\n \n x\n \n \n \n \n 2\n \n \n +\n \n \n \n \n X\n \u00af\n \n \n \n \n 2\n \n \n \n n\n \n \n \n \n \n {\\displaystyle \\operatorname {SE} ({\\bar {X}})={\\sqrt {\\frac {{S_{x}}^{2}+{\\bar {X}}^{2}}{n}}}}\n \n \n(see standard error of the sample mean).\n

                                                                      \n

                                                                      Weighted sum of variables

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n
                                                                      See also: Variance of a weighted arithmetic mean
                                                                      \n
                                                                      Not to be confused with Weighted variance.
                                                                      \n

                                                                      The scaling property and the Bienaym\u00e9 formula, along with the property of the covariance Cov(aX, bY) = ab Cov(X, Y) jointly imply that\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n a\n X\n \u00b1\n b\n Y\n )\n =\n \n a\n \n 2\n \n \n Var\n \u2061\n (\n X\n )\n +\n \n b\n \n 2\n \n \n Var\n \u2061\n (\n Y\n )\n \u00b1\n 2\n a\n b\n \n Cov\n \u2061\n (\n X\n ,\n Y\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (aX\\pm bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)\\pm 2ab\\,\\operatorname {Cov} (X,Y).}\n \n 
                                                                      \n

                                                                      This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y, then the weight of the variance of X will be four times the weight of the variance of Y.\n

                                                                      The expression above can be extended to a weighted sum of multiple variables:\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n \n \n n\n \n \n \n a\n \n i\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n a\n \n i\n \n \n 2\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n +\n 2\n \n \u2211\n \n 1\n \u2264\n i\n \n \n \n \u2211\n \n <\n j\n \u2264\n n\n \n \n \n a\n \n i\n \n \n \n a\n \n j\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i}^{n}a_{i}X_{i}\\right)=\\sum _{i=1}^{n}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i}\\sum _{<j\\leq n}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})}\n \n 
                                                                      \n

                                                                      Product of variables

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      Product of independent variables

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      If two variables X and Y are independent, the variance of their product is given by[9]\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n Y\n )\n =\n [\n E\n \u2061\n (\n X\n )\n \n ]\n \n 2\n \n \n Var\n \u2061\n (\n Y\n )\n +\n [\n E\n \u2061\n (\n Y\n )\n \n ]\n \n 2\n \n \n Var\n \u2061\n (\n X\n )\n +\n Var\n \u2061\n (\n X\n )\n Var\n \u2061\n (\n Y\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (XY)=[\\operatorname {E} (X)]^{2}\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\operatorname {Var} (X)+\\operatorname {Var} (X)\\operatorname {Var} (Y).}\n \n 
                                                                      \n

                                                                      Equivalently, using the basic properties of expectation, it is given by\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n (\n X\n Y\n )\n =\n E\n \u2061\n \n (\n \n X\n \n 2\n \n \n )\n \n E\n \u2061\n \n (\n \n Y\n \n 2\n \n \n )\n \n \u2212\n [\n E\n \u2061\n (\n X\n )\n \n ]\n \n 2\n \n \n [\n E\n \u2061\n (\n Y\n )\n \n ]\n \n 2\n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} (XY)=\\operatorname {E} \\left(X^{2}\\right)\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (X)]^{2}[\\operatorname {E} (Y)]^{2}.}\n \n 
                                                                      \n

                                                                      Product of statistically dependent variables

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      In general, if two variables are statistically dependent, then the variance of their product is given by:\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n Y\n )\n =\n \n\n \n \n \n E\n \u2061\n \n [\n \n \n X\n \n 2\n \n \n \n Y\n \n 2\n \n \n \n ]\n \n \u2212\n [\n E\n \u2061\n (\n X\n Y\n )\n \n ]\n \n 2\n \n \n \n \n \n \n =\n \n\n \n \n \n Cov\n \u2061\n \n (\n \n \n X\n \n 2\n \n \n ,\n \n Y\n \n 2\n \n \n \n )\n \n +\n E\n \u2061\n (\n \n X\n \n 2\n \n \n )\n E\n \u2061\n \n (\n \n Y\n \n 2\n \n \n )\n \n \u2212\n [\n E\n \u2061\n (\n X\n Y\n )\n \n ]\n \n 2\n \n \n \n \n \n \n =\n \n\n \n \n \n Cov\n \u2061\n \n (\n \n \n X\n \n 2\n \n \n ,\n \n Y\n \n 2\n \n \n \n )\n \n +\n \n (\n \n Var\n \u2061\n (\n X\n )\n +\n [\n E\n \u2061\n (\n X\n )\n \n ]\n \n 2\n \n \n \n )\n \n \n (\n \n Var\n \u2061\n (\n Y\n )\n +\n [\n E\n \u2061\n (\n Y\n )\n \n ]\n \n 2\n \n \n \n )\n \n \n \n \n \n \n \n \u2212\n [\n Cov\n \u2061\n (\n X\n ,\n Y\n )\n +\n E\n \u2061\n (\n X\n )\n E\n \u2061\n (\n Y\n )\n \n ]\n \n 2\n \n \n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (XY)={}&\\operatorname {E} \\left[X^{2}Y^{2}\\right]-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\operatorname {E} (X^{2})\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\left(\\operatorname {Var} (X)+[\\operatorname {E} (X)]^{2}\\right)\\left(\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\right)\\\\[5pt]&-[\\operatorname {Cov} (X,Y)+\\operatorname {E} (X)\\operatorname {E} (Y)]^{2}\\end{aligned}}}\n \n 
                                                                      \n

                                                                      Arbitrary functions

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n
                                                                      Main article: Uncertainty propagation
                                                                      \n

                                                                      The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables: see Taylor expansions for the moments of functions of random variables. For example, the approximate variance of a function of one variable is given by\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n [\n \n f\n (\n X\n )\n \n ]\n \n \u2248\n \n \n (\n \n \n f\n \u2032\n \n (\n E\n \u2061\n \n [\n X\n ]\n \n )\n \n )\n \n \n 2\n \n \n Var\n \u2061\n \n [\n X\n ]\n \n \n \n {\\displaystyle \\operatorname {Var} \\left[f(X)\\right]\\approx \\left(f'(\\operatorname {E} \\left[X\\right])\\right)^{2}\\operatorname {Var} \\left[X\\right]}\n \n 
                                                                      \n

                                                                      provided that f is twice differentiable and that the mean and variance of X are finite.\n

                                                                      \n

                                                                      Population variance and sample variance

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      \n

                                                                      \n
                                                                      See also: Unbiased estimation of standard deviation
                                                                      \n

                                                                      Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one estimates the mean and variance from a limited set of observations by using an estimator equation. The estimator is a function of the sample of n observations drawn without observational bias from the whole population of potential observations. In this example, the sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.\n

                                                                      The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance \u2013 these are consistent estimators (they converge to the value of the whole population as the number of samples increases) but can be improved. Most simply, the sample variance is computed as the sum of squared deviations about the (sample) mean, divided by n as the number of samples. However, using values other than n improves the estimator in various ways. Four common values for the denominator are n, n \u2212 1, n + 1, and n \u2212 1.5: n is the simplest (the variance of the sample), n \u2212 1 eliminates bias,[10] n + 1 minimizes mean squared error for the normal distribution,[11] and n \u2212 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.[12]\n

                                                                      Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (n \u2212 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n -1 instead of n, is called Bessel's correction.[10] The resulting estimator is unbiased and is called the (corrected) sample variance or unbiased sample variance. If the mean is determined in some other way than from the same samples used to estimate the variance, then this bias does not arise, and the variance can safely be estimated as that of the samples about the (independently known) mean.\n

                                                                      Secondly, the sample variance does not generally minimize mean squared error between sample variance and population variance. Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance) and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than n \u2212 1) and is a simple example of a shrinkage estimator: one \"shrinks\" the unbiased estimator towards zero. For the normal distribution, dividing by n + 1 (instead of n \u2212 1 or n) minimizes mean squared error.[11] The resulting estimator is biased, however, and is known as the biased sample variation.\n

                                                                      \n

                                                                      Population variance

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      In general, the population variance of a finite population of size N with values xi is given by\n \n \n \n \n \n \n \n \n \u03c3\n \n 2\n \n \n \n \n \n =\n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n \n (\n \n \n x\n \n i\n \n \n \u2212\n \u03bc\n \n )\n \n \n 2\n \n \n =\n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n (\n \n \n x\n \n i\n \n \n 2\n \n \n \u2212\n 2\n \u03bc\n \n x\n \n i\n \n \n +\n \n \u03bc\n \n 2\n \n \n \n )\n \n \n \n \n \n \n \n =\n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n 2\n \n \n \n )\n \n \u2212\n 2\n \u03bc\n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n \n )\n \n +\n \n \u03bc\n \n 2\n \n \n \n \n \n \n \n \n =\n E\n \u2061\n [\n \n x\n \n i\n \n \n 2\n \n \n ]\n \u2212\n \n \u03bc\n \n 2\n \n \n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\sigma ^{2}&={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}-\\mu \\right)^{2}={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}^{2}-2\\mu x_{i}+\\mu ^{2}\\right)\\\\[5pt]&=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-2\\mu \\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)+\\mu ^{2}\\\\[5pt]&=\\operatorname {E} [x_{i}^{2}]-\\mu ^{2}\\end{aligned}}}\n \n where the population mean is \n \n \n \n \u03bc\n =\n E\n \u2061\n [\n \n x\n \n i\n \n \n ]\n =\n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n \n \n {\\textstyle \\mu =\\operatorname {E} [x_{i}]={\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}}\n \n  and \n \n \n \n E\n \u2061\n [\n \n x\n \n i\n \n \n 2\n \n \n ]\n =\n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n 2\n \n \n \n )\n \n \n \n {\\textstyle \\operatorname {E} [x_{i}^{2}]=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)}\n \n , where \n \n \n \n E\n \n \n {\\textstyle \\operatorname {E} }\n \n  is the expectation value operator.\n

                                                                      The population variance can also be computed using[13]\n

                                                                      \n
                                                                      \n \n \n \n \n \u03c3\n \n 2\n \n \n =\n \n \n 1\n \n N\n \n 2\n \n \n \n \n \n \u2211\n \n i\n <\n j\n \n \n \n \n (\n \n \n x\n \n i\n \n \n \u2212\n \n x\n \n j\n \n \n \n )\n \n \n 2\n \n \n =\n \n \n 1\n \n 2\n \n N\n \n 2\n \n \n \n \n \n \n \u2211\n \n i\n ,\n j\n =\n 1\n \n \n N\n \n \n \n \n (\n \n \n x\n \n i\n \n \n \u2212\n \n x\n \n j\n \n \n \n )\n \n \n 2\n \n \n .\n \n \n {\\displaystyle \\sigma ^{2}={\\frac {1}{N^{2}}}\\sum _{i<j}\\left(x_{i}-x_{j}\\right)^{2}={\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}.}\n \n 
                                                                      \n

                                                                      (The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because\n \n \n \n \n \n \n \n \n \n \n 1\n \n 2\n \n N\n \n 2\n \n \n \n \n \n \n \u2211\n \n i\n ,\n j\n =\n 1\n \n \n N\n \n \n \n \n (\n \n \n x\n \n i\n \n \n \u2212\n \n x\n \n j\n \n \n \n )\n \n \n 2\n \n \n \n \n \n \n =\n \n\n \n \n \n \n \n 1\n \n 2\n \n N\n \n 2\n \n \n \n \n \n \n \u2211\n \n i\n ,\n j\n =\n 1\n \n \n N\n \n \n \n (\n \n \n x\n \n i\n \n \n 2\n \n \n \u2212\n 2\n \n x\n \n i\n \n \n \n x\n \n j\n \n \n +\n \n x\n \n j\n \n \n 2\n \n \n \n )\n \n \n \n \n \n =\n \n\n \n \n \n \n \n 1\n \n 2\n N\n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n N\n \n \n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n 2\n \n \n \n )\n \n \u2212\n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n \n )\n \n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n N\n \n \n \n x\n \n j\n \n \n \n )\n \n +\n \n \n 1\n \n 2\n N\n \n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n N\n \n \n \n x\n \n j\n \n \n 2\n \n \n \n )\n \n \n \n \n \n =\n \n\n \n \n \n \n \n 1\n 2\n \n \n \n (\n \n \n \u03c3\n \n 2\n \n \n +\n \n \u03bc\n \n 2\n \n \n \n )\n \n \u2212\n \n \u03bc\n \n 2\n \n \n +\n \n \n 1\n 2\n \n \n \n (\n \n \n \u03c3\n \n 2\n \n \n +\n \n \u03bc\n \n 2\n \n \n \n )\n \n \n \n \n \n =\n \n\n \n \n \n \n \u03c3\n \n 2\n \n \n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}\\\\[5pt]={}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2N}}\\sum _{j=1}^{N}\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}\\right)+{\\frac {1}{2N}}\\sum _{i=1}^{N}\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-\\mu ^{2}+{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\\\[5pt]={}&\\sigma ^{2}.\\end{aligned}}}\n \n The population variance matches the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.\n

                                                                      \n

                                                                      Sample variance

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n
                                                                      See also: Sample standard deviation
                                                                      \n

                                                                      Biased sample variance

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population.[14] This is generally referred to as sample variance or empirical variance. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution.\n

                                                                      We take a sample with replacement of n values Y1, ..., Yn from the population of size \n \n \n \n N\n \n \n {\\textstyle N}\n \n , where n < N, and estimate the variance on the basis of this sample.[15] Directly taking the variance of the sample data gives the average of the squared deviations:\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \n (\n \n \n Y\n \n i\n \n \n \u2212\n \n \n Y\n \u00af\n \n \n \n )\n \n \n 2\n \n \n =\n \n (\n \n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n Y\n \n i\n \n \n 2\n \n \n \n )\n \n \u2212\n \n \n \n Y\n \u00af\n \n \n \n 2\n \n \n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n i\n ,\n j\n \n :\n \n i\n <\n j\n \n \n \n \n (\n \n \n Y\n \n i\n \n \n \u2212\n \n Y\n \n j\n \n \n \n )\n \n \n 2\n \n \n .\n \n \n {\\displaystyle {\\tilde {S}}_{Y}^{2}={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}=\\left({\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}^{2}\\right)-{\\overline {Y}}^{2}={\\frac {1}{n^{2}}}\\sum _{i,j\\,:\\,i<j}\\left(Y_{i}-Y_{j}\\right)^{2}.}\n \n [16]
                                                                      \n

                                                                      (See the section Population variance for the derivation of this formula.) Here, \n \n \n \n \n \n Y\n \u00af\n \n \n \n \n {\\displaystyle {\\overline {Y}}}\n \n  denotes the sample mean:\n

                                                                      \n
                                                                      \n \n \n \n \n \n Y\n \u00af\n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n Y\n \n i\n \n \n .\n \n \n {\\displaystyle {\\overline {Y}}={\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}.}\n \n 
                                                                      \n

                                                                      Since the Yi are selected randomly, both \n \n \n \n \n \n Y\n \u00af\n \n \n \n \n {\\displaystyle {\\overline {Y}}}\n \n  and \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n \n \n {\\displaystyle {\\tilde {S}}_{Y}^{2}}\n \n  are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples {Yi} of size n from the population. For \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n \n \n {\\displaystyle {\\tilde {S}}_{Y}^{2}}\n \n  this gives:\n

                                                                      \n
                                                                      \n \n \n \n \n \n \n \n E\n \u2061\n [\n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n ]\n \n \n \n =\n E\n \u2061\n \n [\n \n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \n (\n \n \n Y\n \n i\n \n \n \u2212\n \n \n 1\n n\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n Y\n \n j\n \n \n \n )\n \n \n 2\n \n \n \n ]\n \n \n \n \n \n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n E\n \u2061\n \n [\n \n \n Y\n \n i\n \n \n 2\n \n \n \u2212\n \n \n 2\n n\n \n \n \n Y\n \n i\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n Y\n \n j\n \n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n Y\n \n j\n \n \n \n \u2211\n \n k\n =\n 1\n \n \n n\n \n \n \n Y\n \n k\n \n \n \n ]\n \n \n \n \n \n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n (\n \n E\n \u2061\n \n [\n \n Y\n \n i\n \n \n 2\n \n \n ]\n \n \u2212\n \n \n 2\n n\n \n \n \n (\n \n \n \u2211\n \n j\n \u2260\n i\n \n \n E\n \u2061\n \n [\n \n \n Y\n \n i\n \n \n \n Y\n \n j\n \n \n \n ]\n \n +\n E\n \u2061\n \n [\n \n Y\n \n i\n \n \n 2\n \n \n ]\n \n \n )\n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n \u2211\n \n k\n \u2260\n j\n \n \n n\n \n \n E\n \u2061\n \n [\n \n \n Y\n \n j\n \n \n \n Y\n \n k\n \n \n \n ]\n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n E\n \u2061\n \n [\n \n Y\n \n j\n \n \n 2\n \n \n ]\n \n \n )\n \n \n \n \n \n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n (\n \n \n \n \n n\n \u2212\n 2\n \n n\n \n \n E\n \u2061\n \n [\n \n Y\n \n i\n \n \n 2\n \n \n ]\n \n \u2212\n \n \n 2\n n\n \n \n \n \u2211\n \n j\n \u2260\n i\n \n \n E\n \u2061\n \n [\n \n \n Y\n \n i\n \n \n \n Y\n \n j\n \n \n \n ]\n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n \u2211\n \n k\n \u2260\n j\n \n \n n\n \n \n E\n \u2061\n \n [\n \n \n Y\n \n j\n \n \n \n Y\n \n k\n \n \n \n ]\n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n E\n \u2061\n \n [\n \n Y\n \n j\n \n \n 2\n \n \n ]\n \n \n )\n \n \n \n \n \n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n [\n \n \n \n \n n\n \u2212\n 2\n \n n\n \n \n \n (\n \n \n \u03c3\n \n 2\n \n \n +\n \n \u03bc\n \n 2\n \n \n \n )\n \n \u2212\n \n \n 2\n n\n \n \n (\n n\n \u2212\n 1\n )\n \n \u03bc\n \n 2\n \n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n n\n (\n n\n \u2212\n 1\n )\n \n \u03bc\n \n 2\n \n \n +\n \n \n 1\n n\n \n \n \n (\n \n \n \u03c3\n \n 2\n \n \n +\n \n \u03bc\n \n 2\n \n \n \n )\n \n \n ]\n \n \n \n \n \n \n \n =\n \n \n \n n\n \u2212\n 1\n \n n\n \n \n \n \u03c3\n \n 2\n \n \n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {E} [{\\tilde {S}}_{Y}^{2}]&=\\operatorname {E} \\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\frac {1}{n}}\\sum _{j=1}^{n}Y_{j}\\right)^{2}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\operatorname {E} \\left[Y_{i}^{2}-{\\frac {2}{n}}Y_{i}\\sum _{j=1}^{n}Y_{j}+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}Y_{j}\\sum _{k=1}^{n}Y_{k}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\left(\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+\\operatorname {E} \\left[Y_{i}^{2}\\right]\\right)+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left({\\frac {n-2}{n}}\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left[{\\frac {n-2}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-{\\frac {2}{n}}(n-1)\\mu ^{2}+{\\frac {1}{n^{2}}}n(n-1)\\mu ^{2}+{\\frac {1}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\right]\\\\[5pt]&={\\frac {n-1}{n}}\\sigma ^{2}.\\end{aligned}}}\n \n 
                                                                      \n

                                                                      Here \n \n \n \n \n \u03c3\n \n 2\n \n \n =\n E\n \u2061\n [\n \n Y\n \n i\n \n \n 2\n \n \n ]\n \u2212\n \n \u03bc\n \n 2\n \n \n \n \n {\\textstyle \\sigma ^{2}=\\operatorname {E} [Y_{i}^{2}]-\\mu ^{2}}\n \n  derived in the section Population variance and \n \n \n \n E\n \u2061\n [\n \n Y\n \n i\n \n \n \n Y\n \n j\n \n \n ]\n =\n E\n \u2061\n [\n \n Y\n \n i\n \n \n ]\n E\n \u2061\n [\n \n Y\n \n j\n \n \n ]\n =\n \n \u03bc\n \n 2\n \n \n \n \n {\\textstyle \\operatorname {E} [Y_{i}Y_{j}]=\\operatorname {E} [Y_{i}]\\operatorname {E} [Y_{j}]=\\mu ^{2}}\n \n  due to independency of \n \n \n \n \n Y\n \n i\n \n \n \n \n {\\textstyle Y_{i}}\n \n  and \n \n \n \n \n Y\n \n j\n \n \n \n \n {\\textstyle Y_{j}}\n \n  are used.\n

                                                                      Hence \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n \n \n {\\textstyle {\\tilde {S}}_{Y}^{2}}\n \n  gives an estimate of the population variance that is biased by a factor of \n \n \n \n \n \n \n n\n \u2212\n 1\n \n n\n \n \n \n \n {\\textstyle {\\frac {n-1}{n}}}\n \n  as the expectation value of \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n \n \n {\\textstyle {\\tilde {S}}_{Y}^{2}}\n \n  is smaller than the population variance (true variance) by that factor. For this reason, \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n \n \n {\\textstyle {\\tilde {S}}_{Y}^{2}}\n \n  is referred to as the biased sample variance.\n

                                                                      \n

                                                                      Unbiased sample variance

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      Correcting for this bias yields the unbiased sample variance, denoted \n \n \n \n \n S\n \n 2\n \n \n \n \n {\\displaystyle S^{2}}\n \n :\n

                                                                      \n
                                                                      \n \n \n \n \n S\n \n 2\n \n \n =\n \n \n n\n \n n\n \u2212\n 1\n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n =\n \n \n n\n \n n\n \u2212\n 1\n \n \n \n \n [\n \n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \n (\n \n \n Y\n \n i\n \n \n \u2212\n \n \n Y\n \u00af\n \n \n \n )\n \n \n 2\n \n \n \n ]\n \n =\n \n \n 1\n \n n\n \u2212\n 1\n \n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \n (\n \n \n Y\n \n i\n \n \n \u2212\n \n \n Y\n \u00af\n \n \n \n )\n \n \n 2\n \n \n \n \n {\\displaystyle S^{2}={\\frac {n}{n-1}}{\\tilde {S}}_{Y}^{2}={\\frac {n}{n-1}}\\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}\\right]={\\frac {1}{n-1}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}}\n \n 
                                                                      \n

                                                                      Either estimator may be simply referred to as the sample variance when the version can be determined by context. The same proof is also applicable for samples taken from a continuous probability distribution.\n

                                                                      The use of the term n \u2212 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation (the square root of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n \u2212 1.5 yields an almost unbiased estimator.\n

                                                                      The unbiased sample variance is a U-statistic for the function \u0192(y1, y2) = (y1 \u2212 y2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.\n

                                                                      \n
                                                                      Example
                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      For a set of numbers {10, 15, 30, 45, 57, 52 63, 72, 81, 93, 102, 105}, if this set is the whole data population for some measurement, then variance is the population variance 932.743 as the sum of the squared deviations about the mean of this set, divided by 12 as the number of the set members. If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.\n

                                                                      \n

                                                                      Distribution of the sample variance

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n
                                                                       
                                                                       
                                                                      Distribution and cumulative distribution of S2/\u03c32, for various values of \u03bd = n \u2212 1, when the yi are independent normally distributed.
                                                                      \n

                                                                      Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Yi are independent observations from a normal distribution, Cochran's theorem shows that the unbiased sample variance S2 follows a scaled chi-squared distribution (see also: asymptotic properties and an elementary proof):[17]\n

                                                                      \n
                                                                      \n \n \n \n (\n n\n \u2212\n 1\n )\n \n \n \n S\n \n 2\n \n \n \n \u03c3\n \n 2\n \n \n \n \n \u223c\n \n \u03c7\n \n n\n \u2212\n 1\n \n \n 2\n \n \n \n \n {\\displaystyle (n-1){\\frac {S^{2}}{\\sigma ^{2}}}\\sim \\chi _{n-1}^{2}}\n \n 
                                                                      \n

                                                                      where \u03c32 is the population variance. As a direct consequence, it follows that\n

                                                                      \n
                                                                      \n \n \n \n E\n \u2061\n \n (\n \n S\n \n 2\n \n \n )\n \n =\n E\n \u2061\n \n (\n \n \n \n \n \u03c3\n \n 2\n \n \n \n n\n \u2212\n 1\n \n \n \n \n \u03c7\n \n n\n \u2212\n 1\n \n \n 2\n \n \n \n )\n \n =\n \n \u03c3\n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {E} \\left(S^{2}\\right)=\\operatorname {E} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)=\\sigma ^{2},}\n \n 
                                                                      \n

                                                                      and[18]\n

                                                                      \n
                                                                      \n \n \n \n Var\n \u2061\n \n [\n \n S\n \n 2\n \n \n ]\n \n =\n Var\n \u2061\n \n (\n \n \n \n \n \u03c3\n \n 2\n \n \n \n n\n \u2212\n 1\n \n \n \n \n \u03c7\n \n n\n \u2212\n 1\n \n \n 2\n \n \n \n )\n \n =\n \n \n \n \u03c3\n \n 4\n \n \n \n (\n n\n \u2212\n 1\n \n )\n \n 2\n \n \n \n \n \n Var\n \u2061\n \n (\n \n \u03c7\n \n n\n \u2212\n 1\n \n \n 2\n \n \n )\n \n =\n \n \n \n 2\n \n \u03c3\n \n 4\n \n \n \n \n n\n \u2212\n 1\n \n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} \\left[S^{2}\\right]=\\operatorname {Var} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)={\\frac {\\sigma ^{4}}{(n-1)^{2}}}\\operatorname {Var} \\left(\\chi _{n-1}^{2}\\right)={\\frac {2\\sigma ^{4}}{n-1}}.}\n \n 
                                                                      \n

                                                                      If Yi are independent and identically distributed, but not necessarily normally distributed, then[19]\n

                                                                      \n
                                                                      \n \n \n \n E\n \u2061\n \n [\n \n S\n \n 2\n \n \n ]\n \n =\n \n \u03c3\n \n 2\n \n \n ,\n \n Var\n \u2061\n \n [\n \n S\n \n 2\n \n \n ]\n \n =\n \n \n \n \u03c3\n \n 4\n \n \n n\n \n \n \n (\n \n \u03ba\n \u2212\n 1\n +\n \n \n 2\n \n n\n \u2212\n 1\n \n \n \n \n )\n \n =\n \n \n 1\n n\n \n \n \n (\n \n \n \u03bc\n \n 4\n \n \n \u2212\n \n \n \n n\n \u2212\n 3\n \n \n n\n \u2212\n 1\n \n \n \n \n \u03c3\n \n 4\n \n \n \n )\n \n ,\n \n \n {\\displaystyle \\operatorname {E} \\left[S^{2}\\right]=\\sigma ^{2},\\quad \\operatorname {Var} \\left[S^{2}\\right]={\\frac {\\sigma ^{4}}{n}}\\left(\\kappa -1+{\\frac {2}{n-1}}\\right)={\\frac {1}{n}}\\left(\\mu _{4}-{\\frac {n-3}{n-1}}\\sigma ^{4}\\right),}\n \n 
                                                                      \n

                                                                      where \u03ba is the kurtosis of the distribution and \u03bc4 is the fourth central moment.\n

                                                                      If the conditions of the law of large numbers hold for the squared observations, S2 is a consistent estimator of \u03c32. One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.).[20][21][22]\n

                                                                      \n

                                                                      Samuelson's inequality

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      Samuelson's inequality is a result that states bounds on the values that individual observations in a sample can take, given that the sample mean and (biased) variance have been calculated.[23] Values must lie within the limits \n \n \n \n \n \n \n y\n \u00af\n \n \n \n \u00b1\n \n \u03c3\n \n Y\n \n \n (\n n\n \u2212\n 1\n \n )\n \n 1\n \n /\n \n 2\n \n \n .\n \n \n {\\displaystyle {\\bar {y}}\\pm \\sigma _{Y}(n-1)^{1/2}.}\n \n \n

                                                                      \n

                                                                      Relations with the harmonic and arithmetic means

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      It has been shown[24] that for a sample {yi} of positive real numbers,\n

                                                                      \n
                                                                      \n \n \n \n \n \u03c3\n \n y\n \n \n 2\n \n \n \u2264\n 2\n \n y\n \n max\n \n \n (\n A\n \u2212\n H\n )\n ,\n \n \n {\\displaystyle \\sigma _{y}^{2}\\leq 2y_{\\max }(A-H),}\n \n 
                                                                      \n

                                                                      where ymax is the maximum of the sample, A is the arithmetic mean, H is the harmonic mean of the sample and \n \n \n \n \n \u03c3\n \n y\n \n \n 2\n \n \n \n \n {\\displaystyle \\sigma _{y}^{2}}\n \n  is the (biased) variance of the sample.\n

                                                                      This bound has been improved, and it is known that variance is bounded by\n

                                                                      \n
                                                                      \n \n \n \n \n \u03c3\n \n y\n \n \n 2\n \n \n \u2264\n \n \n \n \n y\n \n max\n \n \n (\n A\n \u2212\n H\n )\n (\n \n y\n \n max\n \n \n \u2212\n A\n )\n \n \n \n y\n \n max\n \n \n \u2212\n H\n \n \n \n ,\n \n \n {\\displaystyle \\sigma _{y}^{2}\\leq {\\frac {y_{\\max }(A-H)(y_{\\max }-A)}{y_{\\max }-H}},}\n \n 
                                                                      \n
                                                                      \n \n \n \n \n \u03c3\n \n y\n \n \n 2\n \n \n \u2265\n \n \n \n \n y\n \n min\n \n \n (\n A\n \u2212\n H\n )\n (\n A\n \u2212\n \n y\n \n min\n \n \n )\n \n \n H\n \u2212\n \n y\n \n min\n \n \n \n \n \n ,\n \n \n {\\displaystyle \\sigma _{y}^{2}\\geq {\\frac {y_{\\min }(A-H)(A-y_{\\min })}{H-y_{\\min }}},}\n \n 
                                                                      \n

                                                                      where ymin is the minimum of the sample.[25]\n

                                                                      \n

                                                                      Tests of equality of variances

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n

                                                                      The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult.\n

                                                                      Several non parametric tests have been proposed: these include the Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey test, the Capon test, Mood test, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that both medians be known and equal to zero. The Mood, Klotz, Capon and Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey tests also apply to two variances. They allow the median to be unknown but do require that the two medians are equal.\n

                                                                      The Lehmann test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variances include the Box test, the Box\u2013Anderson test and the Moses test.\n

                                                                      Resampling methods, which include the bootstrap and the jackknife, may be used to test the equality of variances.\n

                                                                      \n

                                                                      Moment of inertia

                                                                      \n\n \nedit\n\n\n\n\n
                                                                      \n
                                                                      See also: Moment (physics) \u00a7 Examples
                                                                      \n

                                                                      The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass.[26] It is because of this analogy that such things as the variance are called moments of probability distributions.[26] The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of n points with a covariance matrix of \n \n \n \n \u03a3\n \n \n {\\displaystyle \\Sigma }\n \n  is given by[citation needed]\n

                                                                      \n
                                                                      \n \n \n \n I\n =\n n\n \n (\n \n \n \n 1\n \n \n 3\n \u00d7\n 3\n \n \n tr\n \u2061\n (\n \u03a3\n )\n \u2212\n \u03a3\n \n )\n \n .\n \n \n {\\displaystyle I=n\\left(\\mathbf {1} _{3\\times 3}\\operatorname {tr} (\\Sigma )-\\Sigma \\right).}\n \n 
                                                                      \n

                                                                      This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the x axis and distributed along it. The covariance matrix might look like\n

                                                                      \n
                                                                      \n \n \n \n \u03a3\n =\n \n \n [\n \n \n \n 10\n \n \n 0\n \n \n 0\n \n \n \n \n 0\n \n \n 0.1\n \n \n 0\n \n \n \n \n 0\n \n \n 0\n \n \n 0.1\n \n \n \n ]\n \n \n .\n \n \n {\\displaystyle \\Sigma ={\\begin{bmatrix}10&0&0\\\\0&0.1&0\\\\0&0&0.1\\end{bmatrix}}.}\n \n 
                                                                      \n

                                                                      That is, there is the most variance in the x direction. Physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is\n

                                                                      \n
                                                                      \n \n \n \n I\n =\n n\n \n \n [\n \n \n \n 0.2\n \n \n 0\n \n \n 0\n \n \n \n \n 0\n \n \n 10.1\n \n \n 0\n \n \n \n \n 0\n \n \n 0\n \n \n 10.1\n \n \n \n ]\n \n \n .\n \n \n {\\displaystyle I=n{\\begin{bmatrix}0.2&0&0\\\\0&10.1&0\\\\0&0&10.1\\end{bmatrix}}.}\n \n 
                                                                      \n

                                                                      Semivariance

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                                                                      The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation:\n \n \n \n \n Semivariance\n \n =\n \n \n 1\n \n n\n \n \n \n \n \u2211\n \n i\n :\n \n x\n \n i\n \n \n <\n \u03bc\n \n \n (\n \n x\n \n i\n \n \n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n \n {\\displaystyle {\\text{Semivariance}}={1 \\over {n}}\\sum _{i:x_{i}<\\mu }(x_{i}-\\mu )^{2}}\n \n It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not.[27]\n

                                                                      For inequalities associated with the semivariance, see Chebyshev's inequality \u00a7 Semivariances.\n

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                                                                      Etymology

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                                                                      The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance:[28]\n

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                                                                      The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and, therefore, that the variability may be uniformly measured by the standard deviation corresponding to the square root of the mean square error. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations \n \n \n \n \n \u03c3\n \n 1\n \n \n \n \n {\\displaystyle \\sigma _{1}}\n \n  and \n \n \n \n \n \u03c3\n \n 2\n \n \n \n \n {\\displaystyle \\sigma _{2}}\n \n , it is found that the distribution, when both causes act together, has a standard deviation \n \n \n \n \n \n \n \u03c3\n \n 1\n \n \n 2\n \n \n +\n \n \u03c3\n \n 2\n \n \n 2\n \n \n \n \n \n \n {\\displaystyle {\\sqrt {\\sigma _{1}^{2}+\\sigma _{2}^{2}}}}\n \n . It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance...

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                                                                      Generalizations

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                                                                      For complex variables

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                                                                      If \n \n \n \n x\n \n \n {\\displaystyle x}\n \n  is a scalar complex-valued random variable, with values in \n \n \n \n \n C\n \n ,\n \n \n {\\displaystyle \\mathbb {C} ,}\n \n  then its variance is \n \n \n \n E\n \u2061\n \n [\n \n (\n x\n \u2212\n \u03bc\n )\n (\n x\n \u2212\n \u03bc\n \n )\n \n \u2217\n \n \n \n ]\n \n ,\n \n \n {\\displaystyle \\operatorname {E} \\left[(x-\\mu )(x-\\mu )^{*}\\right],}\n \n  where \n \n \n \n \n x\n \n \u2217\n \n \n \n \n {\\displaystyle x^{*}}\n \n  is the complex conjugate of \n \n \n \n x\n .\n \n \n {\\displaystyle x.}\n \n  This variance is a real scalar.\n

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                                                                      For vector-valued random variables

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                                                                      As a matrix

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                                                                      If \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  is a vector-valued random variable, with values in \n \n \n \n \n \n R\n \n \n n\n \n \n ,\n \n \n {\\displaystyle \\mathbb {R} ^{n},}\n \n  and thought of as a column vector, then a natural generalization of variance is \n \n \n \n E\n \u2061\n \n [\n \n (\n X\n \u2212\n \u03bc\n )\n (\n X\n \u2212\n \u03bc\n \n )\n \n T\n \n \n \n ]\n \n ,\n \n \n {\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\operatorname {T} }\\right],}\n \n  where \n \n \n \n \u03bc\n =\n E\n \u2061\n (\n X\n )\n \n \n {\\displaystyle \\mu =\\operatorname {E} (X)}\n \n  and \n \n \n \n \n X\n \n T\n \n \n \n \n {\\displaystyle X^{\\operatorname {T} }}\n \n  is the transpose of \n \n \n \n X\n ,\n \n \n {\\displaystyle X,}\n \n  and so is a row vector. The result is a positive semi-definite square matrix, commonly referred to as the variance-covariance matrix (or simply as the covariance matrix).\n

                                                                      If \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  is a vector- and complex-valued random variable, with values in \n \n \n \n \n \n C\n \n \n n\n \n \n ,\n \n \n {\\displaystyle \\mathbb {C} ^{n},}\n \n  then the covariance matrix is \n \n \n \n E\n \u2061\n \n [\n \n (\n X\n \u2212\n \u03bc\n )\n (\n X\n \u2212\n \u03bc\n \n )\n \n \u2020\n \n \n \n ]\n \n ,\n \n \n {\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\dagger }\\right],}\n \n  where \n \n \n \n \n X\n \n \u2020\n \n \n \n \n {\\displaystyle X^{\\dagger }}\n \n  is the conjugate transpose of \n \n \n \n X\n .\n \n \n {\\displaystyle X.}\n \n [citation needed] This matrix is also positive semi-definite and square.\n

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                                                                      As a scalar

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                                                                      Another generalization of variance for vector-valued random variables \n \n \n \n X\n \n \n {\\displaystyle X}\n \n , which results in a scalar value rather than in a matrix, is the generalized variance \n \n \n \n det\n (\n C\n )\n \n \n {\\displaystyle \\det(C)}\n \n , the determinant of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.[29]\n

                                                                      A different generalization is obtained by considering the equation for the scalar variance, \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n E\n \u2061\n \n [\n \n (\n X\n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n ]\n \n \n \n {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right]}\n \n , and reinterpreting \n \n \n \n (\n X\n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n \n {\\displaystyle (X-\\mu )^{2}}\n \n  as the squared Euclidean distance between the random variable and its mean, or, simply as the scalar product of the vector \n \n \n \n X\n \u2212\n \u03bc\n \n \n {\\displaystyle X-\\mu }\n \n  with itself. This results in \n \n \n \n E\n \u2061\n \n [\n \n (\n X\n \u2212\n \u03bc\n \n )\n \n T\n \n \n (\n X\n \u2212\n \u03bc\n )\n \n ]\n \n =\n tr\n \u2061\n (\n C\n )\n ,\n \n \n {\\displaystyle \\operatorname {E} \\left[(X-\\mu )^{\\operatorname {T} }(X-\\mu )\\right]=\\operatorname {tr} (C),}\n \n  which is the trace of the covariance matrix.\n

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                                                                      See also

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                                                                      Look up variance in Wiktionary, the free dictionary.
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                                                                      Types of variance

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                                                                      References

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                                                                      1. ^ Wasserman, Larry (2005). All of Statistics: a concise course in statistical inference. Springer texts in statistics. p. 51. ISBN 978-1-4419-2322-6.\n
                                                                        2. \n
                                                                      2. ^ Yuli Zhang; Huaiyu Wu; Lei Cheng (June 2012). Some new deformation formulas about variance and covariance. Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012). pp. 987\u2013992.\n
                                                                        3. \n
                                                                      3. ^ Kagan, A.; Shepp, L. A. (1998). \"Why the variance?\". Statistics & Probability Letters. 38 (4): 329\u2013333. doi:10.1016/S0167-7152(98)00041-8.\n
                                                                        4. \n
                                                                      4. ^ Johnson, Richard; Wichern, Dean (2001). Applied Multivariate Statistical Analysis. Prentice Hall. p. 76. ISBN 0-13-187715-1.\n
                                                                        5. \n
                                                                      5. ^ Lo\u00e8ve, M. (1977) \"Probability Theory\", Graduate Texts in Mathematics, Volume 45, 4th edition, Springer-Verlag, p. 12.\n
                                                                        6. \n
                                                                      6. ^ Bienaym\u00e9, I.-J. (1853) \"Consid\u00e9rations \u00e0 l'appui de la d\u00e9couverte de Laplace sur la loi de probabilit\u00e9 dans la m\u00e9thode des moindres carr\u00e9s\", Comptes rendus de l'Acad\u00e9mie des sciences Paris, 37, p. 309\u2013317; digital copy available [1]\n
                                                                        7. \n
                                                                      7. ^ Bienaym\u00e9, I.-J. (1867) \"Consid\u00e9rations \u00e0 l'appui de la d\u00e9couverte de Laplace sur la loi de probabilit\u00e9 dans la m\u00e9thode des moindres carr\u00e9s\", Journal de Math\u00e9matiques Pures et Appliqu\u00e9es, S\u00e9rie 2, Tome 12, p. 158\u2013167; digital copy available [2][3]\n
                                                                        8. \n
                                                                      8. ^ Cornell, J R, and Benjamin, C A, Probability, Statistics, and Decisions for Civil Engineers, McGraw-Hill, NY, 1970, pp.178-9.\n
                                                                        9. \n
                                                                      9. ^ Goodman, Leo A. (December 1960). \"On the Exact Variance of Products\". Journal of the American Statistical Association. 55 (292): 708\u2013713. doi:10.2307/2281592. JSTOR 2281592.\n
                                                                        10. \n
                                                                      10. ^ a b Reichmann, W. J. (1961). \"Appendix 8\". Use and Abuse of Statistics (Reprinted 1964\u20131970 by Pelican ed.). London: Methuen.\n
                                                                        11. \n
                                                                      11. ^ a b Kourouklis, Stavros (2012). \"A New Estimator of the Variance Based on Minimizing Mean Squared Error\". The American Statistician. 66 (4): 234\u2013236. doi:10.1080/00031305.2012.735209. ISSN 0003-1305. JSTOR 23339501.\n
                                                                        12. \n
                                                                      12. ^ Brugger, R. M. (1969). \"A Note on Unbiased Estimation of the Standard Deviation\". The American Statistician. 23 (4): 32. doi:10.1080/00031305.1969.1048186 (inactive 16 December 2024).{{cite journal}}: CS1 maint: DOI inactive as of December 2024 (link)\n
                                                                        13. \n
                                                                      13. ^ Yuli Zhang; Huaiyu Wu; Lei Cheng (June 2012). Some new deformation formulas about variance and covariance. Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012). pp. 987\u2013992.\n
                                                                        14. \n
                                                                      14. ^ Navidi, William (2006) Statistics for Engineers and Scientists, McGraw-Hill, p. 14.\n
                                                                        15. \n
                                                                      15. ^ Montgomery, D. C. and Runger, G. C. (1994) Applied statistics and probability for engineers, page 201. John Wiley & Sons New York\n
                                                                        16. \n
                                                                      16. ^ Yuli Zhang; Huaiyu Wu; Lei Cheng (June 2012). Some new deformation formulas about variance and covariance. Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012). pp. 987\u2013992.\n
                                                                        17. \n
                                                                      17. ^ Knight K. (2000), Mathematical Statistics, Chapman and Hall, New York. (proposition 2.11)\n
                                                                        18. \n
                                                                      18. ^ Casella and Berger (2002) Statistical Inference, Example 7.3.3, p. 331 [full citation needed]\n
                                                                        19. \n
                                                                      19. ^ Mood, A. M., Graybill, F. A., and Boes, D.C. (1974) Introduction to the Theory of Statistics, 3rd Edition, McGraw-Hill, New York, p. 229\n
                                                                        20. \n
                                                                      20. ^ Kenney, John F.; Keeping, E.S. (1951). Mathematics of Statistics. Part Two (PDF) (2nd ed.). Princeton, New Jersey: D. Van Nostrand Company, Inc. Archived from the original (PDF) on Nov 17, 2018 \u2013 via KrishiKosh.\n
                                                                        21. \n
                                                                      21. ^ Rose, Colin; Smith, Murray D. (2002). \"Mathematical Statistics with Mathematica\". Springer-Verlag, New York.\n
                                                                        22. \n
                                                                      22. ^ Weisstein, Eric W. \"Sample Variance Distribution\". MathWorld Wolfram.\n
                                                                        23. \n
                                                                      23. ^ Samuelson, Paul (1968). \"How Deviant Can You Be?\". Journal of the American Statistical Association. 63 (324): 1522\u20131525. doi:10.1080/01621459.1968.10480944. JSTOR 2285901.\n
                                                                        24. \n
                                                                      24. ^ Mercer, A. McD. (2000). \"Bounds for A\u2013G, A\u2013H, G\u2013H, and a family of inequalities of Ky Fan's type, using a general method\". J. Math. Anal. Appl. 243 (1): 163\u2013173. doi:10.1006/jmaa.1999.6688.\n
                                                                        25. \n
                                                                      25. ^ Sharma, R. (2008). \"Some more inequalities for arithmetic mean, harmonic mean and variance\". Journal of Mathematical Inequalities. 2 (1): 109\u2013114. CiteSeerX 10.1.1.551.9397. doi:10.7153/jmi-02-11.\n
                                                                        26. \n
                                                                      26. ^ a b Magnello, M. Eileen. \"Karl Pearson and the Origins of Modern Statistics: An Elastician becomes a Statistician\". The Rutherford Journal.\n
                                                                        27. \n
                                                                      27. ^ Fama, Eugene F.; French, Kenneth R. (2010-04-21). \"Q&A: Semi-Variance: A Better Risk Measure?\". Fama/French Forum.\n
                                                                        28. \n
                                                                      28. ^ Ronald Fisher (1918) The correlation between relatives on the supposition of Mendelian Inheritance\n
                                                                        29. \n
                                                                      29. ^ Kocherlakota, S.; Kocherlakota, K. (2004). \"Generalized Variance\". Encyclopedia of Statistical Sciences. Wiley Online Library. doi:10.1002/0471667196.ess0869. ISBN 0-471-66719-6.\n
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                                                                      Retrieved from \"https://en.wikipedia.org/w/index.php?title=Variance&oldid=1264758978\"
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                                                                      In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by {\\displaystyle \\sigma ^{2}}, {\\displaystyle s^{2}}, {\\displaystyle \\operatorname {Var} (X)}, {\\displaystyle V(X)}, or {\\displaystyle \\mathbb {V} (X)}.[1]

                                                                      ", "content": [{"c": "In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation(SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by", "t": "text"}, {"c": "{\\displaystyle \\sigma ^{2}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "{\\displaystyle s^{2}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Var} (X)}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "{\\displaystyle V(X)}", "t": "equation-inline"}, {"c": ", or", "t": "text"}, {"c": "{\\displaystyle \\mathbb {V} (X)}", "t": "equation-inline"}, {"c": ".[ 1]", "t": "text"}]}, {"type": "image", "raw_content": "
                                                                      Example of samples from two populations with the same mean but different variances. The red population has mean 100 and variance 100 (SD=10) while the blue population has mean 100 and variance 2500 (SD=50) where SD stands for Standard Deviation.
                                                                      ", "content": {"url": "https://upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Comparison_standard_deviations.svg/400px-Comparison_standard_deviations.svg.png", "data": null, "alt": null, "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

                                                                      An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.\n

                                                                      ", "content": [{"c": "An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      There are two distinct concepts that are both called \"variance\". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.\n

                                                                      ", "content": [{"c": "There are two distinct concepts that are both called \"variance\". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling.\n

                                                                      ", "content": [{"c": "The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling.", "t": "text"}]}, {"type": "image", "raw_content": "
                                                                      Geometric visualisation of the variance of an arbitrary distribution (2, 4, 4, 4, 5, 5, 7, 9):
                                                                      ", "content": {"url": "https://upload.wikimedia.org/wikipedia/commons/thumb/6/64/Variance_visualisation.svg/220px-Variance_visualisation.svg.png", "data": null, "alt": null, "title": null, "caption": null}}, {"type": "list", "raw_content": "
                                                                      1. A frequency distribution is constructed.
                                                                      2. The centroid of the distribution gives its mean.
                                                                      3. A square with sides equal to the difference of each value from the mean is formed for each value.
                                                                      4. Arranging the squares into a rectangle with one side equal to the number of values, n, results in the other side being the distribution's variance, \u03c32.
                                                                      ", "content": {"items": [[[{"c": "A frequency distribution is constructed.", "t": "text"}]], [[{"c": "The centroid of the distribution gives its mean.", "t": "text"}]], [[{"c": "A square with sides equal to the difference of each value from the mean is formed for each value.", "t": "text"}]], [[{"c": "Arranging the squares into a rectangle with one side equal to the number of values, ", "t": "text"}, {"c": "n", "t": "text"}, {"c": ", results in the other side being the distribution's variance, ", "t": "text"}, {"c": "\u03c3", "t": "text"}, {"c": "2", "t": "text"}, {"c": ".", "t": "text"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                                      Definition

                                                                      ", "content": {"title_content": "Definition", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The variance of a random variable {\\displaystyle X} is the expected value of the squared deviation from the mean of {\\displaystyle X}, {\\displaystyle \\mu =\\operatorname {E} [X]}:\n

                                                                      ", "content": [{"c": "The variance of a random variable", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "is the expected value of the squared deviation from the mean of", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "{\\displaystyle \\mu =\\operatorname {E} [X]}", "t": "equation-inline"}, {"c": ":", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right].}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right].}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:\n

                                                                      ", "content": [{"c": "This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X)=\\operatorname {Cov} (X,X).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\operatorname {Cov} (X,X).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      The variance is also equivalent to the second cumulant of a probability distribution that generates {\\displaystyle X}. The variance is typically designated as {\\displaystyle \\operatorname {Var} (X)}, or sometimes as {\\displaystyle V(X)} or {\\displaystyle \\mathbb {V} (X)}, or symbolically as {\\displaystyle \\sigma _{X}^{2}} or simply {\\displaystyle \\sigma ^{2}} (pronounced \"sigma squared\"). The expression for the variance can be expanded as follows:\n

                                                                      ", "content": [{"c": "The variance is also equivalent to the second cumulant of a probability distribution that generates", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": ". The variance is typically designated as", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Var} (X)}", "t": "equation-inline"}, {"c": ", or sometimes as", "t": "text"}, {"c": "{\\displaystyle V(X)}", "t": "equation-inline"}, {"c": "or", "t": "text"}, {"c": "{\\displaystyle \\mathbb {V} (X)}", "t": "equation-inline"}, {"c": ", or symbolically as", "t": "text"}, {"c": "{\\displaystyle \\sigma _{X}^{2}}", "t": "equation-inline"}, {"c": "or simply", "t": "text"}, {"c": "{\\displaystyle \\sigma ^{2}}", "t": "equation-inline"}, {"c": "(pronounced \" sigma squared\"). The expression for the variance can be expanded as follows:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left[(X-\\operatorname {E} [X])^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}-2X\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]^{2}+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}\\end{aligned}}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left[(X-\\operatorname {E} [X])^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}-2X\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]^{2}+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      In other words, the variance of X is equal to the mean of the square of X minus the square of the mean of X. This equation should not be used for computations using floating point arithmetic, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see algorithms for calculating variance.\n

                                                                      ", "content": [{"c": "In other words, the variance of X is equal to the mean of the square of X minus the square of the mean of X. This equation should not be used for computations using floating point arithmetic, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see algorithms for calculating variance.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Discrete random variable

                                                                      ", "content": {"title_content": "Discrete random variable", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      If the generator of random variable {\\displaystyle X} is discrete with probability mass function{\\displaystyle x_{1}\\mapsto p_{1},x_{2}\\mapsto p_{2},\\ldots ,x_{n}\\mapsto p_{n}}, then\n

                                                                      ", "content": [{"c": "If the generator of random variable", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "is discrete with probability mass function", "t": "text"}, {"c": "{\\displaystyle x_{1}\\mapsto p_{1},x_{2}\\mapsto p_{2},\\ldots ,x_{n}\\mapsto p_{n}}", "t": "equation-inline"}, {"c": ", then", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X)=\\sum _{i=1}^{n}p_{i}\\cdot (x_{i}-\\mu )^{2},}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\sum _{i=1}^{n}p_{i}\\cdot (x_{i}-\\mu )^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      where {\\displaystyle \\mu } is the expected value. That is,\n

                                                                      ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\mu }", "t": "equation-inline"}, {"c": "is the expected value. That is,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\mu =\\sum _{i=1}^{n}p_{i}x_{i}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\mu =\\sum _{i=1}^{n}p_{i}x_{i}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      (When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)\n

                                                                      ", "content": [{"c": "(When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The variance of a collection of {\\displaystyle n} equally likely values can be written as\n

                                                                      ", "content": [{"c": "The variance of a collection of", "t": "text"}, {"c": "{\\displaystyle n}", "t": "equation-inline"}, {"c": "equally likely values can be written as", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n}}\\sum _{i=1}^{n}(x_{i}-\\mu )^{2}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n}}\\sum _{i=1}^{n}(x_{i}-\\mu )^{2}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      where {\\displaystyle \\mu } is the average value. That is,\n

                                                                      ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\mu }", "t": "equation-inline"}, {"c": "is the average value. That is,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\mu ={\\frac {1}{n}}\\sum _{i=1}^{n}x_{i}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\mu ={\\frac {1}{n}}\\sum _{i=1}^{n}x_{i}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      The variance of a set of {\\displaystyle n} equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:[2]

                                                                      ", "content": [{"c": "The variance of a set of", "t": "text"}, {"c": "{\\displaystyle n}", "t": "equation-inline"}, {"c": "equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:[ 2]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{\\frac {1}{2}}(x_{i}-x_{j})^{2}={\\frac {1}{n^{2}}}\\sum _{i}\\sum _{j>i}(x_{i}-x_{j})^{2}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{\\frac {1}{2}}(x_{i}-x_{j})^{2}={\\frac {1}{n^{2}}}\\sum _{i}\\sum _{j>i}(x_{i}-x_{j})^{2}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                                      Absolutely continuous random variable

                                                                      ", "content": {"title_content": "Absolutely continuous random variable", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      If the random variable {\\displaystyle X} has a probability density function{\\displaystyle f(x)}, and {\\displaystyle F(x)} is the corresponding cumulative distribution function, then\n

                                                                      ", "content": [{"c": "If the random variable", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "has a probability density function", "t": "text"}, {"c": "{\\displaystyle f(x)}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "{\\displaystyle F(x)}", "t": "equation-inline"}, {"c": "is the corresponding cumulative distribution function, then", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)=\\sigma ^{2}&=\\int _{\\mathbb {R} }(x-\\mu )^{2}f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-2\\mu \\int _{\\mathbb {R} }xf(x)\\,dx+\\mu ^{2}\\int _{\\mathbb {R} }f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\int _{\\mathbb {R} }x\\,dF(x)+\\mu ^{2}\\int _{\\mathbb {R} }\\,dF(x)\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\cdot \\mu +\\mu ^{2}\\cdot 1\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-\\mu ^{2},\\end{aligned}}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)=\\sigma ^{2}&=\\int _{\\mathbb {R} }(x-\\mu )^{2}f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-2\\mu \\int _{\\mathbb {R} }xf(x)\\,dx+\\mu ^{2}\\int _{\\mathbb {R} }f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\int _{\\mathbb {R} }x\\,dF(x)+\\mu ^{2}\\int _{\\mathbb {R} }\\,dF(x)\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\cdot \\mu +\\mu ^{2}\\cdot 1\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-\\mu ^{2},\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      or equivalently,\n

                                                                      ", "content": [{"c": "or equivalently,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X)=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-\\mu ^{2},}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-\\mu ^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      where {\\displaystyle \\mu } is the expected value of {\\displaystyle X} given by\n

                                                                      ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\mu }", "t": "equation-inline"}, {"c": "is the expected value of", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\mu =\\int _{\\mathbb {R} }xf(x)\\,dx=\\int _{\\mathbb {R} }x\\,dF(x).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\mu =\\int _{\\mathbb {R} }xf(x)\\,dx=\\int _{\\mathbb {R} }x\\,dF(x).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      In these formulas, the integrals with respect to {\\displaystyle dx} and {\\displaystyle dF(x)}\nare Lebesgue and Lebesgue\u2013Stieltjes integrals, respectively.\n

                                                                      ", "content": [{"c": "In these formulas, the integrals with respect to", "t": "text"}, {"c": "{\\displaystyle dx}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle dF(x)}", "t": "equation-inline"}, {"c": "are Lebesgue and Lebesgue\u2013Stieltjes integrals, respectively.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      If the function {\\displaystyle x^{2}f(x)} is Riemann-integrable on every finite interval {\\displaystyle [a,b]\\subset \\mathbb {R} ,} then\n

                                                                      ", "content": [{"c": "If the function", "t": "text"}, {"c": "{\\displaystyle x^{2}f(x)}", "t": "equation-inline"}, {"c": "is Riemann-integrable on every finite interval", "t": "text"}, {"c": "{\\displaystyle [a,b]\\subset \\mathbb {R} ,}", "t": "equation-inline"}, {"c": "then", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X)=\\int _{-\\infty }^{+\\infty }x^{2}f(x)\\,dx-\\mu ^{2},}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\int _{-\\infty }^{+\\infty }x^{2}f(x)\\,dx-\\mu ^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      where the integral is an improper Riemann integral.\n

                                                                      ", "content": [{"c": "where the integral is an improper Riemann integral.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Examples

                                                                      ", "content": {"title_content": "Examples", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Exponential distribution

                                                                      ", "content": {"title_content": "Exponential distribution", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The exponential distribution with parameter \u03bb is a continuous distribution whose probability density function is given by\n

                                                                      ", "content": [{"c": "The exponential distribution with parameter \u03bb is a continuous distribution whose probability density function is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle f(x)=\\lambda e^{-\\lambda x}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle f(x)=\\lambda e^{-\\lambda x}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      on the interval [0, \u221e). Its mean can be shown to be\n

                                                                      ", "content": [{"c": "on the interval[0, \u221e). Its mean can be shown to be", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {E} [X]=\\int _{0}^{\\infty }x\\lambda e^{-\\lambda x}\\,dx={\\frac {1}{\\lambda }}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {E} [X]=\\int _{0}^{\\infty }x\\lambda e^{-\\lambda x}\\,dx={\\frac {1}{\\lambda }}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      Using integration by parts and making use of the expected value already calculated, we have:\n

                                                                      ", "content": [{"c": "Using integration by parts and making use of the expected value already calculated, we have:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\begin{aligned}\\operatorname {E} \\left[X^{2}\\right]&=\\int _{0}^{\\infty }x^{2}\\lambda e^{-\\lambda x}\\,dx\\\\&=\\left[-x^{2}e^{-\\lambda x}\\right]_{0}^{\\infty }+\\int _{0}^{\\infty }2xe^{-\\lambda x}\\,dx\\\\&=0+{\\frac {2}{\\lambda }}\\operatorname {E} [X]\\\\&={\\frac {2}{\\lambda ^{2}}}.\\end{aligned}}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {E} \\left[X^{2}\\right]&=\\int _{0}^{\\infty }x^{2}\\lambda e^{-\\lambda x}\\,dx\\\\&=\\left[-x^{2}e^{-\\lambda x}\\right]_{0}^{\\infty }+\\int _{0}^{\\infty }2xe^{-\\lambda x}\\,dx\\\\&=0+{\\frac {2}{\\lambda }}\\operatorname {E} [X]\\\\&={\\frac {2}{\\lambda ^{2}}}.\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      Thus, the variance of X is given by\n

                                                                      ", "content": [{"c": "Thus, the variance of X is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}={\\frac {2}{\\lambda ^{2}}}-\\left({\\frac {1}{\\lambda }}\\right)^{2}={\\frac {1}{\\lambda ^{2}}}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}={\\frac {2}{\\lambda ^{2}}}-\\left({\\frac {1}{\\lambda }}\\right)^{2}={\\frac {1}{\\lambda ^{2}}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                                      Fair dice

                                                                      ", "content": {"title_content": "Fair dice", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      A fair six-sided dice can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is {\\displaystyle (1+2+3+4+5+6)/6=7/2.} Therefore, the variance of X is\n

                                                                      ", "content": [{"c": "A fair six-sided dice can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is", "t": "text"}, {"c": "{\\displaystyle (1+2+3+4+5+6)/6=7/2.}", "t": "equation-inline"}, {"c": "Therefore, the variance of X is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\sum _{i=1}^{6}{\\frac {1}{6}}\\left(i-{\\frac {7}{2}}\\right)^{2}\\\\[5pt]&={\\frac {1}{6}}\\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\\right)\\\\[5pt]&={\\frac {35}{12}}\\approx 2.92.\\end{aligned}}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\sum _{i=1}^{6}{\\frac {1}{6}}\\left(i-{\\frac {7}{2}}\\right)^{2}\\\\[5pt]&={\\frac {1}{6}}\\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\\right)\\\\[5pt]&={\\frac {35}{12}}\\approx 2.92.\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      The general formula for the variance of the outcome, X, of an n-sided die is\n

                                                                      ", "content": [{"c": "The general formula for the variance of the outcome, X, of an n-sided die is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left(X^{2}\\right)-(\\operatorname {E} (X))^{2}\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}i^{2}-\\left({\\frac {1}{n}}\\sum _{i=1}^{n}i\\right)^{2}\\\\[5pt]&={\\frac {(n+1)(2n+1)}{6}}-\\left({\\frac {n+1}{2}}\\right)^{2}\\\\[4pt]&={\\frac {n^{2}-1}{12}}.\\end{aligned}}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left(X^{2}\\right)-(\\operatorname {E} (X))^{2}\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}i^{2}-\\left({\\frac {1}{n}}\\sum _{i=1}^{n}i\\right)^{2}\\\\[5pt]&={\\frac {(n+1)(2n+1)}{6}}-\\left({\\frac {n+1}{2}}\\right)^{2}\\\\[4pt]&={\\frac {n^{2}-1}{12}}.\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                                      Commonly used probability distributions

                                                                      ", "content": {"title_content": "Commonly used probability distributions", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The following table lists the variance for some commonly used probability distributions.\n

                                                                      ", "content": [{"c": "The following table lists the variance for some commonly used probability distributions.", "t": "text"}]}, {"type": "table", "raw_content": "
                                                                      Name of the probability distribution\nProbability distribution function\nMean\nVariance\n
                                                                      Binomial distributionPr(X=k)=(nk)pk(1\u2212p)n\u2212k{\\displaystyle \\Pr \\,(X=k)={\\binom {n}{k}}p^{k}(1-p)^{n-k}}np{\\displaystyle np}np(1\u2212p){\\displaystyle np(1-p)}
                                                                      Geometric distributionPr(X=k)=(1\u2212p)k\u22121p{\\displaystyle \\Pr \\,(X=k)=(1-p)^{k-1}p}1p{\\displaystyle {\\frac {1}{p}}}(1\u2212p)p2{\\displaystyle {\\frac {(1-p)}{p^{2}}}}
                                                                      Normal distributionf(x\u2223\u03bc,\u03c32)=12\u03c0\u03c32e\u2212(x\u2212\u03bc)22\u03c32{\\displaystyle f\\left(x\\mid \\mu ,\\sigma ^{2}\\right)={\\frac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}e^{-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}}\u03bc{\\displaystyle \\mu }\u03c32{\\displaystyle \\sigma ^{2}}
                                                                      Uniform distribution (continuous)f(x\u2223a,b)={1b\u2212afor a\u2264x\u2264b,0for x<a or x>b{\\displaystyle f(x\\mid a,b)={\\begin{cases}{\\frac {1}{b-a}}&{\\text{for }}a\\leq x\\leq b,\\\\[3pt]0&{\\text{for }}x<a{\\text{ or }}x>b\\end{cases}}}a+b2{\\displaystyle {\\frac {a+b}{2}}}(b\u2212a)212{\\displaystyle {\\frac {(b-a)^{2}}{12}}}
                                                                      Exponential distributionf(x\u2223\u03bb)=\u03bbe\u2212\u03bbx{\\displaystyle f(x\\mid \\lambda )=\\lambda e^{-\\lambda x}}1\u03bb{\\displaystyle {\\frac {1}{\\lambda }}}1\u03bb2{\\displaystyle {\\frac {1}{\\lambda ^{2}}}}
                                                                      Poisson distributionf(k\u2223\u03bb)=e\u2212\u03bb\u03bbkk!{\\displaystyle f(k\\mid \\lambda )={\\frac {e^{-\\lambda }\\lambda ^{k}}{k!}}}\u03bb{\\displaystyle \\lambda }\u03bb{\\displaystyle \\lambda }
                                                                      ", "content": {"html": "
                                                                      Name of the probability distributionProbability distribution functionMeanVariance
                                                                      Binomial distribution{\\displaystyle \\Pr \\,(X=k)={\\binom {n}{k}}p^{k}(1-p)^{n-k}}{\\displaystyle np}{\\displaystyle np(1-p)}
                                                                      Geometric distribution{\\displaystyle \\Pr \\,(X=k)=(1-p)^{k-1}p}{\\displaystyle {\\frac {1}{p}}}{\\displaystyle {\\frac {(1-p)}{p^{2}}}}
                                                                      Normal distribution{\\displaystyle f\\left(x\\mid \\mu ,\\sigma ^{2}\\right)={\\frac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}e^{-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}}{\\displaystyle \\mu }{\\displaystyle \\sigma ^{2}}
                                                                      Uniform distribution (continuous){\\displaystyle f(x\\mid a,b)={\\begin{cases}{\\frac {1}{b-a}}&{\\text{for }}a\\leq x\\leq b,\\\\[3pt]0&{\\text{for }}x<a{\\text{ or }}x>b\\end{cases}}}{\\displaystyle {\\frac {a+b}{2}}}{\\displaystyle {\\frac {(b-a)^{2}}{12}}}
                                                                      Exponential distribution{\\displaystyle f(x\\mid \\lambda )=\\lambda e^{-\\lambda x}}{\\displaystyle {\\frac {1}{\\lambda }}}{\\displaystyle {\\frac {1}{\\lambda ^{2}}}}
                                                                      Poisson distribution{\\displaystyle f(k\\mid \\lambda )={\\frac {e^{-\\lambda }\\lambda ^{k}}{k!}}}{\\displaystyle \\lambda }{\\displaystyle \\lambda }
                                                                      ", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

                                                                      Properties

                                                                      ", "content": {"title_content": "Properties", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Basic properties

                                                                      ", "content": {"title_content": "Basic properties", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Variance is non-negative because the squares are positive or zero:\n

                                                                      ", "content": [{"c": "Variance is non-negative because the squares are positive or zero:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X)\\geq 0.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)\\geq 0.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      The variance of a constant is zero.\n

                                                                      ", "content": [{"c": "The variance of a constant is zero.", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (a)=0.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (a)=0.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      Conversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value:\n

                                                                      ", "content": [{"c": "Conversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X)=0\\iff \\exists a:P(X=a)=1.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X)=0\\iff \\exists a:P(X=a)=1.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                                      Issues of finiteness

                                                                      ", "content": {"title_content": "Issues of finiteness", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index{\\displaystyle k} satisfies {\\displaystyle 1<k\\leq 2.}

                                                                      ", "content": [{"c": "If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index", "t": "text"}, {"c": "{\\displaystyle k}", "t": "equation-inline"}, {"c": "satisfies", "t": "text"}, {"c": "{\\displaystyle 1Decomposition", "content": {"title_content": "Decomposition", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The general formula for variance decomposition or the law of total variance is: If {\\displaystyle X} and {\\displaystyle Y} are two random variables, and the variance of {\\displaystyle X} exists, then\n

                                                                      ", "content": [{"c": "The general formula for variance decomposition or the law of total variance is: If", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle Y}", "t": "equation-inline"}, {"c": "are two random variables, and the variance of", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "exists, then", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} [X]=\\operatorname {E} (\\operatorname {Var} [X\\mid Y])+\\operatorname {Var} (\\operatorname {E} [X\\mid Y]).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} [X]=\\operatorname {E} (\\operatorname {Var} [X\\mid Y])+\\operatorname {Var} (\\operatorname {E} [X\\mid Y]).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      The conditional expectation{\\displaystyle \\operatorname {E} (X\\mid Y)} of {\\displaystyle X} given {\\displaystyle Y}, and the conditional variance{\\displaystyle \\operatorname {Var} (X\\mid Y)} may be understood as follows. Given any particular value y of the random variable Y, there is a conditional expectation {\\displaystyle \\operatorname {E} (X\\mid Y=y)} given the event Y = y. This quantity depends on the particular value y; it is a function {\\displaystyle g(y)=\\operatorname {E} (X\\mid Y=y)}. That same function evaluated at the random variable Y is the conditional expectation {\\displaystyle \\operatorname {E} (X\\mid Y)=g(Y).}

                                                                      ", "content": [{"c": "The conditional expectation", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} (X\\mid Y)}", "t": "equation-inline"}, {"c": "of", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "given", "t": "text"}, {"c": "{\\displaystyle Y}", "t": "equation-inline"}, {"c": ", and the conditional variance", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Var} (X\\mid Y)}", "t": "equation-inline"}, {"c": "may be understood as follows. Given any particular value y of the random variable Y, there is a conditional expectation", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} (X\\mid Y=y)}", "t": "equation-inline"}, {"c": "given the event Y= y. This quantity depends on the particular value y; it is a function", "t": "text"}, {"c": "{\\displaystyle g(y)=\\operatorname {E} (X\\mid Y=y)}", "t": "equation-inline"}, {"c": ". That same function evaluated at the random variable Y is the conditional expectation", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} (X\\mid Y)=g(Y).}", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

                                                                      In particular, if {\\displaystyle Y} is a discrete random variable assuming possible values {\\displaystyle y_{1},y_{2},y_{3}\\ldots } with corresponding probabilities {\\displaystyle p_{1},p_{2},p_{3}\\ldots ,}, then in the formula for total variance, the first term on the right-hand side becomes\n

                                                                      ", "content": [{"c": "In particular, if", "t": "text"}, {"c": "{\\displaystyle Y}", "t": "equation-inline"}, {"c": "is a discrete random variable assuming possible values", "t": "text"}, {"c": "{\\displaystyle y_{1},y_{2},y_{3}\\ldots }", "t": "equation-inline"}, {"c": "with corresponding probabilities", "t": "text"}, {"c": "{\\displaystyle p_{1},p_{2},p_{3}\\ldots ,}", "t": "equation-inline"}, {"c": ", then in the formula for total variance, the first term on the right-hand side becomes", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {E} (\\operatorname {Var} [X\\mid Y])=\\sum _{i}p_{i}\\sigma _{i}^{2},}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {E} (\\operatorname {Var} [X\\mid Y])=\\sum _{i}p_{i}\\sigma _{i}^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      where {\\displaystyle \\sigma _{i}^{2}=\\operatorname {Var} [X\\mid Y=y_{i}]}. Similarly, the second term on the right-hand side becomes\n

                                                                      ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\sigma _{i}^{2}=\\operatorname {Var} [X\\mid Y=y_{i}]}", "t": "equation-inline"}, {"c": ". Similarly, the second term on the right-hand side becomes", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (\\operatorname {E} [X\\mid Y])=\\sum _{i}p_{i}\\mu _{i}^{2}-\\left(\\sum _{i}p_{i}\\mu _{i}\\right)^{2}=\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2},}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (\\operatorname {E} [X\\mid Y])=\\sum _{i}p_{i}\\mu _{i}^{2}-\\left(\\sum _{i}p_{i}\\mu _{i}\\right)^{2}=\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      where {\\displaystyle \\mu _{i}=\\operatorname {E} [X\\mid Y=y_{i}]} and {\\displaystyle \\mu =\\sum _{i}p_{i}\\mu _{i}}. Thus the total variance is given by\n

                                                                      ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\mu _{i}=\\operatorname {E} [X\\mid Y=y_{i}]}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle \\mu =\\sum _{i}p_{i}\\mu _{i}}", "t": "equation-inline"}, {"c": ". Thus the total variance is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} [X]=\\sum _{i}p_{i}\\sigma _{i}^{2}+\\left(\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2}\\right).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} [X]=\\sum _{i}p_{i}\\sigma _{i}^{2}+\\left(\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2}\\right).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      A similar formula is applied in analysis of variance, where the corresponding formula is\n

                                                                      ", "content": [{"c": "A similar formula is applied in analysis of variance, where the corresponding formula is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{between}}+{\\mathit {MS}}_{\\text{within}};}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{between}}+{\\mathit {MS}}_{\\text{within}};}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      here {\\displaystyle {\\mathit {MS}}} refers to the Mean of the Squares. In linear regression analysis the corresponding formula is\n

                                                                      ", "content": [{"c": "here", "t": "text"}, {"c": "{\\displaystyle {\\mathit {MS}}}", "t": "equation-inline"}, {"c": "refers to the Mean of the Squares. In linear regression analysis the corresponding formula is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{regression}}+{\\mathit {MS}}_{\\text{residual}}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{regression}}+{\\mathit {MS}}_{\\text{residual}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.\n

                                                                      ", "content": [{"c": "This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Similar decompositions are possible for the sum of squared deviations (sum of squares, {\\displaystyle {\\mathit {SS}}}):\n

                                                                      ", "content": [{"c": "Similar decompositions are possible for the sum of squared deviations (sum of squares,", "t": "text"}, {"c": "{\\displaystyle {\\mathit {SS}}}", "t": "equation-inline"}, {"c": "):", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{between}}+{\\mathit {SS}}_{\\text{within}},}
                                                                      {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{regression}}+{\\mathit {SS}}_{\\text{residual}}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{between}}+{\\mathit {SS}}_{\\text{within}},}", "t": "equation-inline"}]], [[{"c": "{\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{regression}}+{\\mathit {SS}}_{\\text{residual}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                                      Calculation from the CDF

                                                                      ", "content": {"title_content": "Calculation from the CDF", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution functionF using\n

                                                                      ", "content": [{"c": "The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution function F using", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle 2\\int _{0}^{\\infty }u(1-F(u))\\,du-\\left(\\int _{0}^{\\infty }(1-F(u))\\,du\\right)^{2}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle 2\\int _{0}^{\\infty }u(1-F(u))\\,du-\\left(\\int _{0}^{\\infty }(1-F(u))\\,du\\right)^{2}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      This expression can be used to calculate the variance in situations where the CDF, but not the density, can be conveniently expressed.\n

                                                                      ", "content": [{"c": "This expression can be used to calculate the variance in situations where the CDF, but not the density, can be conveniently expressed.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Characteristic property

                                                                      ", "content": {"title_content": "Characteristic property", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} \\left(\\left(X-m\\right)^{2}\\right)=\\mathrm {E} (X)}. Conversely, if a continuous function {\\displaystyle \\varphi } satisfies {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} (\\varphi (X-m))=\\mathrm {E} (X)} for all random variables X, then it is necessarily of the form {\\displaystyle \\varphi (x)=ax^{2}+b}, where a > 0. This also holds in the multidimensional case.[3]

                                                                      ", "content": [{"c": "The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e.", "t": "text"}, {"c": "{\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} \\left(\\left(X-m\\right)^{2}\\right)=\\mathrm {E} (X)}", "t": "equation-inline"}, {"c": ". Conversely, if a continuous function", "t": "text"}, {"c": "{\\displaystyle \\varphi }", "t": "equation-inline"}, {"c": "satisfies", "t": "text"}, {"c": "{\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} (\\varphi (X-m))=\\mathrm {E} (X)}", "t": "equation-inline"}, {"c": "for all random variables X, then it is necessarily of the form", "t": "text"}, {"c": "{\\displaystyle \\varphi (x)=ax^{2}+b}", "t": "equation-inline"}, {"c": ", where a> 0. This also holds in the multidimensional case.[ 3]", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Units of measurement

                                                                      ", "content": {"title_content": "Units of measurement", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is \u221a2.9 \u2248 1.7, slightly larger than the expected absolute deviation of 1.5.\n

                                                                      ", "content": [{"c": "Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is\u221a 2.9 \u2248 1.7, slightly larger than the expected absolute deviation of 1.5.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The standard deviation and the expected absolute deviation can both be used as an indicator of the \"spread\" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution.\n

                                                                      ", "content": [{"c": "The standard deviation and the expected absolute deviation can both be used as an indicator of the \"spread\" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Propagation

                                                                      ", "content": {"title_content": "Propagation", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Addition and multiplication by a constant

                                                                      ", "content": {"title_content": "Addition and multiplication by a constant", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged:\n

                                                                      ", "content": [{"c": "Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X+a)=\\operatorname {Var} (X).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X+a)=\\operatorname {Var} (X).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      If all values are scaled by a constant, the variance is scaled by the square of that constant:\n

                                                                      ", "content": [{"c": "If all values are scaled by a constant, the variance is scaled by the square of that constant:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (aX)=a^{2}\\operatorname {Var} (X).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (aX)=a^{2}\\operatorname {Var} (X).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      The variance of a sum of two random variables is given by\n

                                                                      ", "content": [{"c": "The variance of a sum of two random variables is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (aX+bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)+2ab\\,\\operatorname {Cov} (X,Y)}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (aX+bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)+2ab\\,\\operatorname {Cov} (X,Y)}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (aX-bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)-2ab\\,\\operatorname {Cov} (X,Y)}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (aX-bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)-2ab\\,\\operatorname {Cov} (X,Y)}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      where {\\displaystyle \\operatorname {Cov} (X,Y)} is the covariance.\n

                                                                      ", "content": [{"c": "where", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Cov} (X,Y)}", "t": "equation-inline"}, {"c": "is the covariance.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Linear combinations

                                                                      ", "content": {"title_content": "Linear combinations", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      In general, for the sum of {\\displaystyle N} random variables {\\displaystyle \\{X_{1},\\dots ,X_{N}\\}}, the variance becomes:\n

                                                                      ", "content": [{"c": "In general, for the sum of", "t": "text"}, {"c": "{\\displaystyle N}", "t": "equation-inline"}, {"c": "random variables", "t": "text"}, {"c": "{\\displaystyle \\{X_{1},\\dots ,X_{N}\\}}", "t": "equation-inline"}, {"c": ", the variance becomes:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i,j=1}^{N}\\operatorname {Cov} (X_{i},X_{j})=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i})+\\sum _{i\\neq j}\\operatorname {Cov} (X_{i},X_{j}),}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i,j=1}^{N}\\operatorname {Cov} (X_{i},X_{j})=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i})+\\sum _{i\\neq j}\\operatorname {Cov} (X_{i},X_{j}),}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      see also general Bienaym\u00e9's identity.\n

                                                                      ", "content": [{"c": "see also general Bienaym\u00e9's identity.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      These results lead to the variance of a linear combination as:\n

                                                                      ", "content": [{"c": "These results lead to the variance of a linear combination as:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\begin{aligned}\\operatorname {Var} \\left(\\sum _{i=1}^{N}a_{i}X_{i}\\right)&=\\sum _{i,j=1}^{N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+\\sum _{i\\not =j}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i<j\\leq N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j}).\\end{aligned}}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} \\left(\\sum _{i=1}^{N}a_{i}X_{i}\\right)&=\\sum _{i,j=1}^{N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+\\sum _{i\\not =j}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i

                                                                      If the random variables {\\displaystyle X_{1},\\dots ,X_{N}} are such that\n

                                                                      ", "content": [{"c": "If the random variables", "t": "text"}, {"c": "{\\displaystyle X_{1},\\dots ,X_{N}}", "t": "equation-inline"}, {"c": "are such that", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Cov} (X_{i},X_{j})=0\\ ,\\ \\forall \\ (i\\neq j),}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Cov} (X_{i},X_{j})=0\\ ,\\ \\forall \\ (i\\neq j),}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      then they are said to be uncorrelated. It follows immediately from the expression given earlier that if the random variables {\\displaystyle X_{1},\\dots ,X_{N}} are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:\n

                                                                      ", "content": [{"c": "then they are said to be uncorrelated. It follows immediately from the expression given earlier that if the random variables", "t": "text"}, {"c": "{\\displaystyle X_{1},\\dots ,X_{N}}", "t": "equation-inline"}, {"c": "are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i}).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i}).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      Since independent random variables are always uncorrelated (see Covariance \u00a7 Uncorrelatedness and independence), the equation above holds in particular when the random variables {\\displaystyle X_{1},\\dots ,X_{n}} are independent. Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.\n

                                                                      ", "content": [{"c": "Since independent random variables are always uncorrelated (see Covariance \u00a7 Uncorrelatedness and independence), the equation above holds in particular when the random variables", "t": "text"}, {"c": "{\\displaystyle X_{1},\\dots ,X_{n}}", "t": "equation-inline"}, {"c": "are independent. Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Matrix notation for the variance of a linear combination

                                                                      ", "content": {"title_content": "Matrix notation for the variance of a linear combination", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Define {\\displaystyle X} as a column vector of {\\displaystyle n} random variables {\\displaystyle X_{1},\\ldots ,X_{n}}, and {\\displaystyle c} as a column vector of {\\displaystyle n} scalars {\\displaystyle c_{1},\\ldots ,c_{n}}. Therefore, {\\displaystyle c^{\\mathsf {T}}X} is a linear combination of these random variables, where {\\displaystyle c^{\\mathsf {T}}} denotes the transpose of {\\displaystyle c}. Also let {\\displaystyle \\Sigma } be the covariance matrix of {\\displaystyle X}. The variance of {\\displaystyle c^{\\mathsf {T}}X} is then given by:[4]

                                                                      ", "content": [{"c": "Define", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "as a column vector of", "t": "text"}, {"c": "{\\displaystyle n}", "t": "equation-inline"}, {"c": "random variables", "t": "text"}, {"c": "{\\displaystyle X_{1},\\ldots ,X_{n}}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "{\\displaystyle c}", "t": "equation-inline"}, {"c": "as a column vector of", "t": "text"}, {"c": "{\\displaystyle n}", "t": "equation-inline"}, {"c": "scalars", "t": "text"}, {"c": "{\\displaystyle c_{1},\\ldots ,c_{n}}", "t": "equation-inline"}, {"c": ". Therefore,", "t": "text"}, {"c": "{\\displaystyle c^{\\mathsf {T}}X}", "t": "equation-inline"}, {"c": "is a linear combination of these random variables, where", "t": "text"}, {"c": "{\\displaystyle c^{\\mathsf {T}}}", "t": "equation-inline"}, {"c": "denotes the transpose of", "t": "text"}, {"c": "{\\displaystyle c}", "t": "equation-inline"}, {"c": ". Also let", "t": "text"}, {"c": "{\\displaystyle \\Sigma }", "t": "equation-inline"}, {"c": "be the covariance matrix of", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": ". The variance of", "t": "text"}, {"c": "{\\displaystyle c^{\\mathsf {T}}X}", "t": "equation-inline"}, {"c": "is then given by:[ 4]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left(c^{\\mathsf {T}}X\\right)=c^{\\mathsf {T}}\\Sigma c.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(c^{\\mathsf {T}}X\\right)=c^{\\mathsf {T}}\\Sigma c.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      This implies that the variance of the mean can be written as (with a column vector of ones)\n

                                                                      ", "content": [{"c": "This implies that the variance of the mean can be written as (with a column vector of ones)", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left({\\bar {x}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}1'X\\right)={\\frac {1}{n^{2}}}1'\\Sigma 1.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left({\\bar {x}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}1'X\\right)={\\frac {1}{n^{2}}}1'\\Sigma 1.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                                      Sum of variables

                                                                      ", "content": {"title_content": "Sum of variables", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Sum of uncorrelated variables

                                                                      ", "content": {"title_content": "Sum of uncorrelated variables", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances:\n

                                                                      ", "content": [{"c": "One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} (X_{i}).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} (X_{i}).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      This statement is called the Bienaym\u00e9 formula[5] and was discovered in 1853.[6][7] It is often made with the stronger condition that the variables are independent, but being uncorrelated suffices. So if all the variables have the same variance \u03c32, then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is\n

                                                                      ", "content": [{"c": "This statement is called the Bienaym\u00e9 formula[ 5] and was discovered in 1853.[ 6][ 7] It is often made with the stronger condition that the variables are independent, but being uncorrelated suffices. So if all the variables have the same variance \u03c3 2, then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}\\sum _{i=1}^{n}X_{i}\\right)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)={\\frac {1}{n^{2}}}n\\sigma ^{2}={\\frac {\\sigma ^{2}}{n}}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}\\sum _{i=1}^{n}X_{i}\\right)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)={\\frac {1}{n^{2}}}n\\sigma ^{2}={\\frac {\\sigma ^{2}}{n}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      That is, the variance of the mean decreases when n increases. This formula for the variance of the mean is used in the definition of the standard error of the sample mean, which is used in the central limit theorem.\n

                                                                      ", "content": [{"c": "That is, the variance of the mean decreases when n increases. This formula for the variance of the mean is used in the definition of the standard error of the sample mean, which is used in the central limit theorem.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      To prove the initial statement, it suffices to show that\n

                                                                      ", "content": [{"c": "To prove the initial statement, it suffices to show that", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (X+Y)=\\operatorname {Var} (X)+\\operatorname {Var} (Y).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (X+Y)=\\operatorname {Var} (X)+\\operatorname {Var} (Y).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      The general result then follows by induction. Starting with the definition,\n

                                                                      ", "content": [{"c": "The general result then follows by induction. Starting with the definition,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[(X+Y)^{2}\\right]-(\\operatorname {E} [X+Y])^{2}\\\\[5pt]&=\\operatorname {E} \\left[X^{2}+2XY+Y^{2}\\right]-(\\operatorname {E} [X]+\\operatorname {E} [Y])^{2}.\\end{aligned}}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[(X+Y)^{2}\\right]-(\\operatorname {E} [X+Y])^{2}\\\\[5pt]&=\\operatorname {E} \\left[X^{2}+2XY+Y^{2}\\right]-(\\operatorname {E} [X]+\\operatorname {E} [Y])^{2}.\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y, this further simplifies as follows:\n

                                                                      ", "content": [{"c": "Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y, this further simplifies as follows:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[X^{2}\\right]+2\\operatorname {E} [XY]+\\operatorname {E} \\left[Y^{2}\\right]-\\left(\\operatorname {E} [X]^{2}+2\\operatorname {E} [X]\\operatorname {E} [Y]+\\operatorname {E} [Y]^{2}\\right)\\\\[5pt]&=\\operatorname {E} \\left[X^{2}\\right]+\\operatorname {E} \\left[Y^{2}\\right]-\\operatorname {E} [X]^{2}-\\operatorname {E} [Y]^{2}\\\\[5pt]&=\\operatorname {Var} (X)+\\operatorname {Var} (Y).\\end{aligned}}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[X^{2}\\right]+2\\operatorname {E} [XY]+\\operatorname {E} \\left[Y^{2}\\right]-\\left(\\operatorname {E} [X]^{2}+2\\operatorname {E} [X]\\operatorname {E} [Y]+\\operatorname {E} [Y]^{2}\\right)\\\\[5pt]&=\\operatorname {E} \\left[X^{2}\\right]+\\operatorname {E} \\left[Y^{2}\\right]-\\operatorname {E} [X]^{2}-\\operatorname {E} [Y]^{2}\\\\[5pt]&=\\operatorname {Var} (X)+\\operatorname {Var} (Y).\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                                      Sum of correlated variables

                                                                      ", "content": {"title_content": "Sum of correlated variables", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "
                                                                      Sum of correlated variables with fixed sample size
                                                                      ", "content": {"title_content": "Sum of correlated variables with fixed sample size", "level": "5"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      In general, the variance of the sum of n variables is the sum of their covariances:\n

                                                                      ", "content": [{"c": "In general, the variance of the sum of n variables is the sum of their covariances:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\sum _{j=1}^{n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)+2\\sum _{1\\leq i<j\\leq n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\sum _{j=1}^{n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)+2\\sum _{1\\leq i

                                                                      (Note: The second equality comes from the fact that Cov(Xi,Xi) = Var(Xi).)\n

                                                                      ", "content": [{"c": "(Note: The second equality comes from the fact that Cov( X i, X i) = Var( X i).)", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Here, {\\displaystyle \\operatorname {Cov} (\\cdot ,\\cdot )} is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. The next expression states equivalently that the variance of the sum is the sum of the diagonal of covariance matrix plus two times the sum of its upper triangular elements (or its lower triangular elements); this emphasizes that the covariance matrix is symmetric. This formula is used in the theory of Cronbach's alpha in classical test theory.\n

                                                                      ", "content": [{"c": "Here,", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Cov} (\\cdot ,\\cdot )}", "t": "equation-inline"}, {"c": "is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. The next expression states equivalently that the variance of the sum is the sum of the diagonal of covariance matrix plus two times the sum of its upper triangular elements (or its lower triangular elements); this emphasizes that the covariance matrix is symmetric. This formula is used in the theory of Cronbach's alpha in classical test theory.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      So, if the variables have equal variance \u03c32 and the average correlation of distinct variables is \u03c1, then the variance of their mean is\n

                                                                      ", "content": [{"c": "So, if the variables have equal variance \u03c3 2 and the average correlation of distinct variables is \u03c1, then the variance of their mean is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {\\sigma ^{2}}{n}}+{\\frac {n-1}{n}}\\rho \\sigma ^{2}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {\\sigma ^{2}}{n}}+{\\frac {n-1}{n}}\\rho \\sigma ^{2}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      This implies that the variance of the mean increases with the average of the correlations. In other words, additional correlated observations are not as effective as additional independent observations at reducing the uncertainty of the mean. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to\n

                                                                      ", "content": [{"c": "This implies that the variance of the mean increases with the average of the correlations. In other words, additional correlated observations are not as effective as additional independent observations at reducing the uncertainty of the mean. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {1}{n}}+{\\frac {n-1}{n}}\\rho .}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {1}{n}}+{\\frac {n-1}{n}}\\rho .}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      This formula is used in the Spearman\u2013Brown prediction formula of classical test theory. This converges to \u03c1 if n goes to infinity, provided that the average correlation remains constant or converges too. So for the variance of the mean of standardized variables with equal correlations or converging average correlation we have\n

                                                                      ", "content": [{"c": "This formula is used in the Spearman\u2013Brown prediction formula of classical test theory. This converges to \u03c1 if n goes to infinity, provided that the average correlation remains constant or converges too. So for the variance of the mean of standardized variables with equal correlations or converging average correlation we have", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\lim _{n\\to \\infty }\\operatorname {Var} \\left({\\overline {X}}\\right)=\\rho .}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\lim _{n\\to \\infty }\\operatorname {Var} \\left({\\overline {X}}\\right)=\\rho .}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. This makes clear that the sample mean of correlated variables does not generally converge to the population mean, even though the law of large numbers states that the sample mean will converge for independent variables.\n

                                                                      ", "content": [{"c": "Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. This makes clear that the sample mean of correlated variables does not generally converge to the population mean, even though the law of large numbers states that the sample mean will converge for independent variables.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                                      Sum of uncorrelated variables with random sample size
                                                                      ", "content": {"title_content": "Sum of uncorrelated variables with random sample size", "level": "5"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size N is a random variable whose variation adds to the variation of X, such that,\n

                                                                      ", "content": [{"c": "There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size N is a random variable whose variation adds to the variation of X, such that,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\operatorname {E} \\left[N\\right]\\operatorname {Var} (X)+\\operatorname {Var} (N)(\\operatorname {E} \\left[X\\right])^{2}}[8]
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\operatorname {E} \\left[N\\right]\\operatorname {Var} (X)+\\operatorname {Var} (N)(\\operatorname {E} \\left[X\\right])^{2}}", "t": "equation-inline"}, {"c": "[", "t": "text"}, {"c": "8", "t": "text"}, {"c": "]", "t": "text"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      which follows from the law of total variance.\n

                                                                      ", "content": [{"c": "which follows from the law of total variance.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      If N has a Poisson distribution, then {\\displaystyle \\operatorname {E} [N]=\\operatorname {Var} (N)} with estimator n = N. So, the estimator of {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)} becomes {\\displaystyle n{S_{x}}^{2}+n{\\bar {X}}^{2}}, giving {\\displaystyle \\operatorname {SE} ({\\bar {X}})={\\sqrt {\\frac {{S_{x}}^{2}+{\\bar {X}}^{2}}{n}}}}\n(see standard error of the sample mean).\n

                                                                      ", "content": [{"c": "If N has a Poisson distribution, then", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} [N]=\\operatorname {Var} (N)}", "t": "equation-inline"}, {"c": "with estimator n= N. So, the estimator of", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)}", "t": "equation-inline"}, {"c": "becomes", "t": "text"}, {"c": "{\\displaystyle n{S_{x}}^{2}+n{\\bar {X}}^{2}}", "t": "equation-inline"}, {"c": ", giving", "t": "text"}, {"c": "{\\displaystyle \\operatorname {SE} ({\\bar {X}})={\\sqrt {\\frac {{S_{x}}^{2}+{\\bar {X}}^{2}}{n}}}}", "t": "equation-inline"}, {"c": "(see standard error of the sample mean).", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Weighted sum of variables

                                                                      ", "content": {"title_content": "Weighted sum of variables", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The scaling property and the Bienaym\u00e9 formula, along with the property of the covarianceCov(aX, bY) = ab Cov(X, Y) jointly imply that\n

                                                                      ", "content": [{"c": "The scaling property and the Bienaym\u00e9 formula, along with the property of the covariance Cov( aX, bY) = ab Cov( X, Y) jointly imply that", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (aX\\pm bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)\\pm 2ab\\,\\operatorname {Cov} (X,Y).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (aX\\pm bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)\\pm 2ab\\,\\operatorname {Cov} (X,Y).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y, then the weight of the variance of X will be four times the weight of the variance of Y.\n

                                                                      ", "content": [{"c": "This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y, then the weight of the variance of X will be four times the weight of the variance of Y.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The expression above can be extended to a weighted sum of multiple variables:\n

                                                                      ", "content": [{"c": "The expression above can be extended to a weighted sum of multiple variables:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left(\\sum _{i}^{n}a_{i}X_{i}\\right)=\\sum _{i=1}^{n}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i}\\sum _{<j\\leq n}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left(\\sum _{i}^{n}a_{i}X_{i}\\right)=\\sum _{i=1}^{n}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i}\\sum _{Product of variables", "content": {"title_content": "Product of variables", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Product of independent variables

                                                                      ", "content": {"title_content": "Product of independent variables", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      If two variables X and Y are independent, the variance of their product is given by[9]

                                                                      ", "content": [{"c": "If two variables X and Y are independent, the variance of their product is given by[ 9]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (XY)=[\\operatorname {E} (X)]^{2}\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\operatorname {Var} (X)+\\operatorname {Var} (X)\\operatorname {Var} (Y).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (XY)=[\\operatorname {E} (X)]^{2}\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\operatorname {Var} (X)+\\operatorname {Var} (X)\\operatorname {Var} (Y).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      Equivalently, using the basic properties of expectation, it is given by\n

                                                                      ", "content": [{"c": "Equivalently, using the basic properties of expectation, it is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} (XY)=\\operatorname {E} \\left(X^{2}\\right)\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (X)]^{2}[\\operatorname {E} (Y)]^{2}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} (XY)=\\operatorname {E} \\left(X^{2}\\right)\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (X)]^{2}[\\operatorname {E} (Y)]^{2}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                                      Product of statistically dependent variables

                                                                      ", "content": {"title_content": "Product of statistically dependent variables", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      In general, if two variables are statistically dependent, then the variance of their product is given by:\n

                                                                      ", "content": [{"c": "In general, if two variables are statistically dependent, then the variance of their product is given by:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\begin{aligned}\\operatorname {Var} (XY)={}&\\operatorname {E} \\left[X^{2}Y^{2}\\right]-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\operatorname {E} (X^{2})\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\left(\\operatorname {Var} (X)+[\\operatorname {E} (X)]^{2}\\right)\\left(\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\right)\\\\[5pt]&-[\\operatorname {Cov} (X,Y)+\\operatorname {E} (X)\\operatorname {E} (Y)]^{2}\\end{aligned}}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {Var} (XY)={}&\\operatorname {E} \\left[X^{2}Y^{2}\\right]-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\operatorname {E} (X^{2})\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\left(\\operatorname {Var} (X)+[\\operatorname {E} (X)]^{2}\\right)\\left(\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\right)\\\\[5pt]&-[\\operatorname {Cov} (X,Y)+\\operatorname {E} (X)\\operatorname {E} (Y)]^{2}\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                                      Arbitrary functions

                                                                      ", "content": {"title_content": "Arbitrary functions", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables: see Taylor expansions for the moments of functions of random variables. For example, the approximate variance of a function of one variable is given by\n

                                                                      ", "content": [{"c": "The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables: see Taylor expansions for the moments of functions of random variables. For example, the approximate variance of a function of one variable is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left[f(X)\\right]\\approx \\left(f'(\\operatorname {E} \\left[X\\right])\\right)^{2}\\operatorname {Var} \\left[X\\right]}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left[f(X)\\right]\\approx \\left(f'(\\operatorname {E} \\left[X\\right])\\right)^{2}\\operatorname {Var} \\left[X\\right]}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      provided that f is twice differentiable and that the mean and variance of X are finite.\n

                                                                      ", "content": [{"c": "provided that f is twice differentiable and that the mean and variance of X are finite.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Population variance and sample variance

                                                                      ", "content": {"title_content": "Population variance and sample variance", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one estimates the mean and variance from a limited set of observations by using an estimator equation. The estimator is a function of the sample of nobservations drawn without observational bias from the whole population of potential observations. In this example, the sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.\n

                                                                      ", "content": [{"c": "Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one estimates the mean and variance from a limited set of observations by using an estimator equation. The estimator is a function of the sample of n observations drawn without observational bias from the whole population of potential observations. In this example, the sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance \u2013 these are consistent estimators (they converge to the value of the whole population as the number of samples increases) but can be improved. Most simply, the sample variance is computed as the sum of squared deviations about the (sample) mean, divided by n as the number of samples. However, using values other than n improves the estimator in various ways. Four common values for the denominator are n,n \u2212 1, n + 1, and n \u2212 1.5: n is the simplest (the variance of the sample), n \u2212 1 eliminates bias,[10]n + 1 minimizes mean squared error for the normal distribution,[11] and n \u2212 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.[12]

                                                                      ", "content": [{"c": "The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and(uncorrected) sample variance\u2013 these are consistent estimators(they converge to the value of the whole population as the number of samples increases) but can be improved. Most simply, the sample variance is computed as the sum of squared deviations about the (sample) mean, divided by n as the number of samples. However, using values other than n improves the estimator in various ways. Four common values for the denominator are n, n \u2212 1, n+ 1, and n \u2212 1.5: n is the simplest (the variance of the sample), n \u2212 1 eliminates bias,[ 10] n+ 1 minimizes mean squared error for the normal distribution,[ 11] and n \u2212 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.[ 12]", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (n \u2212 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n -1 instead of n, is called Bessel's correction.[10] The resulting estimator is unbiased and is called the (corrected) sample variance or unbiased sample variance. If the mean is determined in some other way than from the same samples used to estimate the variance, then this bias does not arise, and the variance can safely be estimated as that of the samples about the (independently known) mean.\n

                                                                      ", "content": [{"c": "Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of ( n \u2212 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n-1 instead of n, is called Bessel's correction.[ 10] The resulting estimator is unbiased and is called the(corrected) sample variance or unbiased sample variance. If the mean is determined in some other way than from the same samples used to estimate the variance, then this bias does not arise, and the variance can safely be estimated as that of the samples about the (independently known) mean.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Secondly, the sample variance does not generally minimize mean squared error between sample variance and population variance. Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance) and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than n \u2212 1) and is a simple example of a shrinkage estimator: one \"shrinks\" the unbiased estimator towards zero. For the normal distribution, dividing by n + 1 (instead of n \u2212 1 or n) minimizes mean squared error.[11] The resulting estimator is biased, however, and is known as the biased sample variation.\n

                                                                      ", "content": [{"c": "Secondly, the sample variance does not generally minimize mean squared error between sample variance and population variance. Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance) and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than n \u2212 1) and is a simple example of a shrinkage estimator: one \"shrinks\" the unbiased estimator towards zero. For the normal distribution, dividing by n+ 1 (instead of n \u2212 1 or n) minimizes mean squared error.[ 11] The resulting estimator is biased, however, and is known as the biased sample variation.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Population variance

                                                                      ", "content": {"title_content": "Population variance", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      In general, the population variance of a finitepopulation of size N with values xi is given by

                                                                      ", "content": [{"c": "In general, the population variance of a finite population of size N with values x i is given by", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\u03c32=1N\u2211i=1N(xi\u2212\u03bc)2=1N\u2211i=1N(xi2\u22122\u03bcxi+\u03bc2)=(1N\u2211i=1Nxi2)\u22122\u03bc(1N\u2211i=1Nxi)+\u03bc2=E\u2061[xi2]\u2212\u03bc2{\\displaystyle {\\begin{aligned}\\sigma ^{2}&={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}-\\mu \\right)^{2}={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}^{2}-2\\mu x_{i}+\\mu ^{2}\\right)\\\\[5pt]&=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-2\\mu \\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)+\\mu ^{2}\\\\[5pt]&=\\operatorname {E} [x_{i}^{2}]-\\mu ^{2}\\end{aligned}}}", "content": {"math_content": "{\\displaystyle {\\begin{aligned}\\sigma ^{2}&={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}-\\mu \\right)^{2}={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}^{2}-2\\mu x_{i}+\\mu ^{2}\\right)\\\\[5pt]&=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-2\\mu \\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)+\\mu ^{2}\\\\[5pt]&=\\operatorname {E} [x_{i}^{2}]-\\mu ^{2}\\end{aligned}}}", "math_type": "mathml", "by": "None"}}, {"type": "paragraph", "raw_content": "

                                                                      where the population mean is {\\textstyle \\mu =\\operatorname {E} [x_{i}]={\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}} and {\\textstyle \\operatorname {E} [x_{i}^{2}]=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)}, where {\\textstyle \\operatorname {E} } is the expectation value operator.\n

                                                                      ", "content": [{"c": "where the population mean is", "t": "text"}, {"c": "{\\textstyle \\mu =\\operatorname {E} [x_{i}]={\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\textstyle \\operatorname {E} [x_{i}^{2}]=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)}", "t": "equation-inline"}, {"c": ", where", "t": "text"}, {"c": "{\\textstyle \\operatorname {E} }", "t": "equation-inline"}, {"c": "is the expectation value operator.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The population variance can also be computed using[13]

                                                                      ", "content": [{"c": "The population variance can also be computed using[ 13]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\sigma ^{2}={\\frac {1}{N^{2}}}\\sum _{i<j}\\left(x_{i}-x_{j}\\right)^{2}={\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\sigma ^{2}={\\frac {1}{N^{2}}}\\sum _{i

                                                                      (The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because

                                                                      ", "content": [{"c": "(The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because", "t": "text"}]}, {"type": "equation-interline", "raw_content": "12N2\u2211i,j=1N(xi\u2212xj)2=12N2\u2211i,j=1N(xi2\u22122xixj+xj2)=12N\u2211j=1N(1N\u2211i=1Nxi2)\u2212(1N\u2211i=1Nxi)(1N\u2211j=1Nxj)+12N\u2211i=1N(1N\u2211j=1Nxj2)=12(\u03c32+\u03bc2)\u2212\u03bc2+12(\u03c32+\u03bc2)=\u03c32.{\\displaystyle {\\begin{aligned}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}\\\\[5pt]={}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2N}}\\sum _{j=1}^{N}\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}\\right)+{\\frac {1}{2N}}\\sum _{i=1}^{N}\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-\\mu ^{2}+{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\\\[5pt]={}&\\sigma ^{2}.\\end{aligned}}}", "content": {"math_content": "{\\displaystyle {\\begin{aligned}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}\\\\[5pt]={}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2N}}\\sum _{j=1}^{N}\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}\\right)+{\\frac {1}{2N}}\\sum _{i=1}^{N}\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-\\mu ^{2}+{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\\\[5pt]={}&\\sigma ^{2}.\\end{aligned}}}", "math_type": "mathml", "by": "None"}}, {"type": "paragraph", "raw_content": "

                                                                      The population variance matches the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.\n

                                                                      ", "content": [{"c": "The population variance matches the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Sample variance

                                                                      ", "content": {"title_content": "Sample variance", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Biased sample variance

                                                                      ", "content": {"title_content": "Biased sample variance", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population.[14] This is generally referred to as sample variance or empirical variance. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution.\n

                                                                      ", "content": [{"c": "In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population.[ 14] This is generally referred to as sample variance or empirical variance. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      We take a sample with replacement of n values Y1, ..., Yn from the population of size {\\textstyle N}, where n < N, and estimate the variance on the basis of this sample.[15] Directly taking the variance of the sample data gives the average of the squared deviations:\n

                                                                      ", "content": [{"c": "We take a sample with replacement of n values Y 1, ..., Y n from the population of size", "t": "text"}, {"c": "{\\textstyle N}", "t": "equation-inline"}, {"c": ", where n< N, and estimate the variance on the basis of this sample.[ 15] Directly taking the variance of the sample data gives the average of the squared deviations:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\tilde {S}}_{Y}^{2}={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}=\\left({\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}^{2}\\right)-{\\overline {Y}}^{2}={\\frac {1}{n^{2}}}\\sum _{i,j\\,:\\,i<j}\\left(Y_{i}-Y_{j}\\right)^{2}.}[16]
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\tilde {S}}_{Y}^{2}={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}=\\left({\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}^{2}\\right)-{\\overline {Y}}^{2}={\\frac {1}{n^{2}}}\\sum _{i,j\\,:\\,i

                                                                      (See the section Population variance for the derivation of this formula.) Here, {\\displaystyle {\\overline {Y}}} denotes the sample mean:\n

                                                                      ", "content": [{"c": "(See the section Population variance for the derivation of this formula.) Here,", "t": "text"}, {"c": "{\\displaystyle {\\overline {Y}}}", "t": "equation-inline"}, {"c": "denotes the sample mean:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\overline {Y}}={\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\overline {Y}}={\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      Since the Yi are selected randomly, both {\\displaystyle {\\overline {Y}}} and {\\displaystyle {\\tilde {S}}_{Y}^{2}} are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples {Yi} of size n from the population. For {\\displaystyle {\\tilde {S}}_{Y}^{2}} this gives:\n

                                                                      ", "content": [{"c": "Since the Y i are selected randomly, both", "t": "text"}, {"c": "{\\displaystyle {\\overline {Y}}}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle {\\tilde {S}}_{Y}^{2}}", "t": "equation-inline"}, {"c": "are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples { Y i} of size n from the population. For", "t": "text"}, {"c": "{\\displaystyle {\\tilde {S}}_{Y}^{2}}", "t": "equation-inline"}, {"c": "this gives:", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle {\\begin{aligned}\\operatorname {E} [{\\tilde {S}}_{Y}^{2}]&=\\operatorname {E} \\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\frac {1}{n}}\\sum _{j=1}^{n}Y_{j}\\right)^{2}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\operatorname {E} \\left[Y_{i}^{2}-{\\frac {2}{n}}Y_{i}\\sum _{j=1}^{n}Y_{j}+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}Y_{j}\\sum _{k=1}^{n}Y_{k}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\left(\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+\\operatorname {E} \\left[Y_{i}^{2}\\right]\\right)+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left({\\frac {n-2}{n}}\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left[{\\frac {n-2}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-{\\frac {2}{n}}(n-1)\\mu ^{2}+{\\frac {1}{n^{2}}}n(n-1)\\mu ^{2}+{\\frac {1}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\right]\\\\[5pt]&={\\frac {n-1}{n}}\\sigma ^{2}.\\end{aligned}}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle {\\begin{aligned}\\operatorname {E} [{\\tilde {S}}_{Y}^{2}]&=\\operatorname {E} \\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\frac {1}{n}}\\sum _{j=1}^{n}Y_{j}\\right)^{2}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\operatorname {E} \\left[Y_{i}^{2}-{\\frac {2}{n}}Y_{i}\\sum _{j=1}^{n}Y_{j}+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}Y_{j}\\sum _{k=1}^{n}Y_{k}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\left(\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+\\operatorname {E} \\left[Y_{i}^{2}\\right]\\right)+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left({\\frac {n-2}{n}}\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left[{\\frac {n-2}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-{\\frac {2}{n}}(n-1)\\mu ^{2}+{\\frac {1}{n^{2}}}n(n-1)\\mu ^{2}+{\\frac {1}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\right]\\\\[5pt]&={\\frac {n-1}{n}}\\sigma ^{2}.\\end{aligned}}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      Here {\\textstyle \\sigma ^{2}=\\operatorname {E} [Y_{i}^{2}]-\\mu ^{2}} derived in the section Population variance and {\\textstyle \\operatorname {E} [Y_{i}Y_{j}]=\\operatorname {E} [Y_{i}]\\operatorname {E} [Y_{j}]=\\mu ^{2}} due to independency of {\\textstyle Y_{i}} and {\\textstyle Y_{j}} are used.\n

                                                                      ", "content": [{"c": "Here", "t": "text"}, {"c": "{\\textstyle \\sigma ^{2}=\\operatorname {E} [Y_{i}^{2}]-\\mu ^{2}}", "t": "equation-inline"}, {"c": "derived in the section Population variance and", "t": "text"}, {"c": "{\\textstyle \\operatorname {E} [Y_{i}Y_{j}]=\\operatorname {E} [Y_{i}]\\operatorname {E} [Y_{j}]=\\mu ^{2}}", "t": "equation-inline"}, {"c": "due to independency of", "t": "text"}, {"c": "{\\textstyle Y_{i}}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\textstyle Y_{j}}", "t": "equation-inline"}, {"c": "are used.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Hence {\\textstyle {\\tilde {S}}_{Y}^{2}} gives an estimate of the population variance that is biased by a factor of {\\textstyle {\\frac {n-1}{n}}} as the expectation value of {\\textstyle {\\tilde {S}}_{Y}^{2}} is smaller than the population variance (true variance) by that factor. For this reason, {\\textstyle {\\tilde {S}}_{Y}^{2}} is referred to as the biased sample variance.\n

                                                                      ", "content": [{"c": "Hence", "t": "text"}, {"c": "{\\textstyle {\\tilde {S}}_{Y}^{2}}", "t": "equation-inline"}, {"c": "gives an estimate of the population variance that is biased by a factor of", "t": "text"}, {"c": "{\\textstyle {\\frac {n-1}{n}}}", "t": "equation-inline"}, {"c": "as the expectation value of", "t": "text"}, {"c": "{\\textstyle {\\tilde {S}}_{Y}^{2}}", "t": "equation-inline"}, {"c": "is smaller than the population variance (true variance) by that factor. For this reason,", "t": "text"}, {"c": "{\\textstyle {\\tilde {S}}_{Y}^{2}}", "t": "equation-inline"}, {"c": "is referred to as the biased sample variance.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Unbiased sample variance

                                                                      ", "content": {"title_content": "Unbiased sample variance", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Correcting for this bias yields the unbiased sample variance, denoted {\\displaystyle S^{2}}:\n

                                                                      ", "content": [{"c": "Correcting for this bias yields the unbiased sample variance, denoted", "t": "text"}, {"c": "{\\displaystyle S^{2}}", "t": "equation-inline"}, {"c": ":", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle S^{2}={\\frac {n}{n-1}}{\\tilde {S}}_{Y}^{2}={\\frac {n}{n-1}}\\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}\\right]={\\frac {1}{n-1}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle S^{2}={\\frac {n}{n-1}}{\\tilde {S}}_{Y}^{2}={\\frac {n}{n-1}}\\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}\\right]={\\frac {1}{n-1}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      Either estimator may be simply referred to as the sample variance when the version can be determined by context. The same proof is also applicable for samples taken from a continuous probability distribution.\n

                                                                      ", "content": [{"c": "Either estimator may be simply referred to as the sample variance when the version can be determined by context. The same proof is also applicable for samples taken from a continuous probability distribution.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The use of the term n \u2212 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation (the square root of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n \u2212 1.5 yields an almost unbiased estimator.\n

                                                                      ", "content": [{"c": "The use of the term n \u2212 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation(the square root of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n \u2212 1.5 yields an almost unbiased estimator.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The unbiased sample variance is a U-statistic for the function \u0192(y1, y2) = (y1 \u2212 y2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.\n

                                                                      ", "content": [{"c": "The unbiased sample variance is a U-statistic for the function \u0192( y 1, y 2) = ( y 1 \u2212 y 2) 2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.", "t": "text"}]}, {"type": "title", "raw_content": "
                                                                      Example
                                                                      ", "content": {"title_content": "Example", "level": "5"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      For a set of numbers {10, 15, 30, 45, 57, 52 63, 72, 81, 93, 102, 105}, if this set is the whole data population for some measurement, then variance is the population variance 932.743 as the sum of the squared deviations about the mean of this set, divided by 12 as the number of the set members. If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.\n

                                                                      ", "content": [{"c": "For a set of numbers {10, 15, 30, 45, 57, 52 63, 72, 81, 93, 102, 105}, if this set is the whole data population for some measurement, then variance is the population variance 932.743 as the sum of the squared deviations about the mean of this set, divided by 12 as the number of the set members. If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Distribution of the sample variance

                                                                      ", "content": {"title_content": "Distribution of the sample variance", "level": "4"}}, {"type": "paragraph", "raw_content": "
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                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Yi are independent observations from a normal distribution, Cochran's theorem shows that the unbiased sample varianceS2 follows a scaled chi-squared distribution (see also: asymptotic properties and an elementary proof):[17]

                                                                      ", "content": [{"c": "Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Y i are independent observations from a normal distribution, Cochran's theorem shows that the unbiased sample variance S 2 follows a scaled chi-squared distribution(see also: asymptotic properties and an elementary proof):[ 17]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle (n-1){\\frac {S^{2}}{\\sigma ^{2}}}\\sim \\chi _{n-1}^{2}}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle (n-1){\\frac {S^{2}}{\\sigma ^{2}}}\\sim \\chi _{n-1}^{2}}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      where \u03c32 is the population variance. As a direct consequence, it follows that\n

                                                                      ", "content": [{"c": "where \u03c3 2 is the population variance. As a direct consequence, it follows that", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {E} \\left(S^{2}\\right)=\\operatorname {E} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)=\\sigma ^{2},}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {E} \\left(S^{2}\\right)=\\operatorname {E} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)=\\sigma ^{2},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      and[18]

                                                                      ", "content": [{"c": "and[ 18]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {Var} \\left[S^{2}\\right]=\\operatorname {Var} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)={\\frac {\\sigma ^{4}}{(n-1)^{2}}}\\operatorname {Var} \\left(\\chi _{n-1}^{2}\\right)={\\frac {2\\sigma ^{4}}{n-1}}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {Var} \\left[S^{2}\\right]=\\operatorname {Var} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)={\\frac {\\sigma ^{4}}{(n-1)^{2}}}\\operatorname {Var} \\left(\\chi _{n-1}^{2}\\right)={\\frac {2\\sigma ^{4}}{n-1}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      If Yi are independent and identically distributed, but not necessarily normally distributed, then[19]

                                                                      ", "content": [{"c": "If Y i are independent and identically distributed, but not necessarily normally distributed, then[ 19]", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\operatorname {E} \\left[S^{2}\\right]=\\sigma ^{2},\\quad \\operatorname {Var} \\left[S^{2}\\right]={\\frac {\\sigma ^{4}}{n}}\\left(\\kappa -1+{\\frac {2}{n-1}}\\right)={\\frac {1}{n}}\\left(\\mu _{4}-{\\frac {n-3}{n-1}}\\sigma ^{4}\\right),}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\operatorname {E} \\left[S^{2}\\right]=\\sigma ^{2},\\quad \\operatorname {Var} \\left[S^{2}\\right]={\\frac {\\sigma ^{4}}{n}}\\left(\\kappa -1+{\\frac {2}{n-1}}\\right)={\\frac {1}{n}}\\left(\\mu _{4}-{\\frac {n-3}{n-1}}\\sigma ^{4}\\right),}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      where \u03ba is the kurtosis of the distribution and \u03bc4 is the fourth central moment.\n

                                                                      ", "content": [{"c": "where \u03ba is the kurtosis of the distribution and \u03bc 4 is the fourth central moment.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      If the conditions of the law of large numbers hold for the squared observations, S2 is a consistent estimator of \u03c32. One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.).[20][21][22]

                                                                      ", "content": [{"c": "If the conditions of the law of large numbers hold for the squared observations, S 2 is a consistent estimator of \u03c3 2. One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.).[ 20][ 21][ 22]", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Samuelson's inequality

                                                                      ", "content": {"title_content": "Samuelson's inequality", "level": "4"}}, {"type": "paragraph", "raw_content": "
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                                                                      Samuelson's inequality is a result that states bounds on the values that individual observations in a sample can take, given that the sample mean and (biased) variance have been calculated.[23] Values must lie within the limits {\\displaystyle {\\bar {y}}\\pm \\sigma _{Y}(n-1)^{1/2}.}

                                                                      ", "content": [{"c": "Samuelson's inequality is a result that states bounds on the values that individual observations in a sample can take, given that the sample mean and (biased) variance have been calculated.[ 23] Values must lie within the limits", "t": "text"}, {"c": "{\\displaystyle {\\bar {y}}\\pm \\sigma _{Y}(n-1)^{1/2}.}", "t": "equation-inline"}]}, {"type": "title", "raw_content": "

                                                                      Relations with the harmonic and arithmetic means

                                                                      ", "content": {"title_content": "Relations with the harmonic and arithmetic means", "level": "3"}}, {"type": "paragraph", "raw_content": "
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                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      It has been shown[24] that for a sample {yi} of positive real numbers,\n

                                                                      ", "content": [{"c": "It has been shown[ 24] that for a sample { y i} of positive real numbers,", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\sigma _{y}^{2}\\leq 2y_{\\max }(A-H),}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\sigma _{y}^{2}\\leq 2y_{\\max }(A-H),}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      where ymax is the maximum of the sample, A is the arithmetic mean, H is the harmonic mean of the sample and {\\displaystyle \\sigma _{y}^{2}} is the (biased) variance of the sample.\n

                                                                      ", "content": [{"c": "where y max is the maximum of the sample, A is the arithmetic mean, H is the harmonic mean of the sample and", "t": "text"}, {"c": "{\\displaystyle \\sigma _{y}^{2}}", "t": "equation-inline"}, {"c": "is the (biased) variance of the sample.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      This bound has been improved, and it is known that variance is bounded by\n

                                                                      ", "content": [{"c": "This bound has been improved, and it is known that variance is bounded by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\sigma _{y}^{2}\\leq {\\frac {y_{\\max }(A-H)(y_{\\max }-A)}{y_{\\max }-H}},}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\sigma _{y}^{2}\\leq {\\frac {y_{\\max }(A-H)(y_{\\max }-A)}{y_{\\max }-H}},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\sigma _{y}^{2}\\geq {\\frac {y_{\\min }(A-H)(A-y_{\\min })}{H-y_{\\min }}},}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\sigma _{y}^{2}\\geq {\\frac {y_{\\min }(A-H)(A-y_{\\min })}{H-y_{\\min }}},}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      where ymin is the minimum of the sample.[25]

                                                                      ", "content": [{"c": "where y min is the minimum of the sample.[ 25]", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Tests of equality of variances

                                                                      ", "content": {"title_content": "Tests of equality of variances", "level": "2"}}, {"type": "paragraph", "raw_content": "
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                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult.\n

                                                                      ", "content": [{"c": "The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Several non parametric tests have been proposed: these include the Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey test, the Capon test, Mood test, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that both medians be known and equal to zero. The Mood, Klotz, Capon and Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey tests also apply to two variances. They allow the median to be unknown but do require that the two medians are equal.\n

                                                                      ", "content": [{"c": "Several non parametric tests have been proposed: these include the Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey test, the Capon test, Mood test, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that both medians be known and equal to zero. The Mood, Klotz, Capon and Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey tests also apply to two variances. They allow the median to be unknown but do require that the two medians are equal.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The Lehmann test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variances include the Box test, the Box\u2013Anderson test and the Moses test.\n

                                                                      ", "content": [{"c": "The Lehmann test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variances include the Box test, the Box\u2013Anderson test and the Moses test.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      Resampling methods, which include the bootstrap and the jackknife, may be used to test the equality of variances.\n

                                                                      ", "content": [{"c": "Resampling methods, which include the bootstrap and the jackknife, may be used to test the equality of variances.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Moment of inertia

                                                                      ", "content": {"title_content": "Moment of inertia", "level": "2"}}, {"type": "paragraph", "raw_content": "
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                                                                      The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass.[26] It is because of this analogy that such things as the variance are called moments of probability distributions.[26] The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of n points with a covariance matrix of {\\displaystyle \\Sigma } is given by\n

                                                                      ", "content": [{"c": "The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass.[ 26] It is because of this analogy that such things as the variance are called moments of probability distributions.[ 26] The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of n points with a covariance matrix of", "t": "text"}, {"c": "{\\displaystyle \\Sigma }", "t": "equation-inline"}, {"c": "is given by", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle I=n\\left(\\mathbf {1} _{3\\times 3}\\operatorname {tr} (\\Sigma )-\\Sigma \\right).}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle I=n\\left(\\mathbf {1} _{3\\times 3}\\operatorname {tr} (\\Sigma )-\\Sigma \\right).}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the x axis and distributed along it. The covariance matrix might look like\n

                                                                      ", "content": [{"c": "This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the x axis and distributed along it. The covariance matrix might look like", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle \\Sigma ={\\begin{bmatrix}10&0&0\\\\0&0.1&0\\\\0&0&0.1\\end{bmatrix}}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle \\Sigma ={\\begin{bmatrix}10&0&0\\\\0&0.1&0\\\\0&0&0.1\\end{bmatrix}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "paragraph", "raw_content": "

                                                                      That is, there is the most variance in the x direction. Physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is\n

                                                                      ", "content": [{"c": "That is, there is the most variance in the x direction. Physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is", "t": "text"}]}, {"type": "list", "raw_content": "
                                                                      {\\displaystyle I=n{\\begin{bmatrix}0.2&0&0\\\\0&10.1&0\\\\0&0&10.1\\end{bmatrix}}.}
                                                                      ", "content": {"items": [[[{"c": "{\\displaystyle I=n{\\begin{bmatrix}0.2&0&0\\\\0&10.1&0\\\\0&0&10.1\\end{bmatrix}}.}", "t": "equation-inline"}]]], "ordered": true}}, {"type": "title", "raw_content": "

                                                                      Semivariance

                                                                      ", "content": {"title_content": "Semivariance", "level": "2"}}, {"type": "paragraph", "raw_content": "
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                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation:

                                                                      ", "content": [{"c": "The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "Semivariance=1n\u2211i:xi<\u03bc(xi\u2212\u03bc)2{\\displaystyle {\\text{Semivariance}}={1 \\over {n}}\\sum _{i:x_{i}<\\mu }(x_{i}-\\mu )^{2}}", "content": {"math_content": "{\\displaystyle {\\text{Semivariance}}={1 \\over {n}}\\sum _{i:x_{i}<\\mu }(x_{i}-\\mu )^{2}}", "math_type": "mathml", "by": "None"}}, {"type": "paragraph", "raw_content": "

                                                                      It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not.[27]

                                                                      ", "content": [{"c": "It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not.[ 27]", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      For inequalities associated with the semivariance, see Chebyshev's inequality \u00a7 Semivariances.\n

                                                                      ", "content": [{"c": "For inequalities associated with the semivariance, see Chebyshev's inequality \u00a7 Semivariances.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Etymology

                                                                      ", "content": {"title_content": "Etymology", "level": "2"}}, {"type": "paragraph", "raw_content": "
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                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance:[28]

                                                                      ", "content": [{"c": "The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance:[ 28]", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and, therefore, that the variability may be uniformly measured by the standard deviation corresponding to the square root of the mean square error. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations {\\displaystyle \\sigma _{1}} and {\\displaystyle \\sigma _{2}}, it is found that the distribution, when both causes act together, has a standard deviation {\\displaystyle {\\sqrt {\\sigma _{1}^{2}+\\sigma _{2}^{2}}}}. It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance...

                                                                      ", "content": [{"c": "The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and, therefore, that the variability may be uniformly measured by the standard deviation corresponding to the square root of the mean square error. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations", "t": "text"}, {"c": "{\\displaystyle \\sigma _{1}}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle \\sigma _{2}}", "t": "equation-inline"}, {"c": ", it is found that the distribution, when both causes act together, has a standard deviation", "t": "text"}, {"c": "{\\displaystyle {\\sqrt {\\sigma _{1}^{2}+\\sigma _{2}^{2}}}}", "t": "equation-inline"}, {"c": ". It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance...", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      Generalizations

                                                                      ", "content": {"title_content": "Generalizations", "level": "2"}}, {"type": "paragraph", "raw_content": "
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                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      For complex variables

                                                                      ", "content": {"title_content": "For complex variables", "level": "3"}}, {"type": "paragraph", "raw_content": "
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                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      If {\\displaystyle x} is a scalar complex-valued random variable, with values in {\\displaystyle \\mathbb {C} ,} then its variance is {\\displaystyle \\operatorname {E} \\left[(x-\\mu )(x-\\mu )^{*}\\right],} where {\\displaystyle x^{*}} is the complex conjugate of {\\displaystyle x.} This variance is a real scalar.\n

                                                                      ", "content": [{"c": "If", "t": "text"}, {"c": "{\\displaystyle x}", "t": "equation-inline"}, {"c": "is a scalar complex-valued random variable, with values in", "t": "text"}, {"c": "{\\displaystyle \\mathbb {C} ,}", "t": "equation-inline"}, {"c": "then its variance is", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} \\left[(x-\\mu )(x-\\mu )^{*}\\right],}", "t": "equation-inline"}, {"c": "where", "t": "text"}, {"c": "{\\displaystyle x^{*}}", "t": "equation-inline"}, {"c": "is the complex conjugate of", "t": "text"}, {"c": "{\\displaystyle x.}", "t": "equation-inline"}, {"c": "This variance is a real scalar.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      For vector-valued random variables

                                                                      ", "content": {"title_content": "For vector-valued random variables", "level": "3"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      As a matrix

                                                                      ", "content": {"title_content": "As a matrix", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
                                                                      ", "content": [{"c": "edit", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      If {\\displaystyle X} is a vector-valued random variable, with values in {\\displaystyle \\mathbb {R} ^{n},} and thought of as a column vector, then a natural generalization of variance is {\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\operatorname {T} }\\right],} where {\\displaystyle \\mu =\\operatorname {E} (X)} and {\\displaystyle X^{\\operatorname {T} }} is the transpose of {\\displaystyle X,} and so is a row vector. The result is a positive semi-definite square matrix, commonly referred to as the variance-covariance matrix (or simply as the covariance matrix).\n

                                                                      ", "content": [{"c": "If", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "is a vector-valued random variable, with values in", "t": "text"}, {"c": "{\\displaystyle \\mathbb {R} ^{n},}", "t": "equation-inline"}, {"c": "and thought of as a column vector, then a natural generalization of variance is", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\operatorname {T} }\\right],}", "t": "equation-inline"}, {"c": "where", "t": "text"}, {"c": "{\\displaystyle \\mu =\\operatorname {E} (X)}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "{\\displaystyle X^{\\operatorname {T} }}", "t": "equation-inline"}, {"c": "is the transpose of", "t": "text"}, {"c": "{\\displaystyle X,}", "t": "equation-inline"}, {"c": "and so is a row vector. The result is a positive semi-definite square matrix, commonly referred to as the variance-covariance matrix(or simply as the covariance matrix).", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      If {\\displaystyle X} is a vector- and complex-valued random variable, with values in {\\displaystyle \\mathbb {C} ^{n},} then the covariance matrix is{\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\dagger }\\right],} where {\\displaystyle X^{\\dagger }} is the conjugate transpose of {\\displaystyle X.} This matrix is also positive semi-definite and square.\n

                                                                      ", "content": [{"c": "If", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": "is a vector- and complex-valued random variable, with values in", "t": "text"}, {"c": "{\\displaystyle \\mathbb {C} ^{n},}", "t": "equation-inline"}, {"c": "then the covariance matrix is", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\dagger }\\right],}", "t": "equation-inline"}, {"c": "where", "t": "text"}, {"c": "{\\displaystyle X^{\\dagger }}", "t": "equation-inline"}, {"c": "is the conjugate transpose of", "t": "text"}, {"c": "{\\displaystyle X.}", "t": "equation-inline"}, {"c": "This matrix is also positive semi-definite and square.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      As a scalar

                                                                      ", "content": {"title_content": "As a scalar", "level": "4"}}, {"type": "paragraph", "raw_content": "
                                                                      edit
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                                                                      Another generalization of variance for vector-valued random variables {\\displaystyle X}, which results in a scalar value rather than in a matrix, is the generalized variance{\\displaystyle \\det(C)}, the determinant of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.[29]

                                                                      ", "content": [{"c": "Another generalization of variance for vector-valued random variables", "t": "text"}, {"c": "{\\displaystyle X}", "t": "equation-inline"}, {"c": ", which results in a scalar value rather than in a matrix, is the generalized variance", "t": "text"}, {"c": "{\\displaystyle \\det(C)}", "t": "equation-inline"}, {"c": ", the determinant of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.[ 29]", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                      A different generalization is obtained by considering the equation for the scalar variance, {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right]}, and reinterpreting {\\displaystyle (X-\\mu )^{2}} as the squared Euclidean distance between the random variable and its mean, or, simply as the scalar product of the vector {\\displaystyle X-\\mu } with itself. This results in {\\displaystyle \\operatorname {E} \\left[(X-\\mu )^{\\operatorname {T} }(X-\\mu )\\right]=\\operatorname {tr} (C),} which is the trace of the covariance matrix.\n

                                                                      ", "content": [{"c": "A different generalization is obtained by considering the equation for the scalar variance,", "t": "text"}, {"c": "{\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right]}", "t": "equation-inline"}, {"c": ", and reinterpreting", "t": "text"}, {"c": "{\\displaystyle (X-\\mu )^{2}}", "t": "equation-inline"}, {"c": "as the squared Euclidean distance between the random variable and its mean, or, simply as the scalar product of the vector", "t": "text"}, {"c": "{\\displaystyle X-\\mu }", "t": "equation-inline"}, {"c": "with itself. This results in", "t": "text"}, {"c": "{\\displaystyle \\operatorname {E} \\left[(X-\\mu )^{\\operatorname {T} }(X-\\mu )\\right]=\\operatorname {tr} (C),}", "t": "equation-inline"}, {"c": "which is the trace of the covariance matrix.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                      See also

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                                                                        Types of variance

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                                                                          References

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                                                                            In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by {\\displaystyle \\sigma ^{2}}, {\\displaystyle s^{2}}, {\\displaystyle \\operatorname {Var} (X)}, {\\displaystyle V(X)}, or {\\displaystyle \\mathbb {V} (X)}.[1]

                                                                            Example of samples from two populations with the same mean but different variances. The red population has mean 100 and variance 100 (SD=10) while the blue population has mean 100 and variance 2500 (SD=50) where SD stands for Standard Deviation.

                                                                            An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.\n

                                                                            There are two distinct concepts that are both called \"variance\". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.\n

                                                                            The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling.\n

                                                                            Geometric visualisation of the variance of an arbitrary distribution (2, 4, 4, 4, 5, 5, 7, 9):
                                                                            1. A frequency distribution is constructed.
                                                                            2. The centroid of the distribution gives its mean.
                                                                            3. A square with sides equal to the difference of each value from the mean is formed for each value.
                                                                            4. Arranging the squares into a rectangle with one side equal to the number of values, n, results in the other side being the distribution's variance, \u03c32.

                                                                            Definition

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                                                                            The variance of a random variable {\\displaystyle X} is the expected value of the squared deviation from the mean of {\\displaystyle X}, {\\displaystyle \\mu =\\operatorname {E} [X]}:\n

                                                                            {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right].}

                                                                            This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:\n

                                                                            {\\displaystyle \\operatorname {Var} (X)=\\operatorname {Cov} (X,X).}

                                                                            The variance is also equivalent to the second cumulant of a probability distribution that generates {\\displaystyle X}. The variance is typically designated as {\\displaystyle \\operatorname {Var} (X)}, or sometimes as {\\displaystyle V(X)} or {\\displaystyle \\mathbb {V} (X)}, or symbolically as {\\displaystyle \\sigma _{X}^{2}} or simply {\\displaystyle \\sigma ^{2}} (pronounced \"sigma squared\"). The expression for the variance can be expanded as follows:\n

                                                                            {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left[(X-\\operatorname {E} [X])^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}-2X\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]^{2}+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}\\end{aligned}}}

                                                                            In other words, the variance of X is equal to the mean of the square of X minus the square of the mean of X. This equation should not be used for computations using floating point arithmetic, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see algorithms for calculating variance.\n

                                                                            Discrete random variable

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                                                                            If the generator of random variable {\\displaystyle X} is discrete with probability mass function{\\displaystyle x_{1}\\mapsto p_{1},x_{2}\\mapsto p_{2},\\ldots ,x_{n}\\mapsto p_{n}}, then\n

                                                                            {\\displaystyle \\operatorname {Var} (X)=\\sum _{i=1}^{n}p_{i}\\cdot (x_{i}-\\mu )^{2},}

                                                                            where {\\displaystyle \\mu } is the expected value. That is,\n

                                                                            {\\displaystyle \\mu =\\sum _{i=1}^{n}p_{i}x_{i}.}

                                                                            (When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)\n

                                                                            The variance of a collection of {\\displaystyle n} equally likely values can be written as\n

                                                                            {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n}}\\sum _{i=1}^{n}(x_{i}-\\mu )^{2}}

                                                                            where {\\displaystyle \\mu } is the average value. That is,\n

                                                                            {\\displaystyle \\mu ={\\frac {1}{n}}\\sum _{i=1}^{n}x_{i}.}

                                                                            The variance of a set of {\\displaystyle n} equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:[2]

                                                                            {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{\\frac {1}{2}}(x_{i}-x_{j})^{2}={\\frac {1}{n^{2}}}\\sum _{i}\\sum _{j>i}(x_{i}-x_{j})^{2}.}

                                                                            Absolutely continuous random variable

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                                                                            If the random variable {\\displaystyle X} has a probability density function{\\displaystyle f(x)}, and {\\displaystyle F(x)} is the corresponding cumulative distribution function, then\n

                                                                            {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)=\\sigma ^{2}&=\\int _{\\mathbb {R} }(x-\\mu )^{2}f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-2\\mu \\int _{\\mathbb {R} }xf(x)\\,dx+\\mu ^{2}\\int _{\\mathbb {R} }f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\int _{\\mathbb {R} }x\\,dF(x)+\\mu ^{2}\\int _{\\mathbb {R} }\\,dF(x)\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\cdot \\mu +\\mu ^{2}\\cdot 1\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-\\mu ^{2},\\end{aligned}}}

                                                                            or equivalently,\n

                                                                            {\\displaystyle \\operatorname {Var} (X)=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-\\mu ^{2},}

                                                                            where {\\displaystyle \\mu } is the expected value of {\\displaystyle X} given by\n

                                                                            {\\displaystyle \\mu =\\int _{\\mathbb {R} }xf(x)\\,dx=\\int _{\\mathbb {R} }x\\,dF(x).}

                                                                            In these formulas, the integrals with respect to {\\displaystyle dx} and {\\displaystyle dF(x)}\nare Lebesgue and Lebesgue\u2013Stieltjes integrals, respectively.\n

                                                                            If the function {\\displaystyle x^{2}f(x)} is Riemann-integrable on every finite interval {\\displaystyle [a,b]\\subset \\mathbb {R} ,} then\n

                                                                            {\\displaystyle \\operatorname {Var} (X)=\\int _{-\\infty }^{+\\infty }x^{2}f(x)\\,dx-\\mu ^{2},}

                                                                            where the integral is an improper Riemann integral.\n

                                                                            Examples

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                                                                            Exponential distribution

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                                                                            The exponential distribution with parameter \u03bb is a continuous distribution whose probability density function is given by\n

                                                                            {\\displaystyle f(x)=\\lambda e^{-\\lambda x}}

                                                                            on the interval [0, \u221e). Its mean can be shown to be\n

                                                                            {\\displaystyle \\operatorname {E} [X]=\\int _{0}^{\\infty }x\\lambda e^{-\\lambda x}\\,dx={\\frac {1}{\\lambda }}.}

                                                                            Using integration by parts and making use of the expected value already calculated, we have:\n

                                                                            {\\displaystyle {\\begin{aligned}\\operatorname {E} \\left[X^{2}\\right]&=\\int _{0}^{\\infty }x^{2}\\lambda e^{-\\lambda x}\\,dx\\\\&=\\left[-x^{2}e^{-\\lambda x}\\right]_{0}^{\\infty }+\\int _{0}^{\\infty }2xe^{-\\lambda x}\\,dx\\\\&=0+{\\frac {2}{\\lambda }}\\operatorname {E} [X]\\\\&={\\frac {2}{\\lambda ^{2}}}.\\end{aligned}}}

                                                                            Thus, the variance of X is given by\n

                                                                            {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}={\\frac {2}{\\lambda ^{2}}}-\\left({\\frac {1}{\\lambda }}\\right)^{2}={\\frac {1}{\\lambda ^{2}}}.}

                                                                            Fair dice

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                                                                            A fair six-sided dice can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is {\\displaystyle (1+2+3+4+5+6)/6=7/2.} Therefore, the variance of X is\n

                                                                            {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\sum _{i=1}^{6}{\\frac {1}{6}}\\left(i-{\\frac {7}{2}}\\right)^{2}\\\\[5pt]&={\\frac {1}{6}}\\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\\right)\\\\[5pt]&={\\frac {35}{12}}\\approx 2.92.\\end{aligned}}}

                                                                            The general formula for the variance of the outcome, X, of an n-sided die is\n

                                                                            {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left(X^{2}\\right)-(\\operatorname {E} (X))^{2}\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}i^{2}-\\left({\\frac {1}{n}}\\sum _{i=1}^{n}i\\right)^{2}\\\\[5pt]&={\\frac {(n+1)(2n+1)}{6}}-\\left({\\frac {n+1}{2}}\\right)^{2}\\\\[4pt]&={\\frac {n^{2}-1}{12}}.\\end{aligned}}}

                                                                            Commonly used probability distributions

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                                                                            The following table lists the variance for some commonly used probability distributions.\n

                                                                            Name of the probability distribution\nProbability distribution function\nMean\nVariance\n
                                                                            Binomial distributionPr(X=k)=(nk)pk(1\u2212p)n\u2212k{\\displaystyle \\Pr \\,(X=k)={\\binom {n}{k}}p^{k}(1-p)^{n-k}}np{\\displaystyle np}np(1\u2212p){\\displaystyle np(1-p)}
                                                                            Geometric distributionPr(X=k)=(1\u2212p)k\u22121p{\\displaystyle \\Pr \\,(X=k)=(1-p)^{k-1}p}1p{\\displaystyle {\\frac {1}{p}}}(1\u2212p)p2{\\displaystyle {\\frac {(1-p)}{p^{2}}}}
                                                                            Normal distributionf(x\u2223\u03bc,\u03c32)=12\u03c0\u03c32e\u2212(x\u2212\u03bc)22\u03c32{\\displaystyle f\\left(x\\mid \\mu ,\\sigma ^{2}\\right)={\\frac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}e^{-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}}\u03bc{\\displaystyle \\mu }\u03c32{\\displaystyle \\sigma ^{2}}
                                                                            Uniform distribution (continuous)f(x\u2223a,b)={1b\u2212afor a\u2264x\u2264b,0for x<a or x>b{\\displaystyle f(x\\mid a,b)={\\begin{cases}{\\frac {1}{b-a}}&{\\text{for }}a\\leq x\\leq b,\\\\[3pt]0&{\\text{for }}x<a{\\text{ or }}x>b\\end{cases}}}a+b2{\\displaystyle {\\frac {a+b}{2}}}(b\u2212a)212{\\displaystyle {\\frac {(b-a)^{2}}{12}}}
                                                                            Exponential distributionf(x\u2223\u03bb)=\u03bbe\u2212\u03bbx{\\displaystyle f(x\\mid \\lambda )=\\lambda e^{-\\lambda x}}1\u03bb{\\displaystyle {\\frac {1}{\\lambda }}}1\u03bb2{\\displaystyle {\\frac {1}{\\lambda ^{2}}}}
                                                                            Poisson distributionf(k\u2223\u03bb)=e\u2212\u03bb\u03bbkk!{\\displaystyle f(k\\mid \\lambda )={\\frac {e^{-\\lambda }\\lambda ^{k}}{k!}}}\u03bb{\\displaystyle \\lambda }\u03bb{\\displaystyle \\lambda }

                                                                            Properties

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                                                                            Basic properties

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                                                                            Variance is non-negative because the squares are positive or zero:\n

                                                                            {\\displaystyle \\operatorname {Var} (X)\\geq 0.}

                                                                            The variance of a constant is zero.\n

                                                                            {\\displaystyle \\operatorname {Var} (a)=0.}

                                                                            Conversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value:\n

                                                                            {\\displaystyle \\operatorname {Var} (X)=0\\iff \\exists a:P(X=a)=1.}

                                                                            Issues of finiteness

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                                                                            If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index{\\displaystyle k} satisfies {\\displaystyle 1<k\\leq 2.}

                                                                            Decomposition

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                                                                            The general formula for variance decomposition or the law of total variance is: If {\\displaystyle X} and {\\displaystyle Y} are two random variables, and the variance of {\\displaystyle X} exists, then\n

                                                                            {\\displaystyle \\operatorname {Var} [X]=\\operatorname {E} (\\operatorname {Var} [X\\mid Y])+\\operatorname {Var} (\\operatorname {E} [X\\mid Y]).}

                                                                            The conditional expectation{\\displaystyle \\operatorname {E} (X\\mid Y)} of {\\displaystyle X} given {\\displaystyle Y}, and the conditional variance{\\displaystyle \\operatorname {Var} (X\\mid Y)} may be understood as follows. Given any particular value y of the random variable Y, there is a conditional expectation {\\displaystyle \\operatorname {E} (X\\mid Y=y)} given the event Y = y. This quantity depends on the particular value y; it is a function {\\displaystyle g(y)=\\operatorname {E} (X\\mid Y=y)}. That same function evaluated at the random variable Y is the conditional expectation {\\displaystyle \\operatorname {E} (X\\mid Y)=g(Y).}

                                                                            In particular, if {\\displaystyle Y} is a discrete random variable assuming possible values {\\displaystyle y_{1},y_{2},y_{3}\\ldots } with corresponding probabilities {\\displaystyle p_{1},p_{2},p_{3}\\ldots ,}, then in the formula for total variance, the first term on the right-hand side becomes\n

                                                                            {\\displaystyle \\operatorname {E} (\\operatorname {Var} [X\\mid Y])=\\sum _{i}p_{i}\\sigma _{i}^{2},}

                                                                            where {\\displaystyle \\sigma _{i}^{2}=\\operatorname {Var} [X\\mid Y=y_{i}]}. Similarly, the second term on the right-hand side becomes\n

                                                                            {\\displaystyle \\operatorname {Var} (\\operatorname {E} [X\\mid Y])=\\sum _{i}p_{i}\\mu _{i}^{2}-\\left(\\sum _{i}p_{i}\\mu _{i}\\right)^{2}=\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2},}

                                                                            where {\\displaystyle \\mu _{i}=\\operatorname {E} [X\\mid Y=y_{i}]} and {\\displaystyle \\mu =\\sum _{i}p_{i}\\mu _{i}}. Thus the total variance is given by\n

                                                                            {\\displaystyle \\operatorname {Var} [X]=\\sum _{i}p_{i}\\sigma _{i}^{2}+\\left(\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2}\\right).}

                                                                            A similar formula is applied in analysis of variance, where the corresponding formula is\n

                                                                            {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{between}}+{\\mathit {MS}}_{\\text{within}};}

                                                                            here {\\displaystyle {\\mathit {MS}}} refers to the Mean of the Squares. In linear regression analysis the corresponding formula is\n

                                                                            {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{regression}}+{\\mathit {MS}}_{\\text{residual}}.}

                                                                            This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.\n

                                                                            Similar decompositions are possible for the sum of squared deviations (sum of squares, {\\displaystyle {\\mathit {SS}}}):\n

                                                                            {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{between}}+{\\mathit {SS}}_{\\text{within}},}
                                                                            {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{regression}}+{\\mathit {SS}}_{\\text{residual}}.}

                                                                            Calculation from the CDF

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                                                                            The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution functionF using\n

                                                                            {\\displaystyle 2\\int _{0}^{\\infty }u(1-F(u))\\,du-\\left(\\int _{0}^{\\infty }(1-F(u))\\,du\\right)^{2}.}

                                                                            This expression can be used to calculate the variance in situations where the CDF, but not the density, can be conveniently expressed.\n

                                                                            Characteristic property

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                                                                            The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} \\left(\\left(X-m\\right)^{2}\\right)=\\mathrm {E} (X)}. Conversely, if a continuous function {\\displaystyle \\varphi } satisfies {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} (\\varphi (X-m))=\\mathrm {E} (X)} for all random variables X, then it is necessarily of the form {\\displaystyle \\varphi (x)=ax^{2}+b}, where a > 0. This also holds in the multidimensional case.[3]

                                                                            Units of measurement

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                                                                            Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is \u221a2.9 \u2248 1.7, slightly larger than the expected absolute deviation of 1.5.\n

                                                                            The standard deviation and the expected absolute deviation can both be used as an indicator of the \"spread\" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution.\n

                                                                            Propagation

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                                                                            Addition and multiplication by a constant

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                                                                            Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged:\n

                                                                            {\\displaystyle \\operatorname {Var} (X+a)=\\operatorname {Var} (X).}

                                                                            If all values are scaled by a constant, the variance is scaled by the square of that constant:\n

                                                                            {\\displaystyle \\operatorname {Var} (aX)=a^{2}\\operatorname {Var} (X).}

                                                                            The variance of a sum of two random variables is given by\n

                                                                            {\\displaystyle \\operatorname {Var} (aX+bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)+2ab\\,\\operatorname {Cov} (X,Y)}
                                                                            {\\displaystyle \\operatorname {Var} (aX-bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)-2ab\\,\\operatorname {Cov} (X,Y)}

                                                                            where {\\displaystyle \\operatorname {Cov} (X,Y)} is the covariance.\n

                                                                            Linear combinations

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                                                                            In general, for the sum of {\\displaystyle N} random variables {\\displaystyle \\{X_{1},\\dots ,X_{N}\\}}, the variance becomes:\n

                                                                            {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i,j=1}^{N}\\operatorname {Cov} (X_{i},X_{j})=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i})+\\sum _{i\\neq j}\\operatorname {Cov} (X_{i},X_{j}),}

                                                                            see also general Bienaym\u00e9's identity.\n

                                                                            These results lead to the variance of a linear combination as:\n

                                                                            {\\displaystyle {\\begin{aligned}\\operatorname {Var} \\left(\\sum _{i=1}^{N}a_{i}X_{i}\\right)&=\\sum _{i,j=1}^{N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+\\sum _{i\\not =j}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i<j\\leq N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j}).\\end{aligned}}}

                                                                            If the random variables {\\displaystyle X_{1},\\dots ,X_{N}} are such that\n

                                                                            {\\displaystyle \\operatorname {Cov} (X_{i},X_{j})=0\\ ,\\ \\forall \\ (i\\neq j),}

                                                                            then they are said to be uncorrelated. It follows immediately from the expression given earlier that if the random variables {\\displaystyle X_{1},\\dots ,X_{N}} are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:\n

                                                                            {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i}).}

                                                                            Since independent random variables are always uncorrelated (see Covariance \u00a7 Uncorrelatedness and independence), the equation above holds in particular when the random variables {\\displaystyle X_{1},\\dots ,X_{n}} are independent. Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.\n

                                                                            Matrix notation for the variance of a linear combination

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                                                                            Define {\\displaystyle X} as a column vector of {\\displaystyle n} random variables {\\displaystyle X_{1},\\ldots ,X_{n}}, and {\\displaystyle c} as a column vector of {\\displaystyle n} scalars {\\displaystyle c_{1},\\ldots ,c_{n}}. Therefore, {\\displaystyle c^{\\mathsf {T}}X} is a linear combination of these random variables, where {\\displaystyle c^{\\mathsf {T}}} denotes the transpose of {\\displaystyle c}. Also let {\\displaystyle \\Sigma } be the covariance matrix of {\\displaystyle X}. The variance of {\\displaystyle c^{\\mathsf {T}}X} is then given by:[4]

                                                                            {\\displaystyle \\operatorname {Var} \\left(c^{\\mathsf {T}}X\\right)=c^{\\mathsf {T}}\\Sigma c.}

                                                                            This implies that the variance of the mean can be written as (with a column vector of ones)\n

                                                                            {\\displaystyle \\operatorname {Var} \\left({\\bar {x}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}1'X\\right)={\\frac {1}{n^{2}}}1'\\Sigma 1.}

                                                                            Sum of variables

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                                                                            Sum of uncorrelated variables

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                                                                            One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances:\n

                                                                            {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} (X_{i}).}

                                                                            This statement is called the Bienaym\u00e9 formula[5] and was discovered in 1853.[6][7] It is often made with the stronger condition that the variables are independent, but being uncorrelated suffices. So if all the variables have the same variance \u03c32, then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is\n

                                                                            {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}\\sum _{i=1}^{n}X_{i}\\right)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)={\\frac {1}{n^{2}}}n\\sigma ^{2}={\\frac {\\sigma ^{2}}{n}}.}

                                                                            That is, the variance of the mean decreases when n increases. This formula for the variance of the mean is used in the definition of the standard error of the sample mean, which is used in the central limit theorem.\n

                                                                            To prove the initial statement, it suffices to show that\n

                                                                            {\\displaystyle \\operatorname {Var} (X+Y)=\\operatorname {Var} (X)+\\operatorname {Var} (Y).}

                                                                            The general result then follows by induction. Starting with the definition,\n

                                                                            {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[(X+Y)^{2}\\right]-(\\operatorname {E} [X+Y])^{2}\\\\[5pt]&=\\operatorname {E} \\left[X^{2}+2XY+Y^{2}\\right]-(\\operatorname {E} [X]+\\operatorname {E} [Y])^{2}.\\end{aligned}}}

                                                                            Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y, this further simplifies as follows:\n

                                                                            {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[X^{2}\\right]+2\\operatorname {E} [XY]+\\operatorname {E} \\left[Y^{2}\\right]-\\left(\\operatorname {E} [X]^{2}+2\\operatorname {E} [X]\\operatorname {E} [Y]+\\operatorname {E} [Y]^{2}\\right)\\\\[5pt]&=\\operatorname {E} \\left[X^{2}\\right]+\\operatorname {E} \\left[Y^{2}\\right]-\\operatorname {E} [X]^{2}-\\operatorname {E} [Y]^{2}\\\\[5pt]&=\\operatorname {Var} (X)+\\operatorname {Var} (Y).\\end{aligned}}}

                                                                            Sum of correlated variables

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                                                                            Sum of correlated variables with fixed sample size
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                                                                            In general, the variance of the sum of n variables is the sum of their covariances:\n

                                                                            {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\sum _{j=1}^{n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)+2\\sum _{1\\leq i<j\\leq n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right).}

                                                                            (Note: The second equality comes from the fact that Cov(Xi,Xi) = Var(Xi).)\n

                                                                            Here, {\\displaystyle \\operatorname {Cov} (\\cdot ,\\cdot )} is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. The next expression states equivalently that the variance of the sum is the sum of the diagonal of covariance matrix plus two times the sum of its upper triangular elements (or its lower triangular elements); this emphasizes that the covariance matrix is symmetric. This formula is used in the theory of Cronbach's alpha in classical test theory.\n

                                                                            So, if the variables have equal variance \u03c32 and the average correlation of distinct variables is \u03c1, then the variance of their mean is\n

                                                                            {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {\\sigma ^{2}}{n}}+{\\frac {n-1}{n}}\\rho \\sigma ^{2}.}

                                                                            This implies that the variance of the mean increases with the average of the correlations. In other words, additional correlated observations are not as effective as additional independent observations at reducing the uncertainty of the mean. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to\n

                                                                            {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {1}{n}}+{\\frac {n-1}{n}}\\rho .}

                                                                            This formula is used in the Spearman\u2013Brown prediction formula of classical test theory. This converges to \u03c1 if n goes to infinity, provided that the average correlation remains constant or converges too. So for the variance of the mean of standardized variables with equal correlations or converging average correlation we have\n

                                                                            {\\displaystyle \\lim _{n\\to \\infty }\\operatorname {Var} \\left({\\overline {X}}\\right)=\\rho .}

                                                                            Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. This makes clear that the sample mean of correlated variables does not generally converge to the population mean, even though the law of large numbers states that the sample mean will converge for independent variables.\n

                                                                            Sum of uncorrelated variables with random sample size
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                                                                            There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size N is a random variable whose variation adds to the variation of X, such that,\n

                                                                            {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\operatorname {E} \\left[N\\right]\\operatorname {Var} (X)+\\operatorname {Var} (N)(\\operatorname {E} \\left[X\\right])^{2}}[8]

                                                                            which follows from the law of total variance.\n

                                                                            If N has a Poisson distribution, then {\\displaystyle \\operatorname {E} [N]=\\operatorname {Var} (N)} with estimator n = N. So, the estimator of {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)} becomes {\\displaystyle n{S_{x}}^{2}+n{\\bar {X}}^{2}}, giving {\\displaystyle \\operatorname {SE} ({\\bar {X}})={\\sqrt {\\frac {{S_{x}}^{2}+{\\bar {X}}^{2}}{n}}}}\n(see standard error of the sample mean).\n

                                                                            Weighted sum of variables

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                                                                            The scaling property and the Bienaym\u00e9 formula, along with the property of the covarianceCov(aX, bY) = ab Cov(X, Y) jointly imply that\n

                                                                            {\\displaystyle \\operatorname {Var} (aX\\pm bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)\\pm 2ab\\,\\operatorname {Cov} (X,Y).}

                                                                            This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y, then the weight of the variance of X will be four times the weight of the variance of Y.\n

                                                                            The expression above can be extended to a weighted sum of multiple variables:\n

                                                                            {\\displaystyle \\operatorname {Var} \\left(\\sum _{i}^{n}a_{i}X_{i}\\right)=\\sum _{i=1}^{n}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i}\\sum _{<j\\leq n}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})}

                                                                            Product of variables

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                                                                            Product of independent variables

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                                                                            If two variables X and Y are independent, the variance of their product is given by[9]

                                                                            {\\displaystyle \\operatorname {Var} (XY)=[\\operatorname {E} (X)]^{2}\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\operatorname {Var} (X)+\\operatorname {Var} (X)\\operatorname {Var} (Y).}

                                                                            Equivalently, using the basic properties of expectation, it is given by\n

                                                                            {\\displaystyle \\operatorname {Var} (XY)=\\operatorname {E} \\left(X^{2}\\right)\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (X)]^{2}[\\operatorname {E} (Y)]^{2}.}

                                                                            Product of statistically dependent variables

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                                                                            In general, if two variables are statistically dependent, then the variance of their product is given by:\n

                                                                            {\\displaystyle {\\begin{aligned}\\operatorname {Var} (XY)={}&\\operatorname {E} \\left[X^{2}Y^{2}\\right]-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\operatorname {E} (X^{2})\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\left(\\operatorname {Var} (X)+[\\operatorname {E} (X)]^{2}\\right)\\left(\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\right)\\\\[5pt]&-[\\operatorname {Cov} (X,Y)+\\operatorname {E} (X)\\operatorname {E} (Y)]^{2}\\end{aligned}}}

                                                                            Arbitrary functions

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                                                                            The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables: see Taylor expansions for the moments of functions of random variables. For example, the approximate variance of a function of one variable is given by\n

                                                                            {\\displaystyle \\operatorname {Var} \\left[f(X)\\right]\\approx \\left(f'(\\operatorname {E} \\left[X\\right])\\right)^{2}\\operatorname {Var} \\left[X\\right]}

                                                                            provided that f is twice differentiable and that the mean and variance of X are finite.\n

                                                                            Population variance and sample variance

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                                                                            Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one estimates the mean and variance from a limited set of observations by using an estimator equation. The estimator is a function of the sample of nobservations drawn without observational bias from the whole population of potential observations. In this example, the sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.\n

                                                                            The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance \u2013 these are consistent estimators (they converge to the value of the whole population as the number of samples increases) but can be improved. Most simply, the sample variance is computed as the sum of squared deviations about the (sample) mean, divided by n as the number of samples. However, using values other than n improves the estimator in various ways. Four common values for the denominator are n,n \u2212 1, n + 1, and n \u2212 1.5: n is the simplest (the variance of the sample), n \u2212 1 eliminates bias,[10]n + 1 minimizes mean squared error for the normal distribution,[11] and n \u2212 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.[12]

                                                                            Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (n \u2212 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n -1 instead of n, is called Bessel's correction.[10] The resulting estimator is unbiased and is called the (corrected) sample variance or unbiased sample variance. If the mean is determined in some other way than from the same samples used to estimate the variance, then this bias does not arise, and the variance can safely be estimated as that of the samples about the (independently known) mean.\n

                                                                            Secondly, the sample variance does not generally minimize mean squared error between sample variance and population variance. Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance) and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than n \u2212 1) and is a simple example of a shrinkage estimator: one \"shrinks\" the unbiased estimator towards zero. For the normal distribution, dividing by n + 1 (instead of n \u2212 1 or n) minimizes mean squared error.[11] The resulting estimator is biased, however, and is known as the biased sample variation.\n

                                                                            Population variance

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                                                                            In general, the population variance of a finitepopulation of size N with values xi is given by

                                                                            \u03c32=1N\u2211i=1N(xi\u2212\u03bc)2=1N\u2211i=1N(xi2\u22122\u03bcxi+\u03bc2)=(1N\u2211i=1Nxi2)\u22122\u03bc(1N\u2211i=1Nxi)+\u03bc2=E\u2061[xi2]\u2212\u03bc2{\\displaystyle {\\begin{aligned}\\sigma ^{2}&={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}-\\mu \\right)^{2}={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}^{2}-2\\mu x_{i}+\\mu ^{2}\\right)\\\\[5pt]&=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-2\\mu \\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)+\\mu ^{2}\\\\[5pt]&=\\operatorname {E} [x_{i}^{2}]-\\mu ^{2}\\end{aligned}}}

                                                                            where the population mean is {\\textstyle \\mu =\\operatorname {E} [x_{i}]={\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}} and {\\textstyle \\operatorname {E} [x_{i}^{2}]=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)}, where {\\textstyle \\operatorname {E} } is the expectation value operator.\n

                                                                            The population variance can also be computed using[13]

                                                                            {\\displaystyle \\sigma ^{2}={\\frac {1}{N^{2}}}\\sum _{i<j}\\left(x_{i}-x_{j}\\right)^{2}={\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}.}

                                                                            (The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because

                                                                            12N2\u2211i,j=1N(xi\u2212xj)2=12N2\u2211i,j=1N(xi2\u22122xixj+xj2)=12N\u2211j=1N(1N\u2211i=1Nxi2)\u2212(1N\u2211i=1Nxi)(1N\u2211j=1Nxj)+12N\u2211i=1N(1N\u2211j=1Nxj2)=12(\u03c32+\u03bc2)\u2212\u03bc2+12(\u03c32+\u03bc2)=\u03c32.{\\displaystyle {\\begin{aligned}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}\\\\[5pt]={}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2N}}\\sum _{j=1}^{N}\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}\\right)+{\\frac {1}{2N}}\\sum _{i=1}^{N}\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-\\mu ^{2}+{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\\\[5pt]={}&\\sigma ^{2}.\\end{aligned}}}

                                                                            The population variance matches the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.\n

                                                                            Sample variance

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                                                                            Biased sample variance

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                                                                            In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population.[14] This is generally referred to as sample variance or empirical variance. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution.\n

                                                                            We take a sample with replacement of n values Y1, ..., Yn from the population of size {\\textstyle N}, where n < N, and estimate the variance on the basis of this sample.[15] Directly taking the variance of the sample data gives the average of the squared deviations:\n

                                                                            {\\displaystyle {\\tilde {S}}_{Y}^{2}={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}=\\left({\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}^{2}\\right)-{\\overline {Y}}^{2}={\\frac {1}{n^{2}}}\\sum _{i,j\\,:\\,i<j}\\left(Y_{i}-Y_{j}\\right)^{2}.}[16]

                                                                            (See the section Population variance for the derivation of this formula.) Here, {\\displaystyle {\\overline {Y}}} denotes the sample mean:\n

                                                                            {\\displaystyle {\\overline {Y}}={\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}.}

                                                                            Since the Yi are selected randomly, both {\\displaystyle {\\overline {Y}}} and {\\displaystyle {\\tilde {S}}_{Y}^{2}} are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples {Yi} of size n from the population. For {\\displaystyle {\\tilde {S}}_{Y}^{2}} this gives:\n

                                                                            {\\displaystyle {\\begin{aligned}\\operatorname {E} [{\\tilde {S}}_{Y}^{2}]&=\\operatorname {E} \\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\frac {1}{n}}\\sum _{j=1}^{n}Y_{j}\\right)^{2}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\operatorname {E} \\left[Y_{i}^{2}-{\\frac {2}{n}}Y_{i}\\sum _{j=1}^{n}Y_{j}+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}Y_{j}\\sum _{k=1}^{n}Y_{k}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\left(\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+\\operatorname {E} \\left[Y_{i}^{2}\\right]\\right)+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left({\\frac {n-2}{n}}\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left[{\\frac {n-2}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-{\\frac {2}{n}}(n-1)\\mu ^{2}+{\\frac {1}{n^{2}}}n(n-1)\\mu ^{2}+{\\frac {1}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\right]\\\\[5pt]&={\\frac {n-1}{n}}\\sigma ^{2}.\\end{aligned}}}

                                                                            Here {\\textstyle \\sigma ^{2}=\\operatorname {E} [Y_{i}^{2}]-\\mu ^{2}} derived in the section Population variance and {\\textstyle \\operatorname {E} [Y_{i}Y_{j}]=\\operatorname {E} [Y_{i}]\\operatorname {E} [Y_{j}]=\\mu ^{2}} due to independency of {\\textstyle Y_{i}} and {\\textstyle Y_{j}} are used.\n

                                                                            Hence {\\textstyle {\\tilde {S}}_{Y}^{2}} gives an estimate of the population variance that is biased by a factor of {\\textstyle {\\frac {n-1}{n}}} as the expectation value of {\\textstyle {\\tilde {S}}_{Y}^{2}} is smaller than the population variance (true variance) by that factor. For this reason, {\\textstyle {\\tilde {S}}_{Y}^{2}} is referred to as the biased sample variance.\n

                                                                            Unbiased sample variance

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                                                                            Correcting for this bias yields the unbiased sample variance, denoted {\\displaystyle S^{2}}:\n

                                                                            {\\displaystyle S^{2}={\\frac {n}{n-1}}{\\tilde {S}}_{Y}^{2}={\\frac {n}{n-1}}\\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}\\right]={\\frac {1}{n-1}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}}

                                                                            Either estimator may be simply referred to as the sample variance when the version can be determined by context. The same proof is also applicable for samples taken from a continuous probability distribution.\n

                                                                            The use of the term n \u2212 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation (the square root of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n \u2212 1.5 yields an almost unbiased estimator.\n

                                                                            The unbiased sample variance is a U-statistic for the function \u0192(y1, y2) = (y1 \u2212 y2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.\n

                                                                            Example
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                                                                            For a set of numbers {10, 15, 30, 45, 57, 52 63, 72, 81, 93, 102, 105}, if this set is the whole data population for some measurement, then variance is the population variance 932.743 as the sum of the squared deviations about the mean of this set, divided by 12 as the number of the set members. If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.\n

                                                                            Distribution of the sample variance

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                                                                            Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Yi are independent observations from a normal distribution, Cochran's theorem shows that the unbiased sample varianceS2 follows a scaled chi-squared distribution (see also: asymptotic properties and an elementary proof):[17]

                                                                            {\\displaystyle (n-1){\\frac {S^{2}}{\\sigma ^{2}}}\\sim \\chi _{n-1}^{2}}

                                                                            where \u03c32 is the population variance. As a direct consequence, it follows that\n

                                                                            {\\displaystyle \\operatorname {E} \\left(S^{2}\\right)=\\operatorname {E} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)=\\sigma ^{2},}

                                                                            and[18]

                                                                            {\\displaystyle \\operatorname {Var} \\left[S^{2}\\right]=\\operatorname {Var} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)={\\frac {\\sigma ^{4}}{(n-1)^{2}}}\\operatorname {Var} \\left(\\chi _{n-1}^{2}\\right)={\\frac {2\\sigma ^{4}}{n-1}}.}

                                                                            If Yi are independent and identically distributed, but not necessarily normally distributed, then[19]

                                                                            {\\displaystyle \\operatorname {E} \\left[S^{2}\\right]=\\sigma ^{2},\\quad \\operatorname {Var} \\left[S^{2}\\right]={\\frac {\\sigma ^{4}}{n}}\\left(\\kappa -1+{\\frac {2}{n-1}}\\right)={\\frac {1}{n}}\\left(\\mu _{4}-{\\frac {n-3}{n-1}}\\sigma ^{4}\\right),}

                                                                            where \u03ba is the kurtosis of the distribution and \u03bc4 is the fourth central moment.\n

                                                                            If the conditions of the law of large numbers hold for the squared observations, S2 is a consistent estimator of \u03c32. One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.).[20][21][22]

                                                                            Samuelson's inequality

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                                                                            Samuelson's inequality is a result that states bounds on the values that individual observations in a sample can take, given that the sample mean and (biased) variance have been calculated.[23] Values must lie within the limits {\\displaystyle {\\bar {y}}\\pm \\sigma _{Y}(n-1)^{1/2}.}

                                                                            Relations with the harmonic and arithmetic means

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                                                                            It has been shown[24] that for a sample {yi} of positive real numbers,\n

                                                                            {\\displaystyle \\sigma _{y}^{2}\\leq 2y_{\\max }(A-H),}

                                                                            where ymax is the maximum of the sample, A is the arithmetic mean, H is the harmonic mean of the sample and {\\displaystyle \\sigma _{y}^{2}} is the (biased) variance of the sample.\n

                                                                            This bound has been improved, and it is known that variance is bounded by\n

                                                                            {\\displaystyle \\sigma _{y}^{2}\\leq {\\frac {y_{\\max }(A-H)(y_{\\max }-A)}{y_{\\max }-H}},}
                                                                            {\\displaystyle \\sigma _{y}^{2}\\geq {\\frac {y_{\\min }(A-H)(A-y_{\\min })}{H-y_{\\min }}},}

                                                                            where ymin is the minimum of the sample.[25]

                                                                            Tests of equality of variances

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                                                                            The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult.\n

                                                                            Several non parametric tests have been proposed: these include the Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey test, the Capon test, Mood test, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that both medians be known and equal to zero. The Mood, Klotz, Capon and Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey tests also apply to two variances. They allow the median to be unknown but do require that the two medians are equal.\n

                                                                            The Lehmann test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variances include the Box test, the Box\u2013Anderson test and the Moses test.\n

                                                                            Resampling methods, which include the bootstrap and the jackknife, may be used to test the equality of variances.\n

                                                                            Moment of inertia

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                                                                            The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass.[26] It is because of this analogy that such things as the variance are called moments of probability distributions.[26] The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of n points with a covariance matrix of {\\displaystyle \\Sigma } is given by\n

                                                                            {\\displaystyle I=n\\left(\\mathbf {1} _{3\\times 3}\\operatorname {tr} (\\Sigma )-\\Sigma \\right).}

                                                                            This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the x axis and distributed along it. The covariance matrix might look like\n

                                                                            {\\displaystyle \\Sigma ={\\begin{bmatrix}10&0&0\\\\0&0.1&0\\\\0&0&0.1\\end{bmatrix}}.}

                                                                            That is, there is the most variance in the x direction. Physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is\n

                                                                            {\\displaystyle I=n{\\begin{bmatrix}0.2&0&0\\\\0&10.1&0\\\\0&0&10.1\\end{bmatrix}}.}

                                                                            Semivariance

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                                                                            The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation:

                                                                            Semivariance=1n\u2211i:xi<\u03bc(xi\u2212\u03bc)2{\\displaystyle {\\text{Semivariance}}={1 \\over {n}}\\sum _{i:x_{i}<\\mu }(x_{i}-\\mu )^{2}}

                                                                            It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not.[27]

                                                                            For inequalities associated with the semivariance, see Chebyshev's inequality \u00a7 Semivariances.\n

                                                                            Etymology

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                                                                            The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance:[28]

                                                                            The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and, therefore, that the variability may be uniformly measured by the standard deviation corresponding to the square root of the mean square error. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations {\\displaystyle \\sigma _{1}} and {\\displaystyle \\sigma _{2}}, it is found that the distribution, when both causes act together, has a standard deviation {\\displaystyle {\\sqrt {\\sigma _{1}^{2}+\\sigma _{2}^{2}}}}. It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance...

                                                                            Generalizations

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                                                                            For complex variables

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                                                                            If {\\displaystyle x} is a scalar complex-valued random variable, with values in {\\displaystyle \\mathbb {C} ,} then its variance is {\\displaystyle \\operatorname {E} \\left[(x-\\mu )(x-\\mu )^{*}\\right],} where {\\displaystyle x^{*}} is the complex conjugate of {\\displaystyle x.} This variance is a real scalar.\n

                                                                            For vector-valued random variables

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                                                                            As a matrix

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                                                                            If {\\displaystyle X} is a vector-valued random variable, with values in {\\displaystyle \\mathbb {R} ^{n},} and thought of as a column vector, then a natural generalization of variance is {\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\operatorname {T} }\\right],} where {\\displaystyle \\mu =\\operatorname {E} (X)} and {\\displaystyle X^{\\operatorname {T} }} is the transpose of {\\displaystyle X,} and so is a row vector. The result is a positive semi-definite square matrix, commonly referred to as the variance-covariance matrix (or simply as the covariance matrix).\n

                                                                            If {\\displaystyle X} is a vector- and complex-valued random variable, with values in {\\displaystyle \\mathbb {C} ^{n},} then the covariance matrix is{\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\dagger }\\right],} where {\\displaystyle X^{\\dagger }} is the conjugate transpose of {\\displaystyle X.} This matrix is also positive semi-definite and square.\n

                                                                            As a scalar

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                                                                            Another generalization of variance for vector-valued random variables {\\displaystyle X}, which results in a scalar value rather than in a matrix, is the generalized variance{\\displaystyle \\det(C)}, the determinant of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.[29]

                                                                            A different generalization is obtained by considering the equation for the scalar variance, {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right]}, and reinterpreting {\\displaystyle (X-\\mu )^{2}} as the squared Euclidean distance between the random variable and its mean, or, simply as the scalar product of the vector {\\displaystyle X-\\mu } with itself. This results in {\\displaystyle \\operatorname {E} \\left[(X-\\mu )^{\\operatorname {T} }(X-\\mu )\\right]=\\operatorname {tr} (C),} which is the trace of the covariance matrix.\n

                                                                            See also

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                                                                              Types of variance

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                                                                                References

                                                                                edit
                                                                                  ", "statics": {"paragraph": 178, "paragraph.text": 297, "paragraph.equation-inline": 122, "image": 2, "list": 74, "list.text": 15, "title": 49, "list.equation-inline": 71, "table": 1, "equation-interline": 3}, "url": "https://en.m.wikipedia.org/wiki/Variance", "content": "In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation(SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by ${\\displaystyle \\sigma ^{2}}$ , ${\\displaystyle s^{2}}$ , ${\\displaystyle \\operatorname {Var} (X)}$ , ${\\displaystyle V(X)}$ , or ${\\displaystyle \\mathbb {V} (X)}$ .[ 1]\n\nAn advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.\n\nThere are two distinct concepts that are both called \"variance\". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.\n\nThe two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling.\n\n1. A frequency distribution is constructed.\n2. The centroid of the distribution gives its mean.\n3. A square with sides equal to the difference of each value from the mean is formed for each value.\n4. Arranging the squares into a rectangle with one side equal to the number of values, n , results in the other side being the distribution's variance, \u03c3 2 .\n\n## Definition\n\nedit\n\nThe variance of a random variable ${\\displaystyle X}$ is the expected value of the squared deviation from the mean of ${\\displaystyle X}$ , ${\\displaystyle \\mu =\\operatorname {E} [X]}$ :\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right].}$\n\nThis definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\operatorname {Cov} (X,X).}$\n\nThe variance is also equivalent to the second cumulant of a probability distribution that generates ${\\displaystyle X}$ . The variance is typically designated as ${\\displaystyle \\operatorname {Var} (X)}$ , or sometimes as ${\\displaystyle V(X)}$ or ${\\displaystyle \\mathbb {V} (X)}$ , or symbolically as ${\\displaystyle \\sigma _{X}^{2}}$ or simply ${\\displaystyle \\sigma ^{2}}$ (pronounced \" sigma squared\"). The expression for the variance can be expanded as follows:\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left[(X-\\operatorname {E} [X])^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}-2X\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]^{2}+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}\\end{aligned}}}$\n\nIn other words, the variance of X is equal to the mean of the square of X minus the square of the mean of X. This equation should not be used for computations using floating point arithmetic, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see algorithms for calculating variance.\n\n### Discrete random variable\n\nedit\n\nIf the generator of random variable ${\\displaystyle X}$ is discrete with probability mass function ${\\displaystyle x_{1}\\mapsto p_{1},x_{2}\\mapsto p_{2},\\ldots ,x_{n}\\mapsto p_{n}}$ , then\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\sum _{i=1}^{n}p_{i}\\cdot (x_{i}-\\mu )^{2},}$\n\nwhere ${\\displaystyle \\mu }$ is the expected value. That is,\n\n1. ${\\displaystyle \\mu =\\sum _{i=1}^{n}p_{i}x_{i}.}$\n\n(When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)\n\nThe variance of a collection of ${\\displaystyle n}$ equally likely values can be written as\n\n1. ${\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n}}\\sum _{i=1}^{n}(x_{i}-\\mu )^{2}}$\n\nwhere ${\\displaystyle \\mu }$ is the average value. That is,\n\n1. ${\\displaystyle \\mu ={\\frac {1}{n}}\\sum _{i=1}^{n}x_{i}.}$\n\nThe variance of a set of ${\\displaystyle n}$ equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:[ 2]\n\n1. ${\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{\\frac {1}{2}}(x_{i}-x_{j})^{2}={\\frac {1}{n^{2}}}\\sum _{i}\\sum _{j>i}(x_{i}-x_{j})^{2}.}$\n\n### Absolutely continuous random variable\n\nedit\n\nIf the random variable ${\\displaystyle X}$ has a probability density function ${\\displaystyle f(x)}$ , and ${\\displaystyle F(x)}$ is the corresponding cumulative distribution function, then\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)=\\sigma ^{2}&=\\int _{\\mathbb {R} }(x-\\mu )^{2}f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-2\\mu \\int _{\\mathbb {R} }xf(x)\\,dx+\\mu ^{2}\\int _{\\mathbb {R} }f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\int _{\\mathbb {R} }x\\,dF(x)+\\mu ^{2}\\int _{\\mathbb {R} }\\,dF(x)\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\cdot \\mu +\\mu ^{2}\\cdot 1\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-\\mu ^{2},\\end{aligned}}}$\n\nor equivalently,\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-\\mu ^{2},}$\n\nwhere ${\\displaystyle \\mu }$ is the expected value of ${\\displaystyle X}$ given by\n\n1. ${\\displaystyle \\mu =\\int _{\\mathbb {R} }xf(x)\\,dx=\\int _{\\mathbb {R} }x\\,dF(x).}$\n\nIn these formulas, the integrals with respect to ${\\displaystyle dx}$ and ${\\displaystyle dF(x)}$ are Lebesgue and Lebesgue\u2013Stieltjes integrals, respectively.\n\nIf the function ${\\displaystyle x^{2}f(x)}$ is Riemann-integrable on every finite interval ${\\displaystyle [a,b]\\subset \\mathbb {R} ,}$ then\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\int _{-\\infty }^{+\\infty }x^{2}f(x)\\,dx-\\mu ^{2},}$\n\nwhere the integral is an improper Riemann integral.\n\n## Examples\n\nedit\n\n### Exponential distribution\n\nedit\n\nThe exponential distribution with parameter \u03bb is a continuous distribution whose probability density function is given by\n\n1. ${\\displaystyle f(x)=\\lambda e^{-\\lambda x}}$\n\non the interval[0, \u221e). Its mean can be shown to be\n\n1. ${\\displaystyle \\operatorname {E} [X]=\\int _{0}^{\\infty }x\\lambda e^{-\\lambda x}\\,dx={\\frac {1}{\\lambda }}.}$\n\nUsing integration by parts and making use of the expected value already calculated, we have:\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {E} \\left[X^{2}\\right]&=\\int _{0}^{\\infty }x^{2}\\lambda e^{-\\lambda x}\\,dx\\\\&=\\left[-x^{2}e^{-\\lambda x}\\right]_{0}^{\\infty }+\\int _{0}^{\\infty }2xe^{-\\lambda x}\\,dx\\\\&=0+{\\frac {2}{\\lambda }}\\operatorname {E} [X]\\\\&={\\frac {2}{\\lambda ^{2}}}.\\end{aligned}}}$\n\nThus, the variance of X is given by\n\n1. ${\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}={\\frac {2}{\\lambda ^{2}}}-\\left({\\frac {1}{\\lambda }}\\right)^{2}={\\frac {1}{\\lambda ^{2}}}.}$\n\n### Fair dice\n\nedit\n\nA fair six-sided dice can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is ${\\displaystyle (1+2+3+4+5+6)/6=7/2.}$ Therefore, the variance of X is\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\sum _{i=1}^{6}{\\frac {1}{6}}\\left(i-{\\frac {7}{2}}\\right)^{2}\\\\[5pt]&={\\frac {1}{6}}\\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\\right)\\\\[5pt]&={\\frac {35}{12}}\\approx 2.92.\\end{aligned}}}$\n\nThe general formula for the variance of the outcome, X, of an n-sided die is\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left(X^{2}\\right)-(\\operatorname {E} (X))^{2}\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}i^{2}-\\left({\\frac {1}{n}}\\sum _{i=1}^{n}i\\right)^{2}\\\\[5pt]&={\\frac {(n+1)(2n+1)}{6}}-\\left({\\frac {n+1}{2}}\\right)^{2}\\\\[4pt]&={\\frac {n^{2}-1}{12}}.\\end{aligned}}}$\n\n### Commonly used probability distributions\n\nedit\n\nThe following table lists the variance for some commonly used probability distributions.\n\n| Name of the probability distribution | Probability distribution function | Mean | Variance |\n|---|---|---|---|\n| Binomial distribution | {\\displaystyle \\Pr \\,(X=k)={\\binom {n}{k}}p^{k}(1-p)^{n-k}} | {\\displaystyle np} | {\\displaystyle np(1-p)} |\n| Geometric distribution | {\\displaystyle \\Pr \\,(X=k)=(1-p)^{k-1}p} | {\\displaystyle {\\frac {1}{p}}} | {\\displaystyle {\\frac {(1-p)}{p^{2}}}} |\n| Normal distribution | {\\displaystyle f\\left(x\\mid \\mu ,\\sigma ^{2}\\right)={\\frac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}e^{-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}} | {\\displaystyle \\mu } | {\\displaystyle \\sigma ^{2}} |\n| Uniform distribution (continuous) | {\\displaystyle f(x\\mid a,b)={\\begin{cases}{\\frac {1}{b-a}}&{\\text{for }}a\\leq x\\leq b,\\\\[3pt]0&{\\text{for }}xb\\end{cases}}} | {\\displaystyle {\\frac {a+b}{2}}} | {\\displaystyle {\\frac {(b-a)^{2}}{12}}} |\n| Exponential distribution | {\\displaystyle f(x\\mid \\lambda )=\\lambda e^{-\\lambda x}} | {\\displaystyle {\\frac {1}{\\lambda }}} | {\\displaystyle {\\frac {1}{\\lambda ^{2}}}} |\n| Poisson distribution | {\\displaystyle f(k\\mid \\lambda )={\\frac {e^{-\\lambda }\\lambda ^{k}}{k!}}} | {\\displaystyle \\lambda } | {\\displaystyle \\lambda } |\n\n## Properties\n\nedit\n\n### Basic properties\n\nedit\n\nVariance is non-negative because the squares are positive or zero:\n\n1. ${\\displaystyle \\operatorname {Var} (X)\\geq 0.}$\n\nThe variance of a constant is zero.\n\n1. ${\\displaystyle \\operatorname {Var} (a)=0.}$\n\nConversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value:\n\n1. ${\\displaystyle \\operatorname {Var} (X)=0\\iff \\exists a:P(X=a)=1.}$\n\n### Issues of finiteness\n\nedit\n\nIf a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index ${\\displaystyle k}$ satisfies ${\\displaystyle 1 0. This also holds in the multidimensional case.[ 3]\n\n### Units of measurement\n\nedit\n\nUnlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is\u221a 2.9 \u2248 1.7, slightly larger than the expected absolute deviation of 1.5.\n\nThe standard deviation and the expected absolute deviation can both be used as an indicator of the \"spread\" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution.\n\n## Propagation\n\nedit\n\n### Addition and multiplication by a constant\n\nedit\n\nVariance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged:\n\n1. ${\\displaystyle \\operatorname {Var} (X+a)=\\operatorname {Var} (X).}$\n\nIf all values are scaled by a constant, the variance is scaled by the square of that constant:\n\n1. ${\\displaystyle \\operatorname {Var} (aX)=a^{2}\\operatorname {Var} (X).}$\n\nThe variance of a sum of two random variables is given by\n\n1. ${\\displaystyle \\operatorname {Var} (aX+bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)+2ab\\,\\operatorname {Cov} (X,Y)}$\n\n1. ${\\displaystyle \\operatorname {Var} (aX-bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)-2ab\\,\\operatorname {Cov} (X,Y)}$\n\nwhere ${\\displaystyle \\operatorname {Cov} (X,Y)}$ is the covariance.\n\n### Linear combinations\n\nedit\n\nIn general, for the sum of ${\\displaystyle N}$ random variables ${\\displaystyle \\{X_{1},\\dots ,X_{N}\\}}$ , the variance becomes:\n\n1. ${\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i,j=1}^{N}\\operatorname {Cov} (X_{i},X_{j})=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i})+\\sum _{i\\neq j}\\operatorname {Cov} (X_{i},X_{j}),}$\n\nsee also general Bienaym\u00e9's identity.\n\nThese results lead to the variance of a linear combination as:\n\n1. ${\\displaystyle {\\begin{aligned}\\operatorname {Var} \\left(\\sum _{i=1}^{N}a_{i}X_{i}\\right)&=\\sum _{i,j=1}^{N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+\\sum _{i\\not =j}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i\n\n\n\nVariance - Wikipedia\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
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                                                                                  This article is about the mathematical concept. For other uses, see Variance (disambiguation).
                                                                                  \n

                                                                                  In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \n \n \n \n \n \u03c3\n \n 2\n \n \n \n \n {\\displaystyle \\sigma ^{2}}\n \n\"{\\displaystyle, \n \n \n \n \n s\n \n 2\n \n \n \n \n {\\displaystyle s^{2}}\n \n\"{\\displaystyle, \n \n \n \n Var\n \u2061\n (\n X\n )\n \n \n {\\displaystyle \\operatorname {Var} (X)}\n \n\"{\\displaystyle, \n \n \n \n V\n (\n X\n )\n \n \n {\\displaystyle V(X)}\n \n\"{\\displaystyle, or \n \n \n \n \n V\n \n (\n X\n )\n \n \n {\\displaystyle \\mathbb {V} (X)}\n \n\"{\\displaystyle.[1]\n

                                                                                  Example of samples from two populations with the same mean but different variances. The red population has mean 100 and variance 100 (SD=10) while the blue population has mean 100 and variance 2500 (SD=50) where SD stands for Standard Deviation.
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                                                                                  An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.\n

                                                                                  There are two distinct concepts that are both called \"variance\". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.\n

                                                                                  The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling.\n

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                                                                                  Contents

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                                                                                  \n
                                                                                  Geometric visualisation of the variance of an arbitrary distribution (2, 4, 4, 4, 5, 5, 7, 9):
                                                                                  1. A frequency distribution is constructed.
                                                                                  2. The centroid of the distribution gives its mean.
                                                                                  3. A square with sides equal to the difference of each value from the mean is formed for each value.
                                                                                  4. Arranging the squares into a rectangle with one side equal to the number of values, n, results in the other side being the distribution's variance, \u03c32.
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                                                                                  Definition

                                                                                  \n\n \nedit\n\n\n\n\n
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                                                                                  The variance of a random variable \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  is the expected value of the squared deviation from the mean of \n \n \n \n X\n \n \n {\\displaystyle X}\n \n , \n \n \n \n \u03bc\n =\n E\n \u2061\n [\n X\n ]\n \n \n {\\displaystyle \\mu =\\operatorname {E} [X]}\n \n :\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n E\n \u2061\n \n [\n \n (\n X\n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n ]\n \n .\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right].}\n \n 
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                                                                                  This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:\n

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                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n Cov\n \u2061\n (\n X\n ,\n X\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\operatorname {Cov} (X,X).}\n \n 
                                                                                  \n

                                                                                  The variance is also equivalent to the second cumulant of a probability distribution that generates \n \n \n \n X\n \n \n {\\displaystyle X}\n \n . The variance is typically designated as \n \n \n \n Var\n \u2061\n (\n X\n )\n \n \n {\\displaystyle \\operatorname {Var} (X)}\n \n , or sometimes as \n \n \n \n V\n (\n X\n )\n \n \n {\\displaystyle V(X)}\n \n  or \n \n \n \n \n V\n \n (\n X\n )\n \n \n {\\displaystyle \\mathbb {V} (X)}\n \n , or symbolically as \n \n \n \n \n \u03c3\n \n X\n \n \n 2\n \n \n \n \n {\\displaystyle \\sigma _{X}^{2}}\n \n  or simply \n \n \n \n \n \u03c3\n \n 2\n \n \n \n \n {\\displaystyle \\sigma ^{2}}\n \n  (pronounced \"sigma squared\"). The expression for the variance can be expanded as follows:\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n )\n \n \n \n =\n E\n \u2061\n \n [\n \n (\n X\n \u2212\n E\n \u2061\n [\n X\n ]\n \n )\n \n 2\n \n \n \n ]\n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n \n X\n \n 2\n \n \n \u2212\n 2\n X\n E\n \u2061\n [\n X\n ]\n +\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n \n ]\n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n \u2212\n 2\n E\n \u2061\n [\n X\n ]\n E\n \u2061\n [\n X\n ]\n +\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n \u2212\n 2\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n +\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n \u2212\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left[(X-\\operatorname {E} [X])^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}-2X\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\right]\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]\\operatorname {E} [X]+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-2\\operatorname {E} [X]^{2}+\\operatorname {E} [X]^{2}\\\\[4pt]&=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}\\end{aligned}}}\n \n 
                                                                                  \n

                                                                                  In other words, the variance of X is equal to the mean of the square of X minus the square of the mean of X. This equation should not be used for computations using floating point arithmetic, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see algorithms for calculating variance.\n

                                                                                  \n

                                                                                  Discrete random variable

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  If the generator of random variable \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  is discrete with probability mass function \n \n \n \n \n x\n \n 1\n \n \n \u21a6\n \n p\n \n 1\n \n \n ,\n \n x\n \n 2\n \n \n \u21a6\n \n p\n \n 2\n \n \n ,\n \u2026\n ,\n \n x\n \n n\n \n \n \u21a6\n \n p\n \n n\n \n \n \n \n {\\displaystyle x_{1}\\mapsto p_{1},x_{2}\\mapsto p_{2},\\ldots ,x_{n}\\mapsto p_{n}}\n \n , then\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n p\n \n i\n \n \n \u22c5\n (\n \n x\n \n i\n \n \n \u2212\n \u03bc\n \n )\n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\sum _{i=1}^{n}p_{i}\\cdot (x_{i}-\\mu )^{2},}\n \n 
                                                                                  \n

                                                                                  where \n \n \n \n \u03bc\n \n \n {\\displaystyle \\mu }\n \n  is the expected value. That is,\n

                                                                                  \n
                                                                                  \n \n \n \n \u03bc\n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n p\n \n i\n \n \n \n x\n \n i\n \n \n .\n \n \n {\\displaystyle \\mu =\\sum _{i=1}^{n}p_{i}x_{i}.}\n \n 
                                                                                  \n

                                                                                  (When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)\n

                                                                                  The variance of a collection of \n \n \n \n n\n \n \n {\\displaystyle n}\n \n  equally likely values can be written as\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n (\n \n x\n \n i\n \n \n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n \n {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n}}\\sum _{i=1}^{n}(x_{i}-\\mu )^{2}}\n \n 
                                                                                  \n

                                                                                  where \n \n \n \n \u03bc\n \n \n {\\displaystyle \\mu }\n \n  is the average value. That is,\n

                                                                                  \n
                                                                                  \n \n \n \n \u03bc\n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n x\n \n i\n \n \n .\n \n \n {\\displaystyle \\mu ={\\frac {1}{n}}\\sum _{i=1}^{n}x_{i}.}\n \n 
                                                                                  \n

                                                                                  The variance of a set of \n \n \n \n n\n \n \n {\\displaystyle n}\n \n  equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:[2]\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n \n 1\n 2\n \n \n (\n \n x\n \n i\n \n \n \u2212\n \n x\n \n j\n \n \n \n )\n \n 2\n \n \n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n i\n \n \n \n \u2211\n \n j\n >\n i\n \n \n (\n \n x\n \n i\n \n \n \u2212\n \n x\n \n j\n \n \n \n )\n \n 2\n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} (X)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{\\frac {1}{2}}(x_{i}-x_{j})^{2}={\\frac {1}{n^{2}}}\\sum _{i}\\sum _{j>i}(x_{i}-x_{j})^{2}.}\n \n 
                                                                                  \n

                                                                                  Absolutely continuous random variable

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  If the random variable \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  has a probability density function \n \n \n \n f\n (\n x\n )\n \n \n {\\displaystyle f(x)}\n \n , and \n \n \n \n F\n (\n x\n )\n \n \n {\\displaystyle F(x)}\n \n  is the corresponding cumulative distribution function, then\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \u03c3\n \n 2\n \n \n \n \n \n =\n \n \u222b\n \n \n R\n \n \n \n (\n x\n \u2212\n \u03bc\n \n )\n \n 2\n \n \n f\n (\n x\n )\n \n d\n x\n \n \n \n \n \n \n =\n \n \u222b\n \n \n R\n \n \n \n \n x\n \n 2\n \n \n f\n (\n x\n )\n \n d\n x\n \u2212\n 2\n \u03bc\n \n \u222b\n \n \n R\n \n \n \n x\n f\n (\n x\n )\n \n d\n x\n +\n \n \u03bc\n \n 2\n \n \n \n \u222b\n \n \n R\n \n \n \n f\n (\n x\n )\n \n d\n x\n \n \n \n \n \n \n =\n \n \u222b\n \n \n R\n \n \n \n \n x\n \n 2\n \n \n \n d\n F\n (\n x\n )\n \u2212\n 2\n \u03bc\n \n \u222b\n \n \n R\n \n \n \n x\n \n d\n F\n (\n x\n )\n +\n \n \u03bc\n \n 2\n \n \n \n \u222b\n \n \n R\n \n \n \n \n d\n F\n (\n x\n )\n \n \n \n \n \n \n =\n \n \u222b\n \n \n R\n \n \n \n \n x\n \n 2\n \n \n \n d\n F\n (\n x\n )\n \u2212\n 2\n \u03bc\n \u22c5\n \u03bc\n +\n \n \u03bc\n \n 2\n \n \n \u22c5\n 1\n \n \n \n \n \n \n =\n \n \u222b\n \n \n R\n \n \n \n \n x\n \n 2\n \n \n \n d\n F\n (\n x\n )\n \u2212\n \n \u03bc\n \n 2\n \n \n ,\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)=\\sigma ^{2}&=\\int _{\\mathbb {R} }(x-\\mu )^{2}f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-2\\mu \\int _{\\mathbb {R} }xf(x)\\,dx+\\mu ^{2}\\int _{\\mathbb {R} }f(x)\\,dx\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\int _{\\mathbb {R} }x\\,dF(x)+\\mu ^{2}\\int _{\\mathbb {R} }\\,dF(x)\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-2\\mu \\cdot \\mu +\\mu ^{2}\\cdot 1\\\\[4pt]&=\\int _{\\mathbb {R} }x^{2}\\,dF(x)-\\mu ^{2},\\end{aligned}}}\n \n 
                                                                                  \n

                                                                                  or equivalently,\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \u222b\n \n \n R\n \n \n \n \n x\n \n 2\n \n \n f\n (\n x\n )\n \n d\n x\n \u2212\n \n \u03bc\n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\int _{\\mathbb {R} }x^{2}f(x)\\,dx-\\mu ^{2},}\n \n 
                                                                                  \n

                                                                                  where \n \n \n \n \u03bc\n \n \n {\\displaystyle \\mu }\n \n  is the expected value of \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  given by\n

                                                                                  \n
                                                                                  \n \n \n \n \u03bc\n =\n \n \u222b\n \n \n R\n \n \n \n x\n f\n (\n x\n )\n \n d\n x\n =\n \n \u222b\n \n \n R\n \n \n \n x\n \n d\n F\n (\n x\n )\n .\n \n \n {\\displaystyle \\mu =\\int _{\\mathbb {R} }xf(x)\\,dx=\\int _{\\mathbb {R} }x\\,dF(x).}\n \n 
                                                                                  \n

                                                                                  In these formulas, the integrals with respect to \n \n \n \n d\n x\n \n \n {\\displaystyle dx}\n \n  and \n \n \n \n d\n F\n (\n x\n )\n \n \n {\\displaystyle dF(x)}\n \n \nare Lebesgue and Lebesgue\u2013Stieltjes integrals, respectively.\n

                                                                                  If the function \n \n \n \n \n x\n \n 2\n \n \n f\n (\n x\n )\n \n \n {\\displaystyle x^{2}f(x)}\n \n  is Riemann-integrable on every finite interval \n \n \n \n [\n a\n ,\n b\n ]\n \u2282\n \n R\n \n ,\n \n \n {\\displaystyle [a,b]\\subset \\mathbb {R} ,}\n \n  then\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n \n \u222b\n \n \u2212\n \u221e\n \n \n +\n \u221e\n \n \n \n x\n \n 2\n \n \n f\n (\n x\n )\n \n d\n x\n \u2212\n \n \u03bc\n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\int _{-\\infty }^{+\\infty }x^{2}f(x)\\,dx-\\mu ^{2},}\n \n 
                                                                                  \n

                                                                                  where the integral is an improper Riemann integral.\n

                                                                                  \n

                                                                                  Examples

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  Exponential distribution

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  The exponential distribution with parameter \u03bb is a continuous distribution whose probability density function is given by\n

                                                                                  \n
                                                                                  \n \n \n \n f\n (\n x\n )\n =\n \u03bb\n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n \n {\\displaystyle f(x)=\\lambda e^{-\\lambda x}}\n \n 
                                                                                  \n

                                                                                  on the interval [0, \u221e). Its mean can be shown to be\n

                                                                                  \n
                                                                                  \n \n \n \n E\n \u2061\n [\n X\n ]\n =\n \n \u222b\n \n 0\n \n \n \u221e\n \n \n x\n \u03bb\n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n d\n x\n =\n \n \n 1\n \u03bb\n \n \n .\n \n \n {\\displaystyle \\operatorname {E} [X]=\\int _{0}^{\\infty }x\\lambda e^{-\\lambda x}\\,dx={\\frac {1}{\\lambda }}.}\n \n 
                                                                                  \n

                                                                                  Using integration by parts and making use of the expected value already calculated, we have:\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n \n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n \n \n \n =\n \n \u222b\n \n 0\n \n \n \u221e\n \n \n \n x\n \n 2\n \n \n \u03bb\n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n d\n x\n \n \n \n \n \n \n =\n \n \n [\n \n \u2212\n \n x\n \n 2\n \n \n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n ]\n \n \n 0\n \n \n \u221e\n \n \n +\n \n \u222b\n \n 0\n \n \n \u221e\n \n \n 2\n x\n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n d\n x\n \n \n \n \n \n \n =\n 0\n +\n \n \n 2\n \u03bb\n \n \n E\n \u2061\n [\n X\n ]\n \n \n \n \n \n \n =\n \n \n 2\n \n \u03bb\n \n 2\n \n \n \n \n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {E} \\left[X^{2}\\right]&=\\int _{0}^{\\infty }x^{2}\\lambda e^{-\\lambda x}\\,dx\\\\&=\\left[-x^{2}e^{-\\lambda x}\\right]_{0}^{\\infty }+\\int _{0}^{\\infty }2xe^{-\\lambda x}\\,dx\\\\&=0+{\\frac {2}{\\lambda }}\\operatorname {E} [X]\\\\&={\\frac {2}{\\lambda ^{2}}}.\\end{aligned}}}\n \n 
                                                                                  \n

                                                                                  Thus, the variance of X is given by\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n \u2212\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n =\n \n \n 2\n \n \u03bb\n \n 2\n \n \n \n \n \u2212\n \n \n (\n \n \n 1\n \u03bb\n \n \n )\n \n \n 2\n \n \n =\n \n \n 1\n \n \u03bb\n \n 2\n \n \n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[X^{2}\\right]-\\operatorname {E} [X]^{2}={\\frac {2}{\\lambda ^{2}}}-\\left({\\frac {1}{\\lambda }}\\right)^{2}={\\frac {1}{\\lambda ^{2}}}.}\n \n 
                                                                                  \n

                                                                                  Fair dice

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  A fair six-sided dice can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is \n \n \n \n (\n 1\n +\n 2\n +\n 3\n +\n 4\n +\n 5\n +\n 6\n )\n \n /\n \n 6\n =\n 7\n \n /\n \n 2.\n \n \n {\\displaystyle (1+2+3+4+5+6)/6=7/2.}\n \n  Therefore, the variance of X is\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n )\n \n \n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n 6\n \n \n \n \n 1\n 6\n \n \n \n \n (\n \n i\n \u2212\n \n \n 7\n 2\n \n \n \n )\n \n \n 2\n \n \n \n \n \n \n \n \n =\n \n \n 1\n 6\n \n \n \n (\n \n (\n \u2212\n 5\n \n /\n \n 2\n \n )\n \n 2\n \n \n +\n (\n \u2212\n 3\n \n /\n \n 2\n \n )\n \n 2\n \n \n +\n (\n \u2212\n 1\n \n /\n \n 2\n \n )\n \n 2\n \n \n +\n (\n 1\n \n /\n \n 2\n \n )\n \n 2\n \n \n +\n (\n 3\n \n /\n \n 2\n \n )\n \n 2\n \n \n +\n (\n 5\n \n /\n \n 2\n \n )\n \n 2\n \n \n \n )\n \n \n \n \n \n \n \n =\n \n \n 35\n 12\n \n \n \u2248\n 2.92.\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\sum _{i=1}^{6}{\\frac {1}{6}}\\left(i-{\\frac {7}{2}}\\right)^{2}\\\\[5pt]&={\\frac {1}{6}}\\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\\right)\\\\[5pt]&={\\frac {35}{12}}\\approx 2.92.\\end{aligned}}}\n \n 
                                                                                  \n

                                                                                  The general formula for the variance of the outcome, X, of an n-sided die is\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n )\n \n \n \n =\n E\n \u2061\n \n (\n \n X\n \n 2\n \n \n )\n \n \u2212\n (\n E\n \u2061\n (\n X\n )\n \n )\n \n 2\n \n \n \n \n \n \n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n i\n \n 2\n \n \n \u2212\n \n \n (\n \n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n i\n \n )\n \n \n 2\n \n \n \n \n \n \n \n \n =\n \n \n \n (\n n\n +\n 1\n )\n (\n 2\n n\n +\n 1\n )\n \n 6\n \n \n \u2212\n \n \n (\n \n \n \n n\n +\n 1\n \n 2\n \n \n )\n \n \n 2\n \n \n \n \n \n \n \n \n =\n \n \n \n \n n\n \n 2\n \n \n \u2212\n 1\n \n 12\n \n \n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X)&=\\operatorname {E} \\left(X^{2}\\right)-(\\operatorname {E} (X))^{2}\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}i^{2}-\\left({\\frac {1}{n}}\\sum _{i=1}^{n}i\\right)^{2}\\\\[5pt]&={\\frac {(n+1)(2n+1)}{6}}-\\left({\\frac {n+1}{2}}\\right)^{2}\\\\[4pt]&={\\frac {n^{2}-1}{12}}.\\end{aligned}}}\n \n 
                                                                                  \n

                                                                                  Commonly used probability distributions

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  The following table lists the variance for some commonly used probability distributions.\n

                                                                                  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                                                                  Name of the probability distribution\nProbability distribution function\nMean\nVariance\n
                                                                                  Binomial distribution\n\n \n \n \n Pr\n \n (\n X\n =\n k\n )\n =\n \n \n \n (\n \n \n n\n k\n \n \n )\n \n \n \n \n p\n \n k\n \n \n (\n 1\n \u2212\n p\n \n )\n \n n\n \u2212\n k\n \n \n \n \n {\\displaystyle \\Pr \\,(X=k)={\\binom {n}{k}}p^{k}(1-p)^{n-k}}\n \n \n\n \n \n \n n\n p\n \n \n {\\displaystyle np}\n \n \n\n \n \n \n n\n p\n (\n 1\n \u2212\n p\n )\n \n \n {\\displaystyle np(1-p)}\n \n \n
                                                                                  Geometric distribution\n\n \n \n \n Pr\n \n (\n X\n =\n k\n )\n =\n (\n 1\n \u2212\n p\n \n )\n \n k\n \u2212\n 1\n \n \n p\n \n \n {\\displaystyle \\Pr \\,(X=k)=(1-p)^{k-1}p}\n \n \n\n \n \n \n \n \n 1\n p\n \n \n \n \n {\\displaystyle {\\frac {1}{p}}}\n \n \n\n \n \n \n \n \n \n (\n 1\n \u2212\n p\n )\n \n \n p\n \n 2\n \n \n \n \n \n \n {\\displaystyle {\\frac {(1-p)}{p^{2}}}}\n \n \n
                                                                                  Normal distribution\n\n \n \n \n f\n \n (\n \n x\n \u2223\n \u03bc\n ,\n \n \u03c3\n \n 2\n \n \n \n )\n \n =\n \n \n 1\n \n 2\n \u03c0\n \n \u03c3\n \n 2\n \n \n \n \n \n \n e\n \n \u2212\n \n \n \n (\n x\n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n \n 2\n \n \u03c3\n \n 2\n \n \n \n \n \n \n \n \n \n {\\displaystyle f\\left(x\\mid \\mu ,\\sigma ^{2}\\right)={\\frac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}e^{-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}}\n \n \n\n \n \n \n \u03bc\n \n \n {\\displaystyle \\mu }\n \n \n\n \n \n \n \n \u03c3\n \n 2\n \n \n \n \n {\\displaystyle \\sigma ^{2}}\n \n \n
                                                                                  Uniform distribution (continuous)\n\n \n \n \n f\n (\n x\n \u2223\n a\n ,\n b\n )\n =\n \n \n {\n \n \n \n \n \n 1\n \n b\n \u2212\n a\n \n \n \n \n \n \n for \n \n a\n \u2264\n x\n \u2264\n b\n ,\n \n \n \n \n 0\n \n \n \n for \n \n x\n <\n a\n \n or \n \n x\n >\n b\n \n \n \n \n \n \n \n \n {\\displaystyle f(x\\mid a,b)={\\begin{cases}{\\frac {1}{b-a}}&{\\text{for }}a\\leq x\\leq b,\\\\[3pt]0&{\\text{for }}x<a{\\text{ or }}x>b\\end{cases}}}\n \n \n\n \n \n \n \n \n \n a\n +\n b\n \n 2\n \n \n \n \n {\\displaystyle {\\frac {a+b}{2}}}\n \n \n\n \n \n \n \n \n \n (\n b\n \u2212\n a\n \n )\n \n 2\n \n \n \n 12\n \n \n \n \n {\\displaystyle {\\frac {(b-a)^{2}}{12}}}\n \n \n
                                                                                  Exponential distribution\n\n \n \n \n f\n (\n x\n \u2223\n \u03bb\n )\n =\n \u03bb\n \n e\n \n \u2212\n \u03bb\n x\n \n \n \n \n {\\displaystyle f(x\\mid \\lambda )=\\lambda e^{-\\lambda x}}\n \n \n\n \n \n \n \n \n 1\n \u03bb\n \n \n \n \n {\\displaystyle {\\frac {1}{\\lambda }}}\n \n \n\n \n \n \n \n \n 1\n \n \u03bb\n \n 2\n \n \n \n \n \n \n {\\displaystyle {\\frac {1}{\\lambda ^{2}}}}\n \n \n
                                                                                  Poisson distribution\n\n \n \n \n f\n (\n k\n \u2223\n \u03bb\n )\n =\n \n \n \n \n e\n \n \u2212\n \u03bb\n \n \n \n \u03bb\n \n k\n \n \n \n \n k\n !\n \n \n \n \n \n {\\displaystyle f(k\\mid \\lambda )={\\frac {e^{-\\lambda }\\lambda ^{k}}{k!}}}\n \n \n\n \n \n \n \u03bb\n \n \n {\\displaystyle \\lambda }\n \n \n\n \n \n \n \u03bb\n \n \n {\\displaystyle \\lambda }\n \n \n
                                                                                  \n

                                                                                  Properties

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                                                                                  Basic properties

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                                                                                  Variance is non-negative because the squares are positive or zero:\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n )\n \u2265\n 0.\n \n \n {\\displaystyle \\operatorname {Var} (X)\\geq 0.}\n \n 
                                                                                  \n

                                                                                  The variance of a constant is zero.\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n a\n )\n =\n 0.\n \n \n {\\displaystyle \\operatorname {Var} (a)=0.}\n \n 
                                                                                  \n

                                                                                  Conversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value:\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n 0\n \n \u27fa\n \n \u2203\n a\n :\n P\n (\n X\n =\n a\n )\n =\n 1.\n \n \n {\\displaystyle \\operatorname {Var} (X)=0\\iff \\exists a:P(X=a)=1.}\n \n 
                                                                                  \n

                                                                                  Issues of finiteness

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                                                                                  If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index \n \n \n \n k\n \n \n {\\displaystyle k}\n \n  satisfies \n \n \n \n 1\n <\n k\n \u2264\n 2.\n \n \n {\\displaystyle 1<k\\leq 2.}\n \n \n

                                                                                  \n

                                                                                  Decomposition

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                                                                                  The general formula for variance decomposition or the law of total variance is: If \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  and \n \n \n \n Y\n \n \n {\\displaystyle Y}\n \n  are two random variables, and the variance of \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  exists, then\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n [\n X\n ]\n =\n E\n \u2061\n (\n Var\n \u2061\n [\n X\n \u2223\n Y\n ]\n )\n +\n Var\n \u2061\n (\n E\n \u2061\n [\n X\n \u2223\n Y\n ]\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} [X]=\\operatorname {E} (\\operatorname {Var} [X\\mid Y])+\\operatorname {Var} (\\operatorname {E} [X\\mid Y]).}\n \n 
                                                                                  \n

                                                                                  The conditional expectation \n \n \n \n E\n \u2061\n (\n X\n \u2223\n Y\n )\n \n \n {\\displaystyle \\operatorname {E} (X\\mid Y)}\n \n  of \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  given \n \n \n \n Y\n \n \n {\\displaystyle Y}\n \n , and the conditional variance \n \n \n \n Var\n \u2061\n (\n X\n \u2223\n Y\n )\n \n \n {\\displaystyle \\operatorname {Var} (X\\mid Y)}\n \n  may be understood as follows. Given any particular value y of the random variable Y, there is a conditional expectation \n \n \n \n E\n \u2061\n (\n X\n \u2223\n Y\n =\n y\n )\n \n \n {\\displaystyle \\operatorname {E} (X\\mid Y=y)}\n \n  given the event Y = y. This quantity depends on the particular value y; it is a function \n \n \n \n g\n (\n y\n )\n =\n E\n \u2061\n (\n X\n \u2223\n Y\n =\n y\n )\n \n \n {\\displaystyle g(y)=\\operatorname {E} (X\\mid Y=y)}\n \n . That same function evaluated at the random variable Y is the conditional expectation \n \n \n \n E\n \u2061\n (\n X\n \u2223\n Y\n )\n =\n g\n (\n Y\n )\n .\n \n \n {\\displaystyle \\operatorname {E} (X\\mid Y)=g(Y).}\n \n \n

                                                                                  In particular, if \n \n \n \n Y\n \n \n {\\displaystyle Y}\n \n  is a discrete random variable assuming possible values \n \n \n \n \n y\n \n 1\n \n \n ,\n \n y\n \n 2\n \n \n ,\n \n y\n \n 3\n \n \n \u2026\n \n \n {\\displaystyle y_{1},y_{2},y_{3}\\ldots }\n \n  with corresponding probabilities \n \n \n \n \n p\n \n 1\n \n \n ,\n \n p\n \n 2\n \n \n ,\n \n p\n \n 3\n \n \n \u2026\n ,\n \n \n {\\displaystyle p_{1},p_{2},p_{3}\\ldots ,}\n \n , then in the formula for total variance, the first term on the right-hand side becomes\n

                                                                                  \n
                                                                                  \n \n \n \n E\n \u2061\n (\n Var\n \u2061\n [\n X\n \u2223\n Y\n ]\n )\n =\n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03c3\n \n i\n \n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {E} (\\operatorname {Var} [X\\mid Y])=\\sum _{i}p_{i}\\sigma _{i}^{2},}\n \n 
                                                                                  \n

                                                                                  where \n \n \n \n \n \u03c3\n \n i\n \n \n 2\n \n \n =\n Var\n \u2061\n [\n X\n \u2223\n Y\n =\n \n y\n \n i\n \n \n ]\n \n \n {\\displaystyle \\sigma _{i}^{2}=\\operatorname {Var} [X\\mid Y=y_{i}]}\n \n . Similarly, the second term on the right-hand side becomes\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n E\n \u2061\n [\n X\n \u2223\n Y\n ]\n )\n =\n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03bc\n \n i\n \n \n 2\n \n \n \u2212\n \n \n (\n \n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03bc\n \n i\n \n \n \n )\n \n \n 2\n \n \n =\n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03bc\n \n i\n \n \n 2\n \n \n \u2212\n \n \u03bc\n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {Var} (\\operatorname {E} [X\\mid Y])=\\sum _{i}p_{i}\\mu _{i}^{2}-\\left(\\sum _{i}p_{i}\\mu _{i}\\right)^{2}=\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2},}\n \n 
                                                                                  \n

                                                                                  where \n \n \n \n \n \u03bc\n \n i\n \n \n =\n E\n \u2061\n [\n X\n \u2223\n Y\n =\n \n y\n \n i\n \n \n ]\n \n \n {\\displaystyle \\mu _{i}=\\operatorname {E} [X\\mid Y=y_{i}]}\n \n  and \n \n \n \n \u03bc\n =\n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03bc\n \n i\n \n \n \n \n {\\displaystyle \\mu =\\sum _{i}p_{i}\\mu _{i}}\n \n . Thus the total variance is given by\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n [\n X\n ]\n =\n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03c3\n \n i\n \n \n 2\n \n \n +\n \n (\n \n \n \u2211\n \n i\n \n \n \n p\n \n i\n \n \n \n \u03bc\n \n i\n \n \n 2\n \n \n \u2212\n \n \u03bc\n \n 2\n \n \n \n )\n \n .\n \n \n {\\displaystyle \\operatorname {Var} [X]=\\sum _{i}p_{i}\\sigma _{i}^{2}+\\left(\\sum _{i}p_{i}\\mu _{i}^{2}-\\mu ^{2}\\right).}\n \n 
                                                                                  \n

                                                                                  A similar formula is applied in analysis of variance, where the corresponding formula is\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n M\n S\n \n \n \n total\n \n \n =\n \n \n \n M\n S\n \n \n \n between\n \n \n +\n \n \n \n M\n S\n \n \n \n within\n \n \n ;\n \n \n {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{between}}+{\\mathit {MS}}_{\\text{within}};}\n \n 
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                                                                                  here \n \n \n \n \n \n M\n S\n \n \n \n \n {\\displaystyle {\\mathit {MS}}}\n \n  refers to the Mean of the Squares. In linear regression analysis the corresponding formula is\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n M\n S\n \n \n \n total\n \n \n =\n \n \n \n M\n S\n \n \n \n regression\n \n \n +\n \n \n \n M\n S\n \n \n \n residual\n \n \n .\n \n \n {\\displaystyle {\\mathit {MS}}_{\\text{total}}={\\mathit {MS}}_{\\text{regression}}+{\\mathit {MS}}_{\\text{residual}}.}\n \n 
                                                                                  \n

                                                                                  This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.\n

                                                                                  Similar decompositions are possible for the sum of squared deviations (sum of squares, \n \n \n \n \n \n S\n S\n \n \n \n \n {\\displaystyle {\\mathit {SS}}}\n \n ):\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n S\n S\n \n \n \n total\n \n \n =\n \n \n \n S\n S\n \n \n \n between\n \n \n +\n \n \n \n S\n S\n \n \n \n within\n \n \n ,\n \n \n {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{between}}+{\\mathit {SS}}_{\\text{within}},}\n \n 
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                                                                                  \n \n \n \n \n \n \n S\n S\n \n \n \n total\n \n \n =\n \n \n \n S\n S\n \n \n \n regression\n \n \n +\n \n \n \n S\n S\n \n \n \n residual\n \n \n .\n \n \n {\\displaystyle {\\mathit {SS}}_{\\text{total}}={\\mathit {SS}}_{\\text{regression}}+{\\mathit {SS}}_{\\text{residual}}.}\n \n 
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                                                                                  Calculation from the CDF

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                                                                                  The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution function F using\n

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                                                                                  \n \n \n \n 2\n \n \u222b\n \n 0\n \n \n \u221e\n \n \n u\n (\n 1\n \u2212\n F\n (\n u\n )\n )\n \n d\n u\n \u2212\n \n \n (\n \n \n \u222b\n \n 0\n \n \n \u221e\n \n \n (\n 1\n \u2212\n F\n (\n u\n )\n )\n \n d\n u\n \n )\n \n \n 2\n \n \n .\n \n \n {\\displaystyle 2\\int _{0}^{\\infty }u(1-F(u))\\,du-\\left(\\int _{0}^{\\infty }(1-F(u))\\,du\\right)^{2}.}\n \n 
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                                                                                  This expression can be used to calculate the variance in situations where the CDF, but not the density, can be conveniently expressed.\n

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                                                                                  Characteristic property

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                                                                                  The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. \n \n \n \n \n \n a\n r\n g\n m\n i\n n\n \n \n m\n \n \n \n \n E\n \n \n (\n \n \n (\n \n X\n \u2212\n m\n \n )\n \n \n 2\n \n \n )\n \n =\n \n E\n \n (\n X\n )\n \n \n {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} \\left(\\left(X-m\\right)^{2}\\right)=\\mathrm {E} (X)}\n \n . Conversely, if a continuous function \n \n \n \n \u03c6\n \n \n {\\displaystyle \\varphi }\n \n  satisfies \n \n \n \n \n \n a\n r\n g\n m\n i\n n\n \n \n m\n \n \n \n \n E\n \n (\n \u03c6\n (\n X\n \u2212\n m\n )\n )\n =\n \n E\n \n (\n X\n )\n \n \n {\\displaystyle \\mathrm {argmin} _{m}\\,\\mathrm {E} (\\varphi (X-m))=\\mathrm {E} (X)}\n \n  for all random variables X, then it is necessarily of the form \n \n \n \n \u03c6\n (\n x\n )\n =\n a\n \n x\n \n 2\n \n \n +\n b\n \n \n {\\displaystyle \\varphi (x)=ax^{2}+b}\n \n , where a > 0. This also holds in the multidimensional case.[3]\n

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                                                                                  Units of measurement

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                                                                                  Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is \u221a2.9 \u2248 1.7, slightly larger than the expected absolute deviation of 1.5.\n

                                                                                  The standard deviation and the expected absolute deviation can both be used as an indicator of the \"spread\" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution.\n

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                                                                                  Propagation

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                                                                                  Addition and multiplication by a constant

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                                                                                  Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged:\n

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                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n +\n a\n )\n =\n Var\n \u2061\n (\n X\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (X+a)=\\operatorname {Var} (X).}\n \n 
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                                                                                  If all values are scaled by a constant, the variance is scaled by the square of that constant:\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n a\n X\n )\n =\n \n a\n \n 2\n \n \n Var\n \u2061\n (\n X\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (aX)=a^{2}\\operatorname {Var} (X).}\n \n 
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                                                                                  The variance of a sum of two random variables is given by\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n a\n X\n +\n b\n Y\n )\n =\n \n a\n \n 2\n \n \n Var\n \u2061\n (\n X\n )\n +\n \n b\n \n 2\n \n \n Var\n \u2061\n (\n Y\n )\n +\n 2\n a\n b\n \n Cov\n \u2061\n (\n X\n ,\n Y\n )\n \n \n {\\displaystyle \\operatorname {Var} (aX+bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)+2ab\\,\\operatorname {Cov} (X,Y)}\n \n 
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                                                                                  \n \n \n \n Var\n \u2061\n (\n a\n X\n \u2212\n b\n Y\n )\n =\n \n a\n \n 2\n \n \n Var\n \u2061\n (\n X\n )\n +\n \n b\n \n 2\n \n \n Var\n \u2061\n (\n Y\n )\n \u2212\n 2\n a\n b\n \n Cov\n \u2061\n (\n X\n ,\n Y\n )\n \n \n {\\displaystyle \\operatorname {Var} (aX-bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)-2ab\\,\\operatorname {Cov} (X,Y)}\n \n 
                                                                                  \n

                                                                                  where \n \n \n \n Cov\n \u2061\n (\n X\n ,\n Y\n )\n \n \n {\\displaystyle \\operatorname {Cov} (X,Y)}\n \n  is the covariance.\n

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                                                                                  Linear combinations

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                                                                                  In general, for the sum of \n \n \n \n N\n \n \n {\\displaystyle N}\n \n  random variables \n \n \n \n {\n \n X\n \n 1\n \n \n ,\n \u2026\n ,\n \n X\n \n N\n \n \n }\n \n \n {\\displaystyle \\{X_{1},\\dots ,X_{N}\\}}\n \n , the variance becomes:\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \u2211\n \n i\n ,\n j\n =\n 1\n \n \n N\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n =\n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n +\n \n \u2211\n \n i\n \u2260\n j\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n ,\n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i,j=1}^{N}\\operatorname {Cov} (X_{i},X_{j})=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i})+\\sum _{i\\neq j}\\operatorname {Cov} (X_{i},X_{j}),}\n \n 
                                                                                  \n

                                                                                  see also general Bienaym\u00e9's identity.\n

                                                                                  These results lead to the variance of a linear combination as:\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n a\n \n i\n \n \n \n X\n \n i\n \n \n \n )\n \n \n \n \n =\n \n \u2211\n \n i\n ,\n j\n =\n 1\n \n \n N\n \n \n \n a\n \n i\n \n \n \n a\n \n j\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n \n \n \n \n \n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n a\n \n i\n \n \n 2\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n +\n \n \u2211\n \n i\n \u2260\n j\n \n \n \n a\n \n i\n \n \n \n a\n \n j\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n \n \n \n \n \n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n a\n \n i\n \n \n 2\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n +\n 2\n \n \u2211\n \n 1\n \u2264\n i\n <\n j\n \u2264\n N\n \n \n \n a\n \n i\n \n \n \n a\n \n j\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} \\left(\\sum _{i=1}^{N}a_{i}X_{i}\\right)&=\\sum _{i,j=1}^{N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+\\sum _{i\\not =j}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})\\\\&=\\sum _{i=1}^{N}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i<j\\leq N}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j}).\\end{aligned}}}\n \n 
                                                                                  \n

                                                                                  If the random variables \n \n \n \n \n X\n \n 1\n \n \n ,\n \u2026\n ,\n \n X\n \n N\n \n \n \n \n {\\displaystyle X_{1},\\dots ,X_{N}}\n \n  are such that\n

                                                                                  \n
                                                                                  \n \n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n =\n 0\n \n ,\n \n \u2200\n \n (\n i\n \u2260\n j\n )\n ,\n \n \n {\\displaystyle \\operatorname {Cov} (X_{i},X_{j})=0\\ ,\\ \\forall \\ (i\\neq j),}\n \n 
                                                                                  \n

                                                                                  then they are said to be uncorrelated. It follows immediately from the expression given earlier that if the random variables \n \n \n \n \n X\n \n 1\n \n \n ,\n \u2026\n ,\n \n X\n \n N\n \n \n \n \n {\\displaystyle X_{1},\\dots ,X_{N}}\n \n  are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n .\n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\sum _{i=1}^{N}\\operatorname {Var} (X_{i}).}\n \n 
                                                                                  \n

                                                                                  Since independent random variables are always uncorrelated (see Covariance \u00a7 Uncorrelatedness and independence), the equation above holds in particular when the random variables \n \n \n \n \n X\n \n 1\n \n \n ,\n \u2026\n ,\n \n X\n \n n\n \n \n \n \n {\\displaystyle X_{1},\\dots ,X_{n}}\n \n  are independent. Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.\n

                                                                                  \n

                                                                                  Matrix notation for the variance of a linear combination

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  Define \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  as a column vector of \n \n \n \n n\n \n \n {\\displaystyle n}\n \n  random variables \n \n \n \n \n X\n \n 1\n \n \n ,\n \u2026\n ,\n \n X\n \n n\n \n \n \n \n {\\displaystyle X_{1},\\ldots ,X_{n}}\n \n , and \n \n \n \n c\n \n \n {\\displaystyle c}\n \n  as a column vector of \n \n \n \n n\n \n \n {\\displaystyle n}\n \n  scalars \n \n \n \n \n c\n \n 1\n \n \n ,\n \u2026\n ,\n \n c\n \n n\n \n \n \n \n {\\displaystyle c_{1},\\ldots ,c_{n}}\n \n . Therefore, \n \n \n \n \n c\n \n \n T\n \n \n \n X\n \n \n {\\displaystyle c^{\\mathsf {T}}X}\n \n  is a linear combination of these random variables, where \n \n \n \n \n c\n \n \n T\n \n \n \n \n \n {\\displaystyle c^{\\mathsf {T}}}\n \n  denotes the transpose of \n \n \n \n c\n \n \n {\\displaystyle c}\n \n . Also let \n \n \n \n \u03a3\n \n \n {\\displaystyle \\Sigma }\n \n  be the covariance matrix of \n \n \n \n X\n \n \n {\\displaystyle X}\n \n . The variance of \n \n \n \n \n c\n \n \n T\n \n \n \n X\n \n \n {\\displaystyle c^{\\mathsf {T}}X}\n \n  is then given by:[4]\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n (\n \n \n c\n \n \n T\n \n \n \n X\n \n )\n \n =\n \n c\n \n \n T\n \n \n \n \u03a3\n c\n .\n \n \n {\\displaystyle \\operatorname {Var} \\left(c^{\\mathsf {T}}X\\right)=c^{\\mathsf {T}}\\Sigma c.}\n \n 
                                                                                  \n

                                                                                  This implies that the variance of the mean can be written as (with a column vector of ones)\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n (\n \n \n \n x\n \u00af\n \n \n \n )\n \n =\n Var\n \u2061\n \n (\n \n \n \n 1\n n\n \n \n \n 1\n \u2032\n \n X\n \n )\n \n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n 1\n \u2032\n \n \u03a3\n 1.\n \n \n {\\displaystyle \\operatorname {Var} \\left({\\bar {x}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}1'X\\right)={\\frac {1}{n^{2}}}1'\\Sigma 1.}\n \n 
                                                                                  \n

                                                                                  Sum of variables

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  Sum of uncorrelated variables

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n
                                                                                  Main article: Bienaym\u00e9's identity
                                                                                  \n
                                                                                  See also: Sum of normally distributed random variables
                                                                                  \n

                                                                                  One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances:\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n .\n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} (X_{i}).}\n \n 
                                                                                  \n

                                                                                  This statement is called the Bienaym\u00e9 formula[5] and was discovered in 1853.[6][7] It is often made with the stronger condition that the variables are independent, but being uncorrelated suffices. So if all the variables have the same variance \u03c32, then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n (\n \n \n X\n \u00af\n \n \n )\n \n =\n Var\n \u2061\n \n (\n \n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n Var\n \u2061\n \n (\n \n X\n \n i\n \n \n )\n \n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n n\n \n \u03c3\n \n 2\n \n \n =\n \n \n \n \u03c3\n \n 2\n \n \n n\n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)=\\operatorname {Var} \\left({\\frac {1}{n}}\\sum _{i=1}^{n}X_{i}\\right)={\\frac {1}{n^{2}}}\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)={\\frac {1}{n^{2}}}n\\sigma ^{2}={\\frac {\\sigma ^{2}}{n}}.}\n \n 
                                                                                  \n

                                                                                  That is, the variance of the mean decreases when n increases. This formula for the variance of the mean is used in the definition of the standard error of the sample mean, which is used in the central limit theorem.\n

                                                                                  To prove the initial statement, it suffices to show that\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n +\n Y\n )\n =\n Var\n \u2061\n (\n X\n )\n +\n Var\n \u2061\n (\n Y\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (X+Y)=\\operatorname {Var} (X)+\\operatorname {Var} (Y).}\n \n 
                                                                                  \n

                                                                                  The general result then follows by induction. Starting with the definition,\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n +\n Y\n )\n \n \n \n =\n E\n \u2061\n \n [\n \n (\n X\n +\n Y\n \n )\n \n 2\n \n \n \n ]\n \n \u2212\n (\n E\n \u2061\n [\n X\n +\n Y\n ]\n \n )\n \n 2\n \n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n \n X\n \n 2\n \n \n +\n 2\n X\n Y\n +\n \n Y\n \n 2\n \n \n \n ]\n \n \u2212\n (\n E\n \u2061\n [\n X\n ]\n +\n E\n \u2061\n [\n Y\n ]\n \n )\n \n 2\n \n \n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[(X+Y)^{2}\\right]-(\\operatorname {E} [X+Y])^{2}\\\\[5pt]&=\\operatorname {E} \\left[X^{2}+2XY+Y^{2}\\right]-(\\operatorname {E} [X]+\\operatorname {E} [Y])^{2}.\\end{aligned}}}\n \n 
                                                                                  \n

                                                                                  Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y, this further simplifies as follows:\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n +\n Y\n )\n \n \n \n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n +\n 2\n E\n \u2061\n [\n X\n Y\n ]\n +\n E\n \u2061\n \n [\n \n Y\n \n 2\n \n \n ]\n \n \u2212\n \n (\n \n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n +\n 2\n E\n \u2061\n [\n X\n ]\n E\n \u2061\n [\n Y\n ]\n +\n E\n \u2061\n [\n Y\n \n ]\n \n 2\n \n \n \n )\n \n \n \n \n \n \n \n =\n E\n \u2061\n \n [\n \n X\n \n 2\n \n \n ]\n \n +\n E\n \u2061\n \n [\n \n Y\n \n 2\n \n \n ]\n \n \u2212\n E\n \u2061\n [\n X\n \n ]\n \n 2\n \n \n \u2212\n E\n \u2061\n [\n Y\n \n ]\n \n 2\n \n \n \n \n \n \n \n \n =\n Var\n \u2061\n (\n X\n )\n +\n Var\n \u2061\n (\n Y\n )\n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (X+Y)&=\\operatorname {E} \\left[X^{2}\\right]+2\\operatorname {E} [XY]+\\operatorname {E} \\left[Y^{2}\\right]-\\left(\\operatorname {E} [X]^{2}+2\\operatorname {E} [X]\\operatorname {E} [Y]+\\operatorname {E} [Y]^{2}\\right)\\\\[5pt]&=\\operatorname {E} \\left[X^{2}\\right]+\\operatorname {E} \\left[Y^{2}\\right]-\\operatorname {E} [X]^{2}-\\operatorname {E} [Y]^{2}\\\\[5pt]&=\\operatorname {Var} (X)+\\operatorname {Var} (Y).\\end{aligned}}}\n \n 
                                                                                  \n

                                                                                  Sum of correlated variables

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n
                                                                                  Sum of correlated variables with fixed sample size
                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n
                                                                                  Main article: Bienaym\u00e9's identity
                                                                                  \n

                                                                                  In general, the variance of the sum of n variables is the sum of their covariances:\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n Cov\n \u2061\n \n (\n \n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n \n )\n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n Var\n \u2061\n \n (\n \n X\n \n i\n \n \n )\n \n +\n 2\n \n \u2211\n \n 1\n \u2264\n i\n <\n j\n \u2264\n n\n \n \n Cov\n \u2061\n \n (\n \n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n \n )\n \n .\n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)=\\sum _{i=1}^{n}\\sum _{j=1}^{n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right)=\\sum _{i=1}^{n}\\operatorname {Var} \\left(X_{i}\\right)+2\\sum _{1\\leq i<j\\leq n}\\operatorname {Cov} \\left(X_{i},X_{j}\\right).}\n \n 
                                                                                  \n

                                                                                  (Note: The second equality comes from the fact that Cov(Xi,Xi) = Var(Xi).)\n

                                                                                  Here, \n \n \n \n Cov\n \u2061\n (\n \u22c5\n ,\n \u22c5\n )\n \n \n {\\displaystyle \\operatorname {Cov} (\\cdot ,\\cdot )}\n \n  is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. The next expression states equivalently that the variance of the sum is the sum of the diagonal of covariance matrix plus two times the sum of its upper triangular elements (or its lower triangular elements); this emphasizes that the covariance matrix is symmetric. This formula is used in the theory of Cronbach's alpha in classical test theory.\n

                                                                                  So, if the variables have equal variance \u03c32 and the average correlation of distinct variables is \u03c1, then the variance of their mean is\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n (\n \n \n X\n \u00af\n \n \n )\n \n =\n \n \n \n \u03c3\n \n 2\n \n \n n\n \n \n +\n \n \n \n n\n \u2212\n 1\n \n n\n \n \n \u03c1\n \n \u03c3\n \n 2\n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {\\sigma ^{2}}{n}}+{\\frac {n-1}{n}}\\rho \\sigma ^{2}.}\n \n 
                                                                                  \n

                                                                                  This implies that the variance of the mean increases with the average of the correlations. In other words, additional correlated observations are not as effective as additional independent observations at reducing the uncertainty of the mean. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n (\n \n \n X\n \u00af\n \n \n )\n \n =\n \n \n 1\n n\n \n \n +\n \n \n \n n\n \u2212\n 1\n \n n\n \n \n \u03c1\n .\n \n \n {\\displaystyle \\operatorname {Var} \\left({\\overline {X}}\\right)={\\frac {1}{n}}+{\\frac {n-1}{n}}\\rho .}\n \n 
                                                                                  \n

                                                                                  This formula is used in the Spearman\u2013Brown prediction formula of classical test theory. This converges to \u03c1 if n goes to infinity, provided that the average correlation remains constant or converges too. So for the variance of the mean of standardized variables with equal correlations or converging average correlation we have\n

                                                                                  \n
                                                                                  \n \n \n \n \n lim\n \n n\n \u2192\n \u221e\n \n \n Var\n \u2061\n \n (\n \n \n X\n \u00af\n \n \n )\n \n =\n \u03c1\n .\n \n \n {\\displaystyle \\lim _{n\\to \\infty }\\operatorname {Var} \\left({\\overline {X}}\\right)=\\rho .}\n \n 
                                                                                  \n

                                                                                  Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. This makes clear that the sample mean of correlated variables does not generally converge to the population mean, even though the law of large numbers states that the sample mean will converge for independent variables.\n

                                                                                  \n
                                                                                  Sum of uncorrelated variables with random sample size
                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size N is a random variable whose variation adds to the variation of X, such that,\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n E\n \u2061\n \n [\n N\n ]\n \n Var\n \u2061\n (\n X\n )\n +\n Var\n \u2061\n (\n N\n )\n (\n E\n \u2061\n \n [\n X\n ]\n \n \n )\n \n 2\n \n \n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{N}X_{i}\\right)=\\operatorname {E} \\left[N\\right]\\operatorname {Var} (X)+\\operatorname {Var} (N)(\\operatorname {E} \\left[X\\right])^{2}}\n \n [8]
                                                                                  \n

                                                                                  which follows from the law of total variance.\n

                                                                                  If N has a Poisson distribution, then \n \n \n \n E\n \u2061\n [\n N\n ]\n =\n Var\n \u2061\n (\n N\n )\n \n \n {\\displaystyle \\operatorname {E} [N]=\\operatorname {Var} (N)}\n \n  with estimator n = N. So, the estimator of \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n X\n \n i\n \n \n \n )\n \n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i=1}^{n}X_{i}\\right)}\n \n  becomes \n \n \n \n n\n \n \n \n S\n \n x\n \n \n \n \n 2\n \n \n +\n n\n \n \n \n \n X\n \u00af\n \n \n \n \n 2\n \n \n \n \n {\\displaystyle n{S_{x}}^{2}+n{\\bar {X}}^{2}}\n \n , giving \n \n \n \n SE\n \u2061\n (\n \n \n \n X\n \u00af\n \n \n \n )\n =\n \n \n \n \n \n \n \n S\n \n x\n \n \n \n \n 2\n \n \n +\n \n \n \n \n X\n \u00af\n \n \n \n \n 2\n \n \n \n n\n \n \n \n \n \n {\\displaystyle \\operatorname {SE} ({\\bar {X}})={\\sqrt {\\frac {{S_{x}}^{2}+{\\bar {X}}^{2}}{n}}}}\n \n \n(see standard error of the sample mean).\n

                                                                                  \n

                                                                                  Weighted sum of variables

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n
                                                                                  See also: Variance of a weighted arithmetic mean
                                                                                  \n
                                                                                  Not to be confused with Weighted variance.
                                                                                  \n

                                                                                  The scaling property and the Bienaym\u00e9 formula, along with the property of the covariance Cov(aX, bY) = ab Cov(X, Y) jointly imply that\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n a\n X\n \u00b1\n b\n Y\n )\n =\n \n a\n \n 2\n \n \n Var\n \u2061\n (\n X\n )\n +\n \n b\n \n 2\n \n \n Var\n \u2061\n (\n Y\n )\n \u00b1\n 2\n a\n b\n \n Cov\n \u2061\n (\n X\n ,\n Y\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (aX\\pm bY)=a^{2}\\operatorname {Var} (X)+b^{2}\\operatorname {Var} (Y)\\pm 2ab\\,\\operatorname {Cov} (X,Y).}\n \n 
                                                                                  \n

                                                                                  This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y, then the weight of the variance of X will be four times the weight of the variance of Y.\n

                                                                                  The expression above can be extended to a weighted sum of multiple variables:\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n (\n \n \n \u2211\n \n i\n \n \n n\n \n \n \n a\n \n i\n \n \n \n X\n \n i\n \n \n \n )\n \n =\n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n a\n \n i\n \n \n 2\n \n \n Var\n \u2061\n (\n \n X\n \n i\n \n \n )\n +\n 2\n \n \u2211\n \n 1\n \u2264\n i\n \n \n \n \u2211\n \n <\n j\n \u2264\n n\n \n \n \n a\n \n i\n \n \n \n a\n \n j\n \n \n Cov\n \u2061\n (\n \n X\n \n i\n \n \n ,\n \n X\n \n j\n \n \n )\n \n \n {\\displaystyle \\operatorname {Var} \\left(\\sum _{i}^{n}a_{i}X_{i}\\right)=\\sum _{i=1}^{n}a_{i}^{2}\\operatorname {Var} (X_{i})+2\\sum _{1\\leq i}\\sum _{<j\\leq n}a_{i}a_{j}\\operatorname {Cov} (X_{i},X_{j})}\n \n 
                                                                                  \n

                                                                                  Product of variables

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  Product of independent variables

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  If two variables X and Y are independent, the variance of their product is given by[9]\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n Y\n )\n =\n [\n E\n \u2061\n (\n X\n )\n \n ]\n \n 2\n \n \n Var\n \u2061\n (\n Y\n )\n +\n [\n E\n \u2061\n (\n Y\n )\n \n ]\n \n 2\n \n \n Var\n \u2061\n (\n X\n )\n +\n Var\n \u2061\n (\n X\n )\n Var\n \u2061\n (\n Y\n )\n .\n \n \n {\\displaystyle \\operatorname {Var} (XY)=[\\operatorname {E} (X)]^{2}\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\operatorname {Var} (X)+\\operatorname {Var} (X)\\operatorname {Var} (Y).}\n \n 
                                                                                  \n

                                                                                  Equivalently, using the basic properties of expectation, it is given by\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n (\n X\n Y\n )\n =\n E\n \u2061\n \n (\n \n X\n \n 2\n \n \n )\n \n E\n \u2061\n \n (\n \n Y\n \n 2\n \n \n )\n \n \u2212\n [\n E\n \u2061\n (\n X\n )\n \n ]\n \n 2\n \n \n [\n E\n \u2061\n (\n Y\n )\n \n ]\n \n 2\n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} (XY)=\\operatorname {E} \\left(X^{2}\\right)\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (X)]^{2}[\\operatorname {E} (Y)]^{2}.}\n \n 
                                                                                  \n

                                                                                  Product of statistically dependent variables

                                                                                  \n\n \nedit\n\n\n\n\n
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                                                                                  In general, if two variables are statistically dependent, then the variance of their product is given by:\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n \n Var\n \u2061\n (\n X\n Y\n )\n =\n \n\n \n \n \n E\n \u2061\n \n [\n \n \n X\n \n 2\n \n \n \n Y\n \n 2\n \n \n \n ]\n \n \u2212\n [\n E\n \u2061\n (\n X\n Y\n )\n \n ]\n \n 2\n \n \n \n \n \n \n =\n \n\n \n \n \n Cov\n \u2061\n \n (\n \n \n X\n \n 2\n \n \n ,\n \n Y\n \n 2\n \n \n \n )\n \n +\n E\n \u2061\n (\n \n X\n \n 2\n \n \n )\n E\n \u2061\n \n (\n \n Y\n \n 2\n \n \n )\n \n \u2212\n [\n E\n \u2061\n (\n X\n Y\n )\n \n ]\n \n 2\n \n \n \n \n \n \n =\n \n\n \n \n \n Cov\n \u2061\n \n (\n \n \n X\n \n 2\n \n \n ,\n \n Y\n \n 2\n \n \n \n )\n \n +\n \n (\n \n Var\n \u2061\n (\n X\n )\n +\n [\n E\n \u2061\n (\n X\n )\n \n ]\n \n 2\n \n \n \n )\n \n \n (\n \n Var\n \u2061\n (\n Y\n )\n +\n [\n E\n \u2061\n (\n Y\n )\n \n ]\n \n 2\n \n \n \n )\n \n \n \n \n \n \n \n \u2212\n [\n Cov\n \u2061\n (\n X\n ,\n Y\n )\n +\n E\n \u2061\n (\n X\n )\n E\n \u2061\n (\n Y\n )\n \n ]\n \n 2\n \n \n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {Var} (XY)={}&\\operatorname {E} \\left[X^{2}Y^{2}\\right]-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\operatorname {E} (X^{2})\\operatorname {E} \\left(Y^{2}\\right)-[\\operatorname {E} (XY)]^{2}\\\\[5pt]={}&\\operatorname {Cov} \\left(X^{2},Y^{2}\\right)+\\left(\\operatorname {Var} (X)+[\\operatorname {E} (X)]^{2}\\right)\\left(\\operatorname {Var} (Y)+[\\operatorname {E} (Y)]^{2}\\right)\\\\[5pt]&-[\\operatorname {Cov} (X,Y)+\\operatorname {E} (X)\\operatorname {E} (Y)]^{2}\\end{aligned}}}\n \n 
                                                                                  \n

                                                                                  Arbitrary functions

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n
                                                                                  Main article: Uncertainty propagation
                                                                                  \n

                                                                                  The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables: see Taylor expansions for the moments of functions of random variables. For example, the approximate variance of a function of one variable is given by\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n [\n \n f\n (\n X\n )\n \n ]\n \n \u2248\n \n \n (\n \n \n f\n \u2032\n \n (\n E\n \u2061\n \n [\n X\n ]\n \n )\n \n )\n \n \n 2\n \n \n Var\n \u2061\n \n [\n X\n ]\n \n \n \n {\\displaystyle \\operatorname {Var} \\left[f(X)\\right]\\approx \\left(f'(\\operatorname {E} \\left[X\\right])\\right)^{2}\\operatorname {Var} \\left[X\\right]}\n \n 
                                                                                  \n

                                                                                  provided that f is twice differentiable and that the mean and variance of X are finite.\n

                                                                                  \n

                                                                                  Population variance and sample variance

                                                                                  \n\n \nedit\n\n\n\n\n
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                                                                                  \n

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                                                                                  See also: Unbiased estimation of standard deviation
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                                                                                  Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one estimates the mean and variance from a limited set of observations by using an estimator equation. The estimator is a function of the sample of n observations drawn without observational bias from the whole population of potential observations. In this example, the sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.\n

                                                                                  The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance \u2013 these are consistent estimators (they converge to the value of the whole population as the number of samples increases) but can be improved. Most simply, the sample variance is computed as the sum of squared deviations about the (sample) mean, divided by n as the number of samples. However, using values other than n improves the estimator in various ways. Four common values for the denominator are n, n \u2212 1, n + 1, and n \u2212 1.5: n is the simplest (the variance of the sample), n \u2212 1 eliminates bias,[10] n + 1 minimizes mean squared error for the normal distribution,[11] and n \u2212 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.[12]\n

                                                                                  Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (n \u2212 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n -1 instead of n, is called Bessel's correction.[10] The resulting estimator is unbiased and is called the (corrected) sample variance or unbiased sample variance. If the mean is determined in some other way than from the same samples used to estimate the variance, then this bias does not arise, and the variance can safely be estimated as that of the samples about the (independently known) mean.\n

                                                                                  Secondly, the sample variance does not generally minimize mean squared error between sample variance and population variance. Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance) and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than n \u2212 1) and is a simple example of a shrinkage estimator: one \"shrinks\" the unbiased estimator towards zero. For the normal distribution, dividing by n + 1 (instead of n \u2212 1 or n) minimizes mean squared error.[11] The resulting estimator is biased, however, and is known as the biased sample variation.\n

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                                                                                  Population variance

                                                                                  \n\n \nedit\n\n\n\n\n
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                                                                                  In general, the population variance of a finite population of size N with values xi is given by\n \n \n \n \n \n \n \n \n \u03c3\n \n 2\n \n \n \n \n \n =\n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n \n (\n \n \n x\n \n i\n \n \n \u2212\n \u03bc\n \n )\n \n \n 2\n \n \n =\n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n (\n \n \n x\n \n i\n \n \n 2\n \n \n \u2212\n 2\n \u03bc\n \n x\n \n i\n \n \n +\n \n \u03bc\n \n 2\n \n \n \n )\n \n \n \n \n \n \n \n =\n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n 2\n \n \n \n )\n \n \u2212\n 2\n \u03bc\n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n \n )\n \n +\n \n \u03bc\n \n 2\n \n \n \n \n \n \n \n \n =\n E\n \u2061\n [\n \n x\n \n i\n \n \n 2\n \n \n ]\n \u2212\n \n \u03bc\n \n 2\n \n \n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\sigma ^{2}&={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}-\\mu \\right)^{2}={\\frac {1}{N}}\\sum _{i=1}^{N}\\left(x_{i}^{2}-2\\mu x_{i}+\\mu ^{2}\\right)\\\\[5pt]&=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-2\\mu \\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)+\\mu ^{2}\\\\[5pt]&=\\operatorname {E} [x_{i}^{2}]-\\mu ^{2}\\end{aligned}}}\n \n where the population mean is \n \n \n \n \u03bc\n =\n E\n \u2061\n [\n \n x\n \n i\n \n \n ]\n =\n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n \n \n {\\textstyle \\mu =\\operatorname {E} [x_{i}]={\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}}\n \n  and \n \n \n \n E\n \u2061\n [\n \n x\n \n i\n \n \n 2\n \n \n ]\n =\n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n 2\n \n \n \n )\n \n \n \n {\\textstyle \\operatorname {E} [x_{i}^{2}]=\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)}\n \n , where \n \n \n \n E\n \n \n {\\textstyle \\operatorname {E} }\n \n  is the expectation value operator.\n

                                                                                  The population variance can also be computed using[13]\n

                                                                                  \n
                                                                                  \n \n \n \n \n \u03c3\n \n 2\n \n \n =\n \n \n 1\n \n N\n \n 2\n \n \n \n \n \n \u2211\n \n i\n <\n j\n \n \n \n \n (\n \n \n x\n \n i\n \n \n \u2212\n \n x\n \n j\n \n \n \n )\n \n \n 2\n \n \n =\n \n \n 1\n \n 2\n \n N\n \n 2\n \n \n \n \n \n \n \u2211\n \n i\n ,\n j\n =\n 1\n \n \n N\n \n \n \n \n (\n \n \n x\n \n i\n \n \n \u2212\n \n x\n \n j\n \n \n \n )\n \n \n 2\n \n \n .\n \n \n {\\displaystyle \\sigma ^{2}={\\frac {1}{N^{2}}}\\sum _{i<j}\\left(x_{i}-x_{j}\\right)^{2}={\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}.}\n \n 
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                                                                                  (The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because\n \n \n \n \n \n \n \n \n \n \n 1\n \n 2\n \n N\n \n 2\n \n \n \n \n \n \n \u2211\n \n i\n ,\n j\n =\n 1\n \n \n N\n \n \n \n \n (\n \n \n x\n \n i\n \n \n \u2212\n \n x\n \n j\n \n \n \n )\n \n \n 2\n \n \n \n \n \n \n =\n \n\n \n \n \n \n \n 1\n \n 2\n \n N\n \n 2\n \n \n \n \n \n \n \u2211\n \n i\n ,\n j\n =\n 1\n \n \n N\n \n \n \n (\n \n \n x\n \n i\n \n \n 2\n \n \n \u2212\n 2\n \n x\n \n i\n \n \n \n x\n \n j\n \n \n +\n \n x\n \n j\n \n \n 2\n \n \n \n )\n \n \n \n \n \n =\n \n\n \n \n \n \n \n 1\n \n 2\n N\n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n N\n \n \n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n 2\n \n \n \n )\n \n \u2212\n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n x\n \n i\n \n \n \n )\n \n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n N\n \n \n \n x\n \n j\n \n \n \n )\n \n +\n \n \n 1\n \n 2\n N\n \n \n \n \n \u2211\n \n i\n =\n 1\n \n \n N\n \n \n \n (\n \n \n \n 1\n N\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n N\n \n \n \n x\n \n j\n \n \n 2\n \n \n \n )\n \n \n \n \n \n =\n \n\n \n \n \n \n \n 1\n 2\n \n \n \n (\n \n \n \u03c3\n \n 2\n \n \n +\n \n \u03bc\n \n 2\n \n \n \n )\n \n \u2212\n \n \u03bc\n \n 2\n \n \n +\n \n \n 1\n 2\n \n \n \n (\n \n \n \u03c3\n \n 2\n \n \n +\n \n \u03bc\n \n 2\n \n \n \n )\n \n \n \n \n \n =\n \n\n \n \n \n \n \u03c3\n \n 2\n \n \n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}-x_{j}\\right)^{2}\\\\[5pt]={}&{\\frac {1}{2N^{2}}}\\sum _{i,j=1}^{N}\\left(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2N}}\\sum _{j=1}^{N}\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}^{2}\\right)-\\left({\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}\\right)\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}\\right)+{\\frac {1}{2N}}\\sum _{i=1}^{N}\\left({\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}^{2}\\right)\\\\[5pt]={}&{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-\\mu ^{2}+{\\frac {1}{2}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\\\[5pt]={}&\\sigma ^{2}.\\end{aligned}}}\n \n The population variance matches the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.\n

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                                                                                  Sample variance

                                                                                  \n\n \nedit\n\n\n\n\n
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                                                                                  See also: Sample standard deviation
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                                                                                  Biased sample variance

                                                                                  \n\n \nedit\n\n\n\n\n
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                                                                                  In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population.[14] This is generally referred to as sample variance or empirical variance. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution.\n

                                                                                  We take a sample with replacement of n values Y1, ..., Yn from the population of size \n \n \n \n N\n \n \n {\\textstyle N}\n \n , where n < N, and estimate the variance on the basis of this sample.[15] Directly taking the variance of the sample data gives the average of the squared deviations:\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \n (\n \n \n Y\n \n i\n \n \n \u2212\n \n \n Y\n \u00af\n \n \n \n )\n \n \n 2\n \n \n =\n \n (\n \n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n Y\n \n i\n \n \n 2\n \n \n \n )\n \n \u2212\n \n \n \n Y\n \u00af\n \n \n \n 2\n \n \n =\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n i\n ,\n j\n \n :\n \n i\n <\n j\n \n \n \n \n (\n \n \n Y\n \n i\n \n \n \u2212\n \n Y\n \n j\n \n \n \n )\n \n \n 2\n \n \n .\n \n \n {\\displaystyle {\\tilde {S}}_{Y}^{2}={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}=\\left({\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}^{2}\\right)-{\\overline {Y}}^{2}={\\frac {1}{n^{2}}}\\sum _{i,j\\,:\\,i<j}\\left(Y_{i}-Y_{j}\\right)^{2}.}\n \n [16]
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                                                                                  (See the section Population variance for the derivation of this formula.) Here, \n \n \n \n \n \n Y\n \u00af\n \n \n \n \n {\\displaystyle {\\overline {Y}}}\n \n  denotes the sample mean:\n

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                                                                                  \n \n \n \n \n \n Y\n \u00af\n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n Y\n \n i\n \n \n .\n \n \n {\\displaystyle {\\overline {Y}}={\\frac {1}{n}}\\sum _{i=1}^{n}Y_{i}.}\n \n 
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                                                                                  Since the Yi are selected randomly, both \n \n \n \n \n \n Y\n \u00af\n \n \n \n \n {\\displaystyle {\\overline {Y}}}\n \n  and \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n \n \n {\\displaystyle {\\tilde {S}}_{Y}^{2}}\n \n  are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples {Yi} of size n from the population. For \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n \n \n {\\displaystyle {\\tilde {S}}_{Y}^{2}}\n \n  this gives:\n

                                                                                  \n
                                                                                  \n \n \n \n \n \n \n \n E\n \u2061\n [\n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n ]\n \n \n \n =\n E\n \u2061\n \n [\n \n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \n (\n \n \n Y\n \n i\n \n \n \u2212\n \n \n 1\n n\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n Y\n \n j\n \n \n \n )\n \n \n 2\n \n \n \n ]\n \n \n \n \n \n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n E\n \u2061\n \n [\n \n \n Y\n \n i\n \n \n 2\n \n \n \u2212\n \n \n 2\n n\n \n \n \n Y\n \n i\n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n Y\n \n j\n \n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n Y\n \n j\n \n \n \n \u2211\n \n k\n =\n 1\n \n \n n\n \n \n \n Y\n \n k\n \n \n \n ]\n \n \n \n \n \n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n (\n \n E\n \u2061\n \n [\n \n Y\n \n i\n \n \n 2\n \n \n ]\n \n \u2212\n \n \n 2\n n\n \n \n \n (\n \n \n \u2211\n \n j\n \u2260\n i\n \n \n E\n \u2061\n \n [\n \n \n Y\n \n i\n \n \n \n Y\n \n j\n \n \n \n ]\n \n +\n E\n \u2061\n \n [\n \n Y\n \n i\n \n \n 2\n \n \n ]\n \n \n )\n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n \u2211\n \n k\n \u2260\n j\n \n \n n\n \n \n E\n \u2061\n \n [\n \n \n Y\n \n j\n \n \n \n Y\n \n k\n \n \n \n ]\n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n E\n \u2061\n \n [\n \n Y\n \n j\n \n \n 2\n \n \n ]\n \n \n )\n \n \n \n \n \n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n (\n \n \n \n \n n\n \u2212\n 2\n \n n\n \n \n E\n \u2061\n \n [\n \n Y\n \n i\n \n \n 2\n \n \n ]\n \n \u2212\n \n \n 2\n n\n \n \n \n \u2211\n \n j\n \u2260\n i\n \n \n E\n \u2061\n \n [\n \n \n Y\n \n i\n \n \n \n Y\n \n j\n \n \n \n ]\n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n \n \u2211\n \n k\n \u2260\n j\n \n \n n\n \n \n E\n \u2061\n \n [\n \n \n Y\n \n j\n \n \n \n Y\n \n k\n \n \n \n ]\n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n \n \u2211\n \n j\n =\n 1\n \n \n n\n \n \n E\n \u2061\n \n [\n \n Y\n \n j\n \n \n 2\n \n \n ]\n \n \n )\n \n \n \n \n \n \n \n =\n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n [\n \n \n \n \n n\n \u2212\n 2\n \n n\n \n \n \n (\n \n \n \u03c3\n \n 2\n \n \n +\n \n \u03bc\n \n 2\n \n \n \n )\n \n \u2212\n \n \n 2\n n\n \n \n (\n n\n \u2212\n 1\n )\n \n \u03bc\n \n 2\n \n \n +\n \n \n 1\n \n n\n \n 2\n \n \n \n \n n\n (\n n\n \u2212\n 1\n )\n \n \u03bc\n \n 2\n \n \n +\n \n \n 1\n n\n \n \n \n (\n \n \n \u03c3\n \n 2\n \n \n +\n \n \u03bc\n \n 2\n \n \n \n )\n \n \n ]\n \n \n \n \n \n \n \n =\n \n \n \n n\n \u2212\n 1\n \n n\n \n \n \n \u03c3\n \n 2\n \n \n .\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}\\operatorname {E} [{\\tilde {S}}_{Y}^{2}]&=\\operatorname {E} \\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\frac {1}{n}}\\sum _{j=1}^{n}Y_{j}\\right)^{2}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\operatorname {E} \\left[Y_{i}^{2}-{\\frac {2}{n}}Y_{i}\\sum _{j=1}^{n}Y_{j}+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}Y_{j}\\sum _{k=1}^{n}Y_{k}\\right]\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\left(\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+\\operatorname {E} \\left[Y_{i}^{2}\\right]\\right)+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left({\\frac {n-2}{n}}\\operatorname {E} \\left[Y_{i}^{2}\\right]-{\\frac {2}{n}}\\sum _{j\\neq i}\\operatorname {E} \\left[Y_{i}Y_{j}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\sum _{k\\neq j}^{n}\\operatorname {E} \\left[Y_{j}Y_{k}\\right]+{\\frac {1}{n^{2}}}\\sum _{j=1}^{n}\\operatorname {E} \\left[Y_{j}^{2}\\right]\\right)\\\\[5pt]&={\\frac {1}{n}}\\sum _{i=1}^{n}\\left[{\\frac {n-2}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)-{\\frac {2}{n}}(n-1)\\mu ^{2}+{\\frac {1}{n^{2}}}n(n-1)\\mu ^{2}+{\\frac {1}{n}}\\left(\\sigma ^{2}+\\mu ^{2}\\right)\\right]\\\\[5pt]&={\\frac {n-1}{n}}\\sigma ^{2}.\\end{aligned}}}\n \n 
                                                                                  \n

                                                                                  Here \n \n \n \n \n \u03c3\n \n 2\n \n \n =\n E\n \u2061\n [\n \n Y\n \n i\n \n \n 2\n \n \n ]\n \u2212\n \n \u03bc\n \n 2\n \n \n \n \n {\\textstyle \\sigma ^{2}=\\operatorname {E} [Y_{i}^{2}]-\\mu ^{2}}\n \n  derived in the section Population variance and \n \n \n \n E\n \u2061\n [\n \n Y\n \n i\n \n \n \n Y\n \n j\n \n \n ]\n =\n E\n \u2061\n [\n \n Y\n \n i\n \n \n ]\n E\n \u2061\n [\n \n Y\n \n j\n \n \n ]\n =\n \n \u03bc\n \n 2\n \n \n \n \n {\\textstyle \\operatorname {E} [Y_{i}Y_{j}]=\\operatorname {E} [Y_{i}]\\operatorname {E} [Y_{j}]=\\mu ^{2}}\n \n  due to independency of \n \n \n \n \n Y\n \n i\n \n \n \n \n {\\textstyle Y_{i}}\n \n  and \n \n \n \n \n Y\n \n j\n \n \n \n \n {\\textstyle Y_{j}}\n \n  are used.\n

                                                                                  Hence \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n \n \n {\\textstyle {\\tilde {S}}_{Y}^{2}}\n \n  gives an estimate of the population variance that is biased by a factor of \n \n \n \n \n \n \n n\n \u2212\n 1\n \n n\n \n \n \n \n {\\textstyle {\\frac {n-1}{n}}}\n \n  as the expectation value of \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n \n \n {\\textstyle {\\tilde {S}}_{Y}^{2}}\n \n  is smaller than the population variance (true variance) by that factor. For this reason, \n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n \n \n {\\textstyle {\\tilde {S}}_{Y}^{2}}\n \n  is referred to as the biased sample variance.\n

                                                                                  \n

                                                                                  Unbiased sample variance

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  Correcting for this bias yields the unbiased sample variance, denoted \n \n \n \n \n S\n \n 2\n \n \n \n \n {\\displaystyle S^{2}}\n \n :\n

                                                                                  \n
                                                                                  \n \n \n \n \n S\n \n 2\n \n \n =\n \n \n n\n \n n\n \u2212\n 1\n \n \n \n \n \n \n \n S\n ~\n \n \n \n \n Y\n \n \n 2\n \n \n =\n \n \n n\n \n n\n \u2212\n 1\n \n \n \n \n [\n \n \n \n 1\n n\n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \n (\n \n \n Y\n \n i\n \n \n \u2212\n \n \n Y\n \u00af\n \n \n \n )\n \n \n 2\n \n \n \n ]\n \n =\n \n \n 1\n \n n\n \u2212\n 1\n \n \n \n \n \u2211\n \n i\n =\n 1\n \n \n n\n \n \n \n \n (\n \n \n Y\n \n i\n \n \n \u2212\n \n \n Y\n \u00af\n \n \n \n )\n \n \n 2\n \n \n \n \n {\\displaystyle S^{2}={\\frac {n}{n-1}}{\\tilde {S}}_{Y}^{2}={\\frac {n}{n-1}}\\left[{\\frac {1}{n}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}\\right]={\\frac {1}{n-1}}\\sum _{i=1}^{n}\\left(Y_{i}-{\\overline {Y}}\\right)^{2}}\n \n 
                                                                                  \n

                                                                                  Either estimator may be simply referred to as the sample variance when the version can be determined by context. The same proof is also applicable for samples taken from a continuous probability distribution.\n

                                                                                  The use of the term n \u2212 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation (the square root of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n \u2212 1.5 yields an almost unbiased estimator.\n

                                                                                  The unbiased sample variance is a U-statistic for the function \u0192(y1, y2) = (y1 \u2212 y2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.\n

                                                                                  \n
                                                                                  Example
                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  For a set of numbers {10, 15, 30, 45, 57, 52 63, 72, 81, 93, 102, 105}, if this set is the whole data population for some measurement, then variance is the population variance 932.743 as the sum of the squared deviations about the mean of this set, divided by 12 as the number of the set members. If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.\n

                                                                                  \n

                                                                                  Distribution of the sample variance

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n
                                                                                   
                                                                                   
                                                                                  Distribution and cumulative distribution of S2/\u03c32, for various values of \u03bd = n \u2212 1, when the yi are independent normally distributed.
                                                                                  \n

                                                                                  Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Yi are independent observations from a normal distribution, Cochran's theorem shows that the unbiased sample variance S2 follows a scaled chi-squared distribution (see also: asymptotic properties and an elementary proof):[17]\n

                                                                                  \n
                                                                                  \n \n \n \n (\n n\n \u2212\n 1\n )\n \n \n \n S\n \n 2\n \n \n \n \u03c3\n \n 2\n \n \n \n \n \u223c\n \n \u03c7\n \n n\n \u2212\n 1\n \n \n 2\n \n \n \n \n {\\displaystyle (n-1){\\frac {S^{2}}{\\sigma ^{2}}}\\sim \\chi _{n-1}^{2}}\n \n 
                                                                                  \n

                                                                                  where \u03c32 is the population variance. As a direct consequence, it follows that\n

                                                                                  \n
                                                                                  \n \n \n \n E\n \u2061\n \n (\n \n S\n \n 2\n \n \n )\n \n =\n E\n \u2061\n \n (\n \n \n \n \n \u03c3\n \n 2\n \n \n \n n\n \u2212\n 1\n \n \n \n \n \u03c7\n \n n\n \u2212\n 1\n \n \n 2\n \n \n \n )\n \n =\n \n \u03c3\n \n 2\n \n \n ,\n \n \n {\\displaystyle \\operatorname {E} \\left(S^{2}\\right)=\\operatorname {E} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)=\\sigma ^{2},}\n \n 
                                                                                  \n

                                                                                  and[18]\n

                                                                                  \n
                                                                                  \n \n \n \n Var\n \u2061\n \n [\n \n S\n \n 2\n \n \n ]\n \n =\n Var\n \u2061\n \n (\n \n \n \n \n \u03c3\n \n 2\n \n \n \n n\n \u2212\n 1\n \n \n \n \n \u03c7\n \n n\n \u2212\n 1\n \n \n 2\n \n \n \n )\n \n =\n \n \n \n \u03c3\n \n 4\n \n \n \n (\n n\n \u2212\n 1\n \n )\n \n 2\n \n \n \n \n \n Var\n \u2061\n \n (\n \n \u03c7\n \n n\n \u2212\n 1\n \n \n 2\n \n \n )\n \n =\n \n \n \n 2\n \n \u03c3\n \n 4\n \n \n \n \n n\n \u2212\n 1\n \n \n \n .\n \n \n {\\displaystyle \\operatorname {Var} \\left[S^{2}\\right]=\\operatorname {Var} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)={\\frac {\\sigma ^{4}}{(n-1)^{2}}}\\operatorname {Var} \\left(\\chi _{n-1}^{2}\\right)={\\frac {2\\sigma ^{4}}{n-1}}.}\n \n 
                                                                                  \n

                                                                                  If Yi are independent and identically distributed, but not necessarily normally distributed, then[19]\n

                                                                                  \n
                                                                                  \n \n \n \n E\n \u2061\n \n [\n \n S\n \n 2\n \n \n ]\n \n =\n \n \u03c3\n \n 2\n \n \n ,\n \n Var\n \u2061\n \n [\n \n S\n \n 2\n \n \n ]\n \n =\n \n \n \n \u03c3\n \n 4\n \n \n n\n \n \n \n (\n \n \u03ba\n \u2212\n 1\n +\n \n \n 2\n \n n\n \u2212\n 1\n \n \n \n \n )\n \n =\n \n \n 1\n n\n \n \n \n (\n \n \n \u03bc\n \n 4\n \n \n \u2212\n \n \n \n n\n \u2212\n 3\n \n \n n\n \u2212\n 1\n \n \n \n \n \u03c3\n \n 4\n \n \n \n )\n \n ,\n \n \n {\\displaystyle \\operatorname {E} \\left[S^{2}\\right]=\\sigma ^{2},\\quad \\operatorname {Var} \\left[S^{2}\\right]={\\frac {\\sigma ^{4}}{n}}\\left(\\kappa -1+{\\frac {2}{n-1}}\\right)={\\frac {1}{n}}\\left(\\mu _{4}-{\\frac {n-3}{n-1}}\\sigma ^{4}\\right),}\n \n 
                                                                                  \n

                                                                                  where \u03ba is the kurtosis of the distribution and \u03bc4 is the fourth central moment.\n

                                                                                  If the conditions of the law of large numbers hold for the squared observations, S2 is a consistent estimator of \u03c32. One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.).[20][21][22]\n

                                                                                  \n

                                                                                  Samuelson's inequality

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  Samuelson's inequality is a result that states bounds on the values that individual observations in a sample can take, given that the sample mean and (biased) variance have been calculated.[23] Values must lie within the limits \n \n \n \n \n \n \n y\n \u00af\n \n \n \n \u00b1\n \n \u03c3\n \n Y\n \n \n (\n n\n \u2212\n 1\n \n )\n \n 1\n \n /\n \n 2\n \n \n .\n \n \n {\\displaystyle {\\bar {y}}\\pm \\sigma _{Y}(n-1)^{1/2}.}\n \n \n

                                                                                  \n

                                                                                  Relations with the harmonic and arithmetic means

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  It has been shown[24] that for a sample {yi} of positive real numbers,\n

                                                                                  \n
                                                                                  \n \n \n \n \n \u03c3\n \n y\n \n \n 2\n \n \n \u2264\n 2\n \n y\n \n max\n \n \n (\n A\n \u2212\n H\n )\n ,\n \n \n {\\displaystyle \\sigma _{y}^{2}\\leq 2y_{\\max }(A-H),}\n \n 
                                                                                  \n

                                                                                  where ymax is the maximum of the sample, A is the arithmetic mean, H is the harmonic mean of the sample and \n \n \n \n \n \u03c3\n \n y\n \n \n 2\n \n \n \n \n {\\displaystyle \\sigma _{y}^{2}}\n \n  is the (biased) variance of the sample.\n

                                                                                  This bound has been improved, and it is known that variance is bounded by\n

                                                                                  \n
                                                                                  \n \n \n \n \n \u03c3\n \n y\n \n \n 2\n \n \n \u2264\n \n \n \n \n y\n \n max\n \n \n (\n A\n \u2212\n H\n )\n (\n \n y\n \n max\n \n \n \u2212\n A\n )\n \n \n \n y\n \n max\n \n \n \u2212\n H\n \n \n \n ,\n \n \n {\\displaystyle \\sigma _{y}^{2}\\leq {\\frac {y_{\\max }(A-H)(y_{\\max }-A)}{y_{\\max }-H}},}\n \n 
                                                                                  \n
                                                                                  \n \n \n \n \n \u03c3\n \n y\n \n \n 2\n \n \n \u2265\n \n \n \n \n y\n \n min\n \n \n (\n A\n \u2212\n H\n )\n (\n A\n \u2212\n \n y\n \n min\n \n \n )\n \n \n H\n \u2212\n \n y\n \n min\n \n \n \n \n \n ,\n \n \n {\\displaystyle \\sigma _{y}^{2}\\geq {\\frac {y_{\\min }(A-H)(A-y_{\\min })}{H-y_{\\min }}},}\n \n 
                                                                                  \n

                                                                                  where ymin is the minimum of the sample.[25]\n

                                                                                  \n

                                                                                  Tests of equality of variances

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n

                                                                                  The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult.\n

                                                                                  Several non parametric tests have been proposed: these include the Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey test, the Capon test, Mood test, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that both medians be known and equal to zero. The Mood, Klotz, Capon and Barton\u2013David\u2013Ansari\u2013Freund\u2013Siegel\u2013Tukey tests also apply to two variances. They allow the median to be unknown but do require that the two medians are equal.\n

                                                                                  The Lehmann test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variances include the Box test, the Box\u2013Anderson test and the Moses test.\n

                                                                                  Resampling methods, which include the bootstrap and the jackknife, may be used to test the equality of variances.\n

                                                                                  \n

                                                                                  Moment of inertia

                                                                                  \n\n \nedit\n\n\n\n\n
                                                                                  \n
                                                                                  See also: Moment (physics) \u00a7 Examples
                                                                                  \n

                                                                                  The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass.[26] It is because of this analogy that such things as the variance are called moments of probability distributions.[26] The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of n points with a covariance matrix of \n \n \n \n \u03a3\n \n \n {\\displaystyle \\Sigma }\n \n  is given by[citation needed]\n

                                                                                  \n
                                                                                  \n \n \n \n I\n =\n n\n \n (\n \n \n \n 1\n \n \n 3\n \u00d7\n 3\n \n \n tr\n \u2061\n (\n \u03a3\n )\n \u2212\n \u03a3\n \n )\n \n .\n \n \n {\\displaystyle I=n\\left(\\mathbf {1} _{3\\times 3}\\operatorname {tr} (\\Sigma )-\\Sigma \\right).}\n \n 
                                                                                  \n

                                                                                  This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the x axis and distributed along it. The covariance matrix might look like\n

                                                                                  \n
                                                                                  \n \n \n \n \u03a3\n =\n \n \n [\n \n \n \n 10\n \n \n 0\n \n \n 0\n \n \n \n \n 0\n \n \n 0.1\n \n \n 0\n \n \n \n \n 0\n \n \n 0\n \n \n 0.1\n \n \n \n ]\n \n \n .\n \n \n {\\displaystyle \\Sigma ={\\begin{bmatrix}10&0&0\\\\0&0.1&0\\\\0&0&0.1\\end{bmatrix}}.}\n \n 
                                                                                  \n

                                                                                  That is, there is the most variance in the x direction. Physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is\n

                                                                                  \n
                                                                                  \n \n \n \n I\n =\n n\n \n \n [\n \n \n \n 0.2\n \n \n 0\n \n \n 0\n \n \n \n \n 0\n \n \n 10.1\n \n \n 0\n \n \n \n \n 0\n \n \n 0\n \n \n 10.1\n \n \n \n ]\n \n \n .\n \n \n {\\displaystyle I=n{\\begin{bmatrix}0.2&0&0\\\\0&10.1&0\\\\0&0&10.1\\end{bmatrix}}.}\n \n 
                                                                                  \n

                                                                                  Semivariance

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                                                                                  The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation:\n \n \n \n \n Semivariance\n \n =\n \n \n 1\n \n n\n \n \n \n \n \u2211\n \n i\n :\n \n x\n \n i\n \n \n <\n \u03bc\n \n \n (\n \n x\n \n i\n \n \n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n \n {\\displaystyle {\\text{Semivariance}}={1 \\over {n}}\\sum _{i:x_{i}<\\mu }(x_{i}-\\mu )^{2}}\n \n It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not.[27]\n

                                                                                  For inequalities associated with the semivariance, see Chebyshev's inequality \u00a7 Semivariances.\n

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                                                                                  Etymology

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                                                                                  The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance:[28]\n

                                                                                  \n

                                                                                  The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and, therefore, that the variability may be uniformly measured by the standard deviation corresponding to the square root of the mean square error. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations \n \n \n \n \n \u03c3\n \n 1\n \n \n \n \n {\\displaystyle \\sigma _{1}}\n \n  and \n \n \n \n \n \u03c3\n \n 2\n \n \n \n \n {\\displaystyle \\sigma _{2}}\n \n , it is found that the distribution, when both causes act together, has a standard deviation \n \n \n \n \n \n \n \u03c3\n \n 1\n \n \n 2\n \n \n +\n \n \u03c3\n \n 2\n \n \n 2\n \n \n \n \n \n \n {\\displaystyle {\\sqrt {\\sigma _{1}^{2}+\\sigma _{2}^{2}}}}\n \n . It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance...

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                                                                                  Generalizations

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                                                                                  For complex variables

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                                                                                  If \n \n \n \n x\n \n \n {\\displaystyle x}\n \n  is a scalar complex-valued random variable, with values in \n \n \n \n \n C\n \n ,\n \n \n {\\displaystyle \\mathbb {C} ,}\n \n  then its variance is \n \n \n \n E\n \u2061\n \n [\n \n (\n x\n \u2212\n \u03bc\n )\n (\n x\n \u2212\n \u03bc\n \n )\n \n \u2217\n \n \n \n ]\n \n ,\n \n \n {\\displaystyle \\operatorname {E} \\left[(x-\\mu )(x-\\mu )^{*}\\right],}\n \n  where \n \n \n \n \n x\n \n \u2217\n \n \n \n \n {\\displaystyle x^{*}}\n \n  is the complex conjugate of \n \n \n \n x\n .\n \n \n {\\displaystyle x.}\n \n  This variance is a real scalar.\n

                                                                                  \n

                                                                                  For vector-valued random variables

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                                                                                  As a matrix

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                                                                                  If \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  is a vector-valued random variable, with values in \n \n \n \n \n \n R\n \n \n n\n \n \n ,\n \n \n {\\displaystyle \\mathbb {R} ^{n},}\n \n  and thought of as a column vector, then a natural generalization of variance is \n \n \n \n E\n \u2061\n \n [\n \n (\n X\n \u2212\n \u03bc\n )\n (\n X\n \u2212\n \u03bc\n \n )\n \n T\n \n \n \n ]\n \n ,\n \n \n {\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\operatorname {T} }\\right],}\n \n  where \n \n \n \n \u03bc\n =\n E\n \u2061\n (\n X\n )\n \n \n {\\displaystyle \\mu =\\operatorname {E} (X)}\n \n  and \n \n \n \n \n X\n \n T\n \n \n \n \n {\\displaystyle X^{\\operatorname {T} }}\n \n  is the transpose of \n \n \n \n X\n ,\n \n \n {\\displaystyle X,}\n \n  and so is a row vector. The result is a positive semi-definite square matrix, commonly referred to as the variance-covariance matrix (or simply as the covariance matrix).\n

                                                                                  If \n \n \n \n X\n \n \n {\\displaystyle X}\n \n  is a vector- and complex-valued random variable, with values in \n \n \n \n \n \n C\n \n \n n\n \n \n ,\n \n \n {\\displaystyle \\mathbb {C} ^{n},}\n \n  then the covariance matrix is \n \n \n \n E\n \u2061\n \n [\n \n (\n X\n \u2212\n \u03bc\n )\n (\n X\n \u2212\n \u03bc\n \n )\n \n \u2020\n \n \n \n ]\n \n ,\n \n \n {\\displaystyle \\operatorname {E} \\left[(X-\\mu )(X-\\mu )^{\\dagger }\\right],}\n \n  where \n \n \n \n \n X\n \n \u2020\n \n \n \n \n {\\displaystyle X^{\\dagger }}\n \n  is the conjugate transpose of \n \n \n \n X\n .\n \n \n {\\displaystyle X.}\n \n [citation needed] This matrix is also positive semi-definite and square.\n

                                                                                  \n

                                                                                  As a scalar

                                                                                  \n\n \nedit\n\n\n\n\n
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                                                                                  Another generalization of variance for vector-valued random variables \n \n \n \n X\n \n \n {\\displaystyle X}\n \n , which results in a scalar value rather than in a matrix, is the generalized variance \n \n \n \n det\n (\n C\n )\n \n \n {\\displaystyle \\det(C)}\n \n , the determinant of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.[29]\n

                                                                                  A different generalization is obtained by considering the equation for the scalar variance, \n \n \n \n Var\n \u2061\n (\n X\n )\n =\n E\n \u2061\n \n [\n \n (\n X\n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n ]\n \n \n \n {\\displaystyle \\operatorname {Var} (X)=\\operatorname {E} \\left[(X-\\mu )^{2}\\right]}\n \n , and reinterpreting \n \n \n \n (\n X\n \u2212\n \u03bc\n \n )\n \n 2\n \n \n \n \n {\\displaystyle (X-\\mu )^{2}}\n \n  as the squared Euclidean distance between the random variable and its mean, or, simply as the scalar product of the vector \n \n \n \n X\n \u2212\n \u03bc\n \n \n {\\displaystyle X-\\mu }\n \n  with itself. This results in \n \n \n \n E\n \u2061\n \n [\n \n (\n X\n \u2212\n \u03bc\n \n )\n \n T\n \n \n (\n X\n \u2212\n \u03bc\n )\n \n ]\n \n =\n tr\n \u2061\n (\n C\n )\n ,\n \n \n {\\displaystyle \\operatorname {E} \\left[(X-\\mu )^{\\operatorname {T} }(X-\\mu )\\right]=\\operatorname {tr} (C),}\n \n  which is the trace of the covariance matrix.\n

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                                                                                  See also

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                                                                                  \n
                                                                                  Look up variance in Wiktionary, the free dictionary.
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                                                                                  Types of variance

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                                                                                  References

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                                                                                  1. ^ Wasserman, Larry (2005). All of Statistics: a concise course in statistical inference. Springer texts in statistics. p. 51. ISBN 978-1-4419-2322-6.\n
                                                                                    2. \n
                                                                                  2. ^ Yuli Zhang; Huaiyu Wu; Lei Cheng (June 2012). Some new deformation formulas about variance and covariance. Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012). pp. 987\u2013992.\n
                                                                                    3. \n
                                                                                  3. ^ Kagan, A.; Shepp, L. A. (1998). \"Why the variance?\". Statistics & Probability Letters. 38 (4): 329\u2013333. doi:10.1016/S0167-7152(98)00041-8.\n
                                                                                    4. \n
                                                                                  4. ^ Johnson, Richard; Wichern, Dean (2001). Applied Multivariate Statistical Analysis. Prentice Hall. p. 76. ISBN 0-13-187715-1.\n
                                                                                    5. \n
                                                                                  5. ^ Lo\u00e8ve, M. (1977) \"Probability Theory\", Graduate Texts in Mathematics, Volume 45, 4th edition, Springer-Verlag, p. 12.\n
                                                                                    6. \n
                                                                                  6. ^ Bienaym\u00e9, I.-J. (1853) \"Consid\u00e9rations \u00e0 l'appui de la d\u00e9couverte de Laplace sur la loi de probabilit\u00e9 dans la m\u00e9thode des moindres carr\u00e9s\", Comptes rendus de l'Acad\u00e9mie des sciences Paris, 37, p. 309\u2013317; digital copy available [1]\n
                                                                                    7. \n
                                                                                  7. ^ Bienaym\u00e9, I.-J. (1867) \"Consid\u00e9rations \u00e0 l'appui de la d\u00e9couverte de Laplace sur la loi de probabilit\u00e9 dans la m\u00e9thode des moindres carr\u00e9s\", Journal de Math\u00e9matiques Pures et Appliqu\u00e9es, S\u00e9rie 2, Tome 12, p. 158\u2013167; digital copy available [2][3]\n
                                                                                    8. \n
                                                                                  8. ^ Cornell, J R, and Benjamin, C A, Probability, Statistics, and Decisions for Civil Engineers, McGraw-Hill, NY, 1970, pp.178-9.\n
                                                                                    9. \n
                                                                                  9. ^ Goodman, Leo A. (December 1960). \"On the Exact Variance of Products\". Journal of the American Statistical Association. 55 (292): 708\u2013713. doi:10.2307/2281592. JSTOR 2281592.\n
                                                                                    10. \n
                                                                                  10. ^ a b Reichmann, W. J. (1961). \"Appendix 8\". Use and Abuse of Statistics (Reprinted 1964\u20131970 by Pelican ed.). London: Methuen.\n
                                                                                    11. \n
                                                                                  11. ^ a b Kourouklis, Stavros (2012). \"A New Estimator of the Variance Based on Minimizing Mean Squared Error\". The American Statistician. 66 (4): 234\u2013236. doi:10.1080/00031305.2012.735209. ISSN 0003-1305. JSTOR 23339501.\n
                                                                                    12. \n
                                                                                  12. ^ Brugger, R. M. (1969). \"A Note on Unbiased Estimation of the Standard Deviation\". The American Statistician. 23 (4): 32. doi:10.1080/00031305.1969.1048186 (inactive 16 December 2024).{{cite journal}}: CS1 maint: DOI inactive as of December 2024 (link)\n
                                                                                    13. \n
                                                                                  13. ^ Yuli Zhang; Huaiyu Wu; Lei Cheng (June 2012). Some new deformation formulas about variance and covariance. Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012). pp. 987\u2013992.\n
                                                                                    14. \n
                                                                                  14. ^ Navidi, William (2006) Statistics for Engineers and Scientists, McGraw-Hill, p. 14.\n
                                                                                    15. \n
                                                                                  15. ^ Montgomery, D. C. and Runger, G. C. (1994) Applied statistics and probability for engineers, page 201. John Wiley & Sons New York\n
                                                                                    16. \n
                                                                                  16. ^ Yuli Zhang; Huaiyu Wu; Lei Cheng (June 2012). Some new deformation formulas about variance and covariance. Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012). pp. 987\u2013992.\n
                                                                                    17. \n
                                                                                  17. ^ Knight K. (2000), Mathematical Statistics, Chapman and Hall, New York. (proposition 2.11)\n
                                                                                    18. \n
                                                                                  18. ^ Casella and Berger (2002) Statistical Inference, Example 7.3.3, p. 331 [full citation needed]\n
                                                                                    19. \n
                                                                                  19. ^ Mood, A. M., Graybill, F. A., and Boes, D.C. (1974) Introduction to the Theory of Statistics, 3rd Edition, McGraw-Hill, New York, p. 229\n
                                                                                    20. \n
                                                                                  20. ^ Kenney, John F.; Keeping, E.S. (1951). Mathematics of Statistics. Part Two (PDF) (2nd ed.). Princeton, New Jersey: D. Van Nostrand Company, Inc. Archived from the original (PDF) on Nov 17, 2018 \u2013 via KrishiKosh.\n
                                                                                    21. \n
                                                                                  21. ^ Rose, Colin; Smith, Murray D. (2002). \"Mathematical Statistics with Mathematica\". Springer-Verlag, New York.\n
                                                                                    22. \n
                                                                                  22. ^ Weisstein, Eric W. \"Sample Variance Distribution\". MathWorld Wolfram.\n
                                                                                    23. \n
                                                                                  23. ^ Samuelson, Paul (1968). \"How Deviant Can You Be?\". Journal of the American Statistical Association. 63 (324): 1522\u20131525. doi:10.1080/01621459.1968.10480944. JSTOR 2285901.\n
                                                                                    24. \n
                                                                                  24. ^ Mercer, A. McD. (2000). \"Bounds for A\u2013G, A\u2013H, G\u2013H, and a family of inequalities of Ky Fan's type, using a general method\". J. Math. Anal. Appl. 243 (1): 163\u2013173. doi:10.1006/jmaa.1999.6688.\n
                                                                                    25. \n
                                                                                  25. ^ Sharma, R. (2008). \"Some more inequalities for arithmetic mean, harmonic mean and variance\". Journal of Mathematical Inequalities. 2 (1): 109\u2013114. CiteSeerX 10.1.1.551.9397. doi:10.7153/jmi-02-11.\n
                                                                                    26. \n
                                                                                  26. ^ a b Magnello, M. Eileen. \"Karl Pearson and the Origins of Modern Statistics: An Elastician becomes a Statistician\". The Rutherford Journal.\n
                                                                                    27. \n
                                                                                  27. ^ Fama, Eugene F.; French, Kenneth R. (2010-04-21). \"Q&A: Semi-Variance: A Better Risk Measure?\". Fama/French Forum.\n
                                                                                    28. \n
                                                                                  28. ^ Ronald Fisher (1918) The correlation between relatives on the supposition of Mendelian Inheritance\n
                                                                                    29. \n
                                                                                  29. ^ Kocherlakota, S.; Kocherlakota, K. (2004). \"Generalized Variance\". Encyclopedia of Statistical Sciences. Wiley Online Library. doi:10.1002/0471667196.ess0869. ISBN 0-471-66719-6.\n
                                                                                    30. \n
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                                                                                  Retrieved from \"https://en.wikipedia.org/w/index.php?title=Variance&oldid=1264758978\"
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                                                                                  \n\t\t\t\tZiet u iets wat niet hoort of niet klopt?\n\t\t\t
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                                                                                  \n\t\t\t\t Rapporteren\n\t\t\t
                                                                                  ", "content": [{"c": "Rapporteren", "t": "text"}]}, {"type": "paragraph", "raw_content": "

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                                                                                  \n\t\t\t\tZondag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
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                                                                                  Is Grondwerk GWY in\n\t\t\t\t\tNoordwolde uw bedrijf?\n\t\t\t\t

                                                                                  ", "content": {"title_content": "Is Grondwerk GWY in Noordwolde uw bedrijf?", "level": "3"}}, {"type": "paragraph", "raw_content": "

                                                                                  Claim uw pagina, vul uw gegevens aan en word beter gevonden.

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                                                                                  Eigenschappen Grondwerk

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                                                                                  Niet bekend
                                                                                  ", "content": [{"c": "Niet bekend", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                                                  Niet bekend
                                                                                  ", "content": [{"c": "Niet bekend", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                                  \n\t\t\tBeoordelingen\n\t\t

                                                                                  ", "content": {"title_content": "Beoordelingen", "level": "2"}}, {"type": "title", "raw_content": "

                                                                                  \n\t\t\t\t\tHoe waardeer jij grondwerk GWY uit Noordwolde?\n\t\t\t\t

                                                                                  ", "content": {"title_content": "Hoe waardeer jij grondwerk GWY uit Noordwolde ?", "level": "4"}}]], "main_html": "
                                                                                  \n\t\t\t\tZiet u iets wat niet hoort of niet klopt?\n\t\t\t
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                                                                                  Openingstijden

                                                                                  \n\t\t\t\tMaandag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tDinsdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tWoensdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tDonderdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tVrijdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tZaterdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tZondag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t

                                                                                  Is Grondwerk GWY in\n\t\t\t\t\tNoordwolde uw bedrijf?\n\t\t\t\t


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                                                                                  Eigenschappen Grondwerk

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                                                                                  \n\t\t\tBeoordelingen\n\t\t

                                                                                  \n\t\t\t\t\tHoe waardeer jij grondwerk GWY uit Noordwolde?\n\t\t\t\t

                                                                                  ", "statics": {"paragraph": 6, "paragraph.text": 6, "title": 5, "table": 1}, "url": "https://grondwerken-overzicht.nl/noordwolde/gwy", "content": "Ziet u iets wat niet hoort of niet klopt?\n\nRapporteren\n\nDit bedrijf heeft helaas nog geen omschrijving. Bent u echter de eigenaar van dit bedrijf? Claim dan uw pagina en vul uw gegevens aan, of help ons en je mede-bezoekers door dit bedrijf te tippen over deze vermelding.\n\n## Openingstijden\n\n| Maandag | - |\n|---|---|\n| Dinsdag | - |\n| Woensdag | - |\n| Donderdag | - |\n| Vrijdag | - |\n| Zaterdag | - |\n| Zondag | - |\n\n### Is Grondwerk GWY in Noordwolde uw bedrijf?\n\nClaim uw pagina, vul uw gegevens aan en word beter gevonden. Binnen een paar minuten geregeld en volledig gratis!\n\n## Eigenschappen Grondwerk\n\nNiet bekend\n\nNiet bekend\n\n## Beoordelingen\n\n#### Hoe waardeer jij grondwerk GWY uit Noordwolde ?\n", "html": "\n\n\n\n\n\n\n\n\n\n\tGrondwerk GWY in Noordwolde - Grondwerkgids grondwerken-overzicht.nl\n\n\t\n\t\n\t\n\t\n\n\t\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n\t\t\n\t\t\n\t\t\t\t\t\n\n\t\t\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n\n\n\t\n \n \n\t\n\t\n \t\n\t\n \n \n \n \n \t\n \n\n\t\n\t\n\n\t\n\t\n\t\n\n\n\n\t\n\t\t\t\t
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                                                                                  \n\t\t\t\t\t\t\t\t\t\tGWY\n\t\t\t\t\t\t\t\t\t

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                                                                                  \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tIndustriestraat 11
                                                                                  \n\t\t\t\t\t\t\t\t\t                                                                                  \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t8391AG\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tNoordwolde\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t
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                                                                                  \n \t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t05614 ... Toon nummer\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t \t\t\t\t\t\t\t

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                                                                                  \n\t\t\t\t\t\t\t0 beoordelingen\n\n\t\t\t\t\t\t\t\t\t\t\t\t\t
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                                                                                  \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t Delen\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t
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                                                                                  \n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tNeem contact op\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tContact opnemen\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tSchrijf een beoordeling\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tSchrijf beoordeling\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t
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                                                                                  \n\t\t\t\n\t\t\t\tZiet u iets wat niet hoort of niet klopt?\n\t\t\t\n\t\t\t\n\t\t\t\t Rapporteren\n\t\t\t\n\t\t
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                                                                                  Dit bedrijf heeft helaas nog geen omschrijving. Bent u echter de eigenaar van dit bedrijf? Claim dan uw pagina en vul uw gegevens aan, of help ons en je mede-bezoekers door dit bedrijf te tippen over deze vermelding.

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                                                                                  Openingstijden

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                                                                                  Is Grondwerk GWY in\n\t\t\t\t\tNoordwolde uw bedrijf?\n\t\t\t\t


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                                                                                  Claim uw pagina, vul uw gegevens aan en word beter gevonden.

                                                                                  \n\t\t\t\tBinnen een paar minuten geregeld en volledig gratis!

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                                                                                  Eigenschappen Grondwerk

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                                                                                  \n\t\t\t\t\t\t\t\t\tNiet bekend\n\t\t\t\t\t\t\t
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                                                                                  \n Facebook\n

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                                                                                  \n Facebook\n

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                                                                                  Berichten laden ..

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                                                                                  \n Geen Facebook gegevens gevonden\n

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                                                                                  \n Claim uw bedrijf
                                                                                  om uw tijdlijn hier weer te geven\n

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                                                                                  \n\t\t\tBeoordelingen\n\t\t

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                                                                                  Waardering van bezoekers

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                                                                                  Overzicht waarderingen

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                                                                                  \n\t\t\t\t\t\t\t\t\t\t\tDeel je mening
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                                                                                  \n\n\t\t\tCommunicatie\n\t
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                                                                                  \n\t\t\t\t\tHoe waardeer jij grondwerk GWY uit Noordwolde?\n\t\t\t\t

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                                                                                  \n\t\t\t\tU dient ingelogd te zijn om een review te plaatsen. Nu inloggen of registreer een account.\n\t\t\t
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                                                                                  \n\t\t\t\t\t\t\t\t\t\tWilt uw als eigenaar van deze vermelding notificaties van nieuwe beoordelingen ontvangen?\n\t\t\t\t\t\t\t\t\t\t

                                                                                  \n\t\t\t\t\t\t\t\t\t\tStap direct over naar een betaalde vermelding,
                                                                                  of bekijk nu alle mogelijkheden.\n\t\t\t\t\t\t\t\t\t
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                                                                                  Informeer GWY over deze vermelding

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                                                                                  Verwijderingsverzoek indienen voor
                                                                                  GWY

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                                                                                  Afbeelding toevoegen

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                                                                                  Bezig met uploaden ..

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                                                                                  Deze bedrijfsvermelding rapporteren

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                                                                                  \n\t\t\t\t\t\tgrondwerken-overzicht.nl streeft ernaar om de vermelde gegevens zo juist, nuttig & netjes mogelijk te houden. We nemen uw bericht zo snel mogelijk in behandeling.\n\t\t\t\t\t                                                                                  \n\t\t\t\t
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                                                                                  \n Je bent niet ingelogd. Log eerst even in of registreer je Crasmedia account.\n
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                                                                                  Inloggen

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                                                                                  \n \n \n \n\n \n \n Wachtwoord vergeten?\n \n

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                                                                                  Nog geen account?

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                                                                                  De voordelen van een account:

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                                                                                  • Plaats reviews bij al uw favoriete bedrijven
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                                                                                  • Volg het laatste nieuws van bedrijven waar uw interesse in heeft
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                                                                                  • Registreer of claim uw eigen bedrijf
                                                                                    • \n
                                                                                  • \n \n Crasmedia account - \u00e9\u00e9n account voor al onze\n gidsen\n
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                                                                                  \n\n Nog geen account? Registeer nu!\n
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                                                                                  \n Je bent niet ingelogd. Log eerst even in of registreer je Crasmedia account.\n
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                                                                                  Inloggen

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                                                                                  \n \n \n\n \n \n Wachtwoord vergeten?\n \n

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                                                                                  Nog geen account?

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                                                                                  De voordelen van een account:

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                                                                                    \n
                                                                                  • Plaats reviews bij al uw favoriete bedrijven
                                                                                    • \n
                                                                                  • Volg het laatste nieuws van bedrijven waar uw interesse in heeft
                                                                                    • \n
                                                                                  • Registreer of claim uw eigen bedrijf
                                                                                    • \n
                                                                                  • \n \n Crasmedia account - \u00e9\u00e9n account voor al onze\n gidsen\n
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                                                                                  Registreren

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                                                                                  \n\t\t\n\t\t\n\t\t* Het wachtwoord dient uit minimaal 8 karakters te bestaan\n\t
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                                                                                  1. \n\t\t\t\t\n\t\t\t\t\tGrondwerken-overzicht.nl\n\t\t\t\t\n\t\t\t\t\n\t\t\t
                                                                                    2. \n\n\n\t\t\t\t\t\t
                                                                                  2. \n\t\t\t\t \u203a \n\t\t\t\t\n\t\t\t\t\tFriesland\n\t\t\t\t\n\t\t\t\t\n\t\t\t
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                                                                                  3. \n\t\t\t\t\t\t \u203a \n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tNoordwolde\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t
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                                                                                  4. \n\t\t\t\t\t\t \u203a \n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tGwy\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t
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                                                                                  • Voor grondwerk bedrijven
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                                                                                  © 2025 Grondwerken-overzicht.nl All rights reserved. Map data \u00a9 OpenStreetMap-auteurs

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                                                                                  \n\t\n\n\t\n\t\t\t\n\n\t\t\n\n\t\t\n\t\t\n\t\t\n\t\t\n\n\n"} +{"content_list": [[{"type": "paragraph", "raw_content": "
                                                                                  \n\t\t\t\tZiet u iets wat niet hoort of niet klopt?\n\t\t\t
                                                                                  ", "content": [{"c": "Ziet u iets wat niet hoort of niet klopt?", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                                                  \n\t\t\t\t Rapporteren\n\t\t\t
                                                                                  ", "content": [{"c": "Rapporteren", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                                  Dit bedrijf heeft helaas nog geen omschrijving. Bent u echter de eigenaar van dit bedrijf? Claim dan uw pagina en vul uw gegevens aan, of help ons en je mede-bezoekers door dit bedrijf te tippen over deze vermelding.

                                                                                  ", "content": [{"c": "Dit bedrijf heeft helaas nog geen omschrijving. Bent u echter de eigenaar van dit bedrijf? Claim dan uw pagina en vul uw gegevens aan, of help ons en je mede-bezoekers door dit bedrijf te tippen over deze vermelding.", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                                  Openingstijden

                                                                                  ", "content": {"title_content": "Openingstijden", "level": "2"}}, {"type": "table", "raw_content": "
                                                                                  \n\t\t\t\tMaandag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tDinsdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tWoensdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tDonderdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tVrijdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tZaterdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tZondag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  ", "content": {"html": "
                                                                                  Maandag-
                                                                                  Dinsdag-
                                                                                  Woensdag-
                                                                                  Donderdag-
                                                                                  Vrijdag-
                                                                                  Zaterdag-
                                                                                  Zondag-
                                                                                  ", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

                                                                                  Is Grondwerk GWY in\n\t\t\t\t\tNoordwolde uw bedrijf?\n\t\t\t\t

                                                                                  ", "content": {"title_content": "Is Grondwerk GWY in Noordwolde uw bedrijf?", "level": "3"}}, {"type": "paragraph", "raw_content": "

                                                                                  Claim uw pagina, vul uw gegevens aan en word beter gevonden.

                                                                                  \n\t\t\t\tBinnen een paar minuten geregeld en volledig gratis!

                                                                                  ", "content": [{"c": "Claim uw pagina, vul uw gegevens aan en word beter gevonden. Binnen een paar minuten geregeld en volledig gratis!", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                                  Eigenschappen Grondwerk

                                                                                  ", "content": {"title_content": "Eigenschappen Grondwerk", "level": "2"}}, {"type": "paragraph", "raw_content": "
                                                                                  Niet bekend
                                                                                  ", "content": [{"c": "Niet bekend", "t": "text"}]}, {"type": "paragraph", "raw_content": "
                                                                                  Niet bekend
                                                                                  ", "content": [{"c": "Niet bekend", "t": "text"}]}, {"type": "title", "raw_content": "

                                                                                  \n\t\t\tBeoordelingen\n\t\t

                                                                                  ", "content": {"title_content": "Beoordelingen", "level": "2"}}, {"type": "title", "raw_content": "

                                                                                  \n\t\t\t\t\tHoe waardeer jij grondwerk GWY uit Noordwolde?\n\t\t\t\t

                                                                                  ", "content": {"title_content": "Hoe waardeer jij grondwerk GWY uit Noordwolde ?", "level": "4"}}]], "main_html": "
                                                                                  \n\t\t\t\tZiet u iets wat niet hoort of niet klopt?\n\t\t\t
                                                                                  \n\t\t\t\t Rapporteren\n\t\t\t

                                                                                  Dit bedrijf heeft helaas nog geen omschrijving. Bent u echter de eigenaar van dit bedrijf? Claim dan uw pagina en vul uw gegevens aan, of help ons en je mede-bezoekers door dit bedrijf te tippen over deze vermelding.

                                                                                  Openingstijden

                                                                                  \n\t\t\t\tMaandag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tDinsdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tWoensdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tDonderdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tVrijdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tZaterdag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\tZondag\n\t\t\t\n\n\t\t\t\t\t\t\t\t\t\t\t\t-\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t

                                                                                  Is Grondwerk GWY in\n\t\t\t\t\tNoordwolde uw bedrijf?\n\t\t\t\t


                                                                                  Claim uw pagina, vul uw gegevens aan en word beter gevonden.

                                                                                  \n\t\t\t\tBinnen een paar minuten geregeld en volledig gratis!

                                                                                  Eigenschappen Grondwerk

                                                                                  Niet bekend
                                                                                  Niet bekend

                                                                                  \n\t\t\tBeoordelingen\n\t\t

                                                                                  \n\t\t\t\t\tHoe waardeer jij grondwerk GWY uit Noordwolde?\n\t\t\t\t

                                                                                  ", "statics": {"paragraph": 6, "paragraph.text": 6, "title": 5, "table": 1}, "url": "https://grondwerken-overzicht.nl/noordwolde/gwy", "content": "Ziet u iets wat niet hoort of niet klopt?\n\nRapporteren\n\nDit bedrijf heeft helaas nog geen omschrijving. Bent u echter de eigenaar van dit bedrijf? Claim dan uw pagina en vul uw gegevens aan, of help ons en je mede-bezoekers door dit bedrijf te tippen over deze vermelding.\n\n## Openingstijden\n\n| Maandag | - |\n|---|---|\n| Dinsdag | - |\n| Woensdag | - |\n| Donderdag | - |\n| Vrijdag | - |\n| Zaterdag | - |\n| Zondag | - |\n\n### Is Grondwerk GWY in Noordwolde uw bedrijf?\n\nClaim uw pagina, vul uw gegevens aan en word beter gevonden. Binnen een paar minuten geregeld en volledig gratis!\n\n## Eigenschappen Grondwerk\n\nNiet bekend\n\nNiet bekend\n\n## Beoordelingen\n\n#### Hoe waardeer jij grondwerk GWY uit Noordwolde ?\n", "html": "\n\n\n\n\n\n\n\n\n\n\tGrondwerk GWY in Noordwolde - Grondwerkgids grondwerken-overzicht.nl\n\n\t\n\t\n\t\n\t\n\n\t\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n\t\t\n\t\t\n\t\t\t\t\t\n\n\t\t\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n\n\n\t\n \n \n\t\n\t\n \t\n\t\n \n \n \n \n \t\n \n\n\t\n\t\n\n\t\n\t\n\t\n\n\n\n\t\n\t\t\t\t
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                                                                                  \n\t\t\t\t\t\t\t\t\t\tGWY\n\t\t\t\t\t\t\t\t\t

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                                                                                  \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tIndustriestraat 11
                                                                                  \n\t\t\t\t\t\t\t\t\t                                                                                  \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t8391AG\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tNoordwolde\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t
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                                                                                  \n \t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t05614 ... Toon nummer\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t \t\t\t\t\t\t\t

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                                                                                  \n\t\t\t\t\t\t\t0 beoordelingen\n\n\t\t\t\t\t\t\t\t\t\t\t\t\t
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                                                                                  \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tVolgen\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\n\t\t\t\t\t\t\t
                                                                                  \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t Delen\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t
                                                                                  \n\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t Tip bedrijf\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t Streetview\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t Route\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t
                                                                                  \n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tNeem contact op\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tContact opnemen\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tSchrijf een beoordeling\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tSchrijf beoordeling\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t
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                                                                                  \n\t\t\t\n\t\t\t\tZiet u iets wat niet hoort of niet klopt?\n\t\t\t\n\t\t\t\n\t\t\t\t Rapporteren\n\t\t\t\n\t\t
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                                                                                  Dit bedrijf heeft helaas nog geen omschrijving. Bent u echter de eigenaar van dit bedrijf? Claim dan uw pagina en vul uw gegevens aan, of help ons en je mede-bezoekers door dit bedrijf te tippen over deze vermelding.

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                                                                                  Openingstijden

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                                                                                  Is Grondwerk GWY in\n\t\t\t\t\tNoordwolde uw bedrijf?\n\t\t\t\t


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                                                                                  Claim uw pagina, vul uw gegevens aan en word beter gevonden.

                                                                                  \n\t\t\t\tBinnen een paar minuten geregeld en volledig gratis!

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                                                                                  \n .. of dien een verwijderingsverzoek in\n\t\t\t
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                                                                                  Eigenschappen Grondwerk

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                                                                                  \n Facebook\n

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                                                                                  \n Facebook\n

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                                                                                  Berichten laden ..

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                                                                                  \n Geen Facebook gegevens gevonden\n

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                                                                                  \n Claim uw bedrijf
                                                                                  om uw tijdlijn hier weer te geven\n

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                                                                                  \n\t\t\tBeoordelingen\n\t\t

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                                                                                  Waardering van bezoekers

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                                                                                  Overzicht waarderingen

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                                                                                  \n\t\t\t\t\t\t\t\t\t\t\tDeel je mening
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                                                                                  \n\t\t\t\t\tHoe waardeer jij grondwerk GWY uit Noordwolde?\n\t\t\t\t

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                                                                                  \n\t\t\t\tU dient ingelogd te zijn om een review te plaatsen. Nu inloggen of registreer een account.\n\t\t\t
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                                                                                  \n\t\t\t\t\t\t\t\t\t\tWilt uw als eigenaar van deze vermelding notificaties van nieuwe beoordelingen ontvangen?\n\t\t\t\t\t\t\t\t\t\t

                                                                                  \n\t\t\t\t\t\t\t\t\t\tStap direct over naar een betaalde vermelding,
                                                                                  of bekijk nu alle mogelijkheden.\n\t\t\t\t\t\t\t\t\t
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                                                                                  Informeer GWY over deze vermelding

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                                                                                  Verwijderingsverzoek indienen voor
                                                                                  GWY

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                                                                                  Afbeelding toevoegen

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                                                                                  Bezig met uploaden ..

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                                                                                  Deze bedrijfsvermelding rapporteren

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                                                                                  \n\t\t\t\t\t\tgrondwerken-overzicht.nl streeft ernaar om de vermelde gegevens zo juist, nuttig & netjes mogelijk te houden. We nemen uw bericht zo snel mogelijk in behandeling.\n\t\t\t\t\t                                                                                  \n\t\t\t\t
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                                                                                  \n Je bent niet ingelogd. Log eerst even in of registreer je Crasmedia account.\n
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                                                                                  Inloggen

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                                                                                  \n \n \n \n\n \n \n Wachtwoord vergeten?\n \n

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                                                                                  Nog geen account?

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                                                                                  De voordelen van een account:

                                                                                  \n\n
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                                                                                  • Plaats reviews bij al uw favoriete bedrijven
                                                                                    • \n
                                                                                  • Volg het laatste nieuws van bedrijven waar uw interesse in heeft
                                                                                    • \n
                                                                                  • Registreer of claim uw eigen bedrijf
                                                                                    • \n
                                                                                  • \n \n Crasmedia account - \u00e9\u00e9n account voor al onze\n gidsen\n
                                                                                    • \n
                                                                                  \n\n Nog geen account? Registeer nu!\n
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                                                                                  \n Je bent niet ingelogd. Log eerst even in of registreer je Crasmedia account.\n
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                                                                                  Inloggen

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                                                                                  \n \n \n\n \n \n Wachtwoord vergeten?\n \n

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                                                                                  © 2025 Grondwerken-overzicht.nl All rights reserved. Map data \u00a9 OpenStreetMap-auteurs

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                                                                                  \n\t\n\n\t\n\t\t\t\n\n\t\t\n\n\t\t\n\t\t\n\t\t\n\t\t\n\n\n"} diff --git a/bench/data/groundtruth/table_2.jsonl b/bench/data/groundtruth/table_2.jsonl index 3c79d5c0..f97bdf06 100644 --- a/bench/data/groundtruth/table_2.jsonl +++ b/bench/data/groundtruth/table_2.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "paragraph", "raw_content": "

                                                                                  Looking for the Best Underbody Underglow Kits that can last long and perform well? Read our review and buying guide on Best Underbody Underglow Kits below.

                                                                                  ", "content": [{"c": "Looking for the Best Underbody Underglow Kits that can last long and perform well? Read our review and buying guide on Best Underbody Underglow Kits below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                                  To help you in your search, we have completed this list of the Best Underbody Underglow Kits.

                                                                                  ", "content": [{"c": "To help you in your search, we have completed this list of the Best Underbody Underglow Kits.", "t": "text"}]}, {"type": "table", "raw_content": "
                                                                                  RankProduct NameScore
                                                                                  1
                                                                                  \"Car
                                                                                  Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color
                                                                                  9.7
                                                                                  2
                                                                                  \"4pcs
                                                                                  4pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent
                                                                                  9.5
                                                                                  3
                                                                                  \"Xprite
                                                                                  Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function
                                                                                  9.1
                                                                                  4
                                                                                  \"Car
                                                                                  Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof
                                                                                  8.8
                                                                                  5
                                                                                  \"GOODRUN
                                                                                  GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent
                                                                                  8.6
                                                                                  6
                                                                                  \"OPT7
                                                                                  OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync
                                                                                  8.3
                                                                                  7
                                                                                  \"Xprite
                                                                                  Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody
                                                                                  8.1
                                                                                  8
                                                                                  \"LEDGlow
                                                                                  LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars
                                                                                  7.8
                                                                                  9
                                                                                  \"KORJO
                                                                                  KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color
                                                                                  7.5
                                                                                  10
                                                                                  \"XTAUTO
                                                                                  XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light
                                                                                  7.2
                                                                                  ", "content": {"html": "
                                                                                  RankProduct NameScore
                                                                                  1Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color9.7
                                                                                  24pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent9.5
                                                                                  3Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function9.1
                                                                                  4Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof8.8
                                                                                  5GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent8.6
                                                                                  6OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync8.3
                                                                                  7Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody8.1
                                                                                  8LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars7.8
                                                                                  9KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color7.5
                                                                                  10XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light7.2
                                                                                  ", "is_complex": false}}]], "main_html": "

                                                                                  Looking for the Best Underbody Underglow Kits that can last long and perform well? Read our review and buying guide on Best Underbody Underglow Kits below.

                                                                                  To help you in your search, we have completed this list of the Best Underbody Underglow Kits.

                                                                                  RankProduct NameScore
                                                                                  1
                                                                                  \"Car
                                                                                  Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color
                                                                                  9.7
                                                                                  2
                                                                                  \"4pcs
                                                                                  4pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent
                                                                                  9.5
                                                                                  3
                                                                                  \"Xprite
                                                                                  Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function
                                                                                  9.1
                                                                                  4
                                                                                  \"Car
                                                                                  Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof
                                                                                  8.8
                                                                                  5
                                                                                  \"GOODRUN
                                                                                  GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent
                                                                                  8.6
                                                                                  6
                                                                                  \"OPT7
                                                                                  OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync
                                                                                  8.3
                                                                                  7
                                                                                  \"Xprite
                                                                                  Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody
                                                                                  8.1
                                                                                  8
                                                                                  \"LEDGlow
                                                                                  LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars
                                                                                  7.8
                                                                                  9
                                                                                  \"KORJO
                                                                                  KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color
                                                                                  7.5
                                                                                  10
                                                                                  \"XTAUTO
                                                                                  XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light
                                                                                  7.2
                                                                                  ", "statics": {"paragraph": 2, "paragraph.text": 2, "table": 1}, "url": "https://10bestreviewz.com/automotive/best-underbody-underglow-kits", "content": "Looking for the Best Underbody Underglow Kits that can last long and perform well? Read our review and buying guide on Best Underbody Underglow Kits below.\n\nTo help you in your search, we have completed this list of the Best Underbody Underglow Kits.\n\n| Rank | Product Name | Score |\n|---|---|---|\n| 1 | Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color | 9.7 |\n| 2 | 4pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent | 9.5 |\n| 3 | Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function | 9.1 |\n| 4 | Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof | 8.8 |\n| 5 | GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent | 8.6 |\n| 6 | OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync | 8.3 |\n| 7 | Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody | 8.1 |\n| 8 | LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars | 7.8 |\n| 9 | KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color | 7.5 |\n| 10 | XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light | 7.2 |\n", "html": "\n\n\n\n \n \n \n \n \n \n \n \n \n \n Top #10 Best Underbody Underglow Kits in 2025 | Reviews by Experts\n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n\n \n \n \n \n \n\n\n\n
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                                                                                  Best Underbody Underglow Kits In January 2025

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                                                                                  Looking for the Best Underbody Underglow Kits that can last long and perform well? Read our review and buying guide on Best Underbody Underglow Kits below.

                                                                                  \n

                                                                                  To help you in your search, we have completed this list of the Best Underbody Underglow Kits.

                                                                                  \n

                                                                                  Top Rated Best Underbody Underglow Kits Of 2025

                                                                                  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                                                  RankProduct NameScore
                                                                                  \n
                                                                                  \n
                                                                                  1
                                                                                  \n
                                                                                  \n \"Car\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  9.7
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  2
                                                                                  \n
                                                                                  \n \"4pcs\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  4pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  9.5
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  3
                                                                                  \n
                                                                                  \n \"Xprite\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  9.1
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
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                                                                                  \n
                                                                                  4
                                                                                  \n
                                                                                  \n \"Car\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  8.8
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
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                                                                                  \n
                                                                                  5
                                                                                  \n
                                                                                  \n \"GOODRUN\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  8.6
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
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                                                                                  \n
                                                                                  6
                                                                                  \n
                                                                                  \n \"OPT7\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  8.3
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
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                                                                                  \n
                                                                                  7
                                                                                  \n
                                                                                  \n \"Xprite\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  8.1
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
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                                                                                  \n
                                                                                  8
                                                                                  \n
                                                                                  \n \"LEDGlow\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
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                                                                                  7.8
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
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                                                                                  \n
                                                                                  9
                                                                                  \n
                                                                                  \n \"KORJO\n
                                                                                  \n
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                                                                                  KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
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                                                                                  7.5
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
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                                                                                  \n
                                                                                  10
                                                                                  \n
                                                                                  \n \"XTAUTO\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light
                                                                                  \n
                                                                                  Buy Now
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                                                                                  7.2
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                                                                                  Buy Now
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                                                                                  1. Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color

                                                                                  \n
                                                                                  \n
                                                                                  \n 1\n
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                                                                                  \n
                                                                                  \n \"Car\n
                                                                                  Buy Now
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                                                                                  9.7/10 our score
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                                                                                  \n Buy Now\n
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                                                                                  • Sound active & music decorationthe car led neon underglow lights has also 8 color flashing modes with sound sensors that lets under car glow system synchronize with music beat.the led music light strip follows the beat and adjusts flashing to fit the strength of the sound,adjustable brightness.
                                                                                  • High quality led chipwireless remote control color changing, flashing mode and music mode, create colorful gradient mode. upgraded high intensity smd 5050 led chip, provide brighter light, add the look and feel of the vehicle with bright lighting effects.
                                                                                  • More color & energy savingthe led under car glow system lights up colorful colors,it can be bent, fold, with your imagination to create your ambience,give a stylishly stunning look to the areas you may want to put them on.even though they're really bright, they're great power savers so you don't have to worry energy consumption.
                                                                                  • Universal compatibilitycar underglow lights its voltage is compatible with 12v. led underbody lights is waterproof and comes with built-in overload protection, so it can be used safely.it can be used by almost all of the car. perfect for under wheel arches, around the front grille, under foot-wells, in the trunk, along the dash.
                                                                                  • Easy installationstrong adhesive led strip lights kit(with black fixed tie), flexible and easy to install.designed with flexible tubing, protected from terrible road conditions and debris. high quality material and low temperature resistance.
                                                                                  \n

                                                                                  2. 4pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent

                                                                                  \n
                                                                                  \n
                                                                                  \n 2\n
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                                                                                  \n
                                                                                  \n \"4pcs\n
                                                                                  Buy Now
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                                                                                  9.5/10 our score
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                                                                                  \n Buy Now\n
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                                                                                  • High brightness led chip: high quality flexible smd 5050 led strip lights, waterproof, anti-collision, anti-corrosion, suitable for outdoor and indoor use. (remote works within 32ft/10m).
                                                                                  • Easy to install: with 3m sticker on the back of the light, soft and flexible, easy to install, no need to change the wire circuit in your car, can be used in a variety of interior and exterior applications including all cars, truck, jeep, suv, vans, motorcycle, boats and much more, installation area:car bumper, inlet grid, cainterior, car rear, car bottom, etc..
                                                                                  • Package included: 2 x 3ft (36 inch) & 2 x 4ft(48 inch) music control led strip lights, remote, ir/voice control box
                                                                                  • Sound functions and multi-color: contains 8 preset static colors. 4 lighting effects: 2 preset jump and 2 preset fading patterns. 4 sound active modes, the led under car glow system produces multi colors (blue, green, purple, red, teal, white, and yellow)
                                                                                  • Waterproof and protection: our underglow lights are entirely sealed by waterproof(ip68) pvc. made of flexible rubber tubing protecting from dust and high/low temperature. can be bent freely and used for more than 50000 hrs lifetime.
                                                                                  \n

                                                                                  3. Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function

                                                                                  \n
                                                                                  \n
                                                                                  \n 3\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"Xprite\n
                                                                                  Buy Now
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                                                                                  \n
                                                                                  \n
                                                                                  9.1/10 our score
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                                                                                  \n Buy Now\n
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                                                                                  • Hight quality - made of flexible rubber tubing, protected from road conditions and debris. it can be bent freely and has the characteristic of high and low temperature resistance.waterproof rate: ip67 note: it is very necessary to use the included fuse holder to install in order to proper protect your vehicle and equipment.
                                                                                  • New remote control design - faster on/off response and more easy adjustable brightness/function button,easier to operate.lightweight and easy to carry.the remote range over 100ft.
                                                                                  • High brightness led chip - high intensity smd 5050 led chips. 3 in 1 rgb chips, brighter output ,uniform lighting. enhances the look and feel of your vehicle with brilliant lighting effects.
                                                                                  • Multi-color function - this undercar strips lighting kits have 8 preset static colors .4 lighting effects: 2 preset jump and 2 preset fading patterns. 4 sound active modes, leds change according to sound. pre-applied adhesive tape.
                                                                                  • Easy installation - with quick connect plugs and pre-applied adhesive tape to apply the led strips under the car. can be used in a variety of interior and exterior applications including all cars, suv,jeep,vans ,truck, boats, motorcycle decoration, etc,.
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                                                                                  4. Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof

                                                                                  \n
                                                                                  \n
                                                                                  \n 4\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"Car\n
                                                                                  Buy Now
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                                                                                  \n
                                                                                  \n
                                                                                  8.8/10 our score
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                                                                                  \n
                                                                                  \n Buy Now\n
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                                                                                  • Addressable led strip & chasing effect ----- special design by adding ic in strip lights, you can control a variety of colors, brightness and chasing speed as you want. built-in 120 kinds of color chasing modes looks like recurrently chasing, skipping, dancing, twinkling on the strip, give you an amazing visual experience. you also can mark your favorite chasing ...
                                                                                  • Easy installationlight strips with strong sturdy self-adhesive tape can be simply peeled off and pasted on any smooth surface (please make sure the surface you intend to stick the light is clean,flat and dry). waterproof rate: ip68
                                                                                  • Car underglow strip lights perfect application for all cars, suv, jeep, truck decoration, etc. low-power design neon lights strip : this car atmosphere lights kit designed to be simple, also provides elegance outside the car when driving at night or enjoying music.using for lighting up the entire under car, enhances the look and feel of your vehicle with brilliant eye-c...
                                                                                  • App controlldownload app in google play or app store, connect bluetooth, no setting or network management required. easily turn on/off, adjust color and brightness. everything controlled by your smartphone, very convenient and easy operation. note: app named: led hue. supports ios 10.0 or above / android 4.4 or above.
                                                                                  • Howtosetuptheappofthecarunderglowlights? https://youtu.be/ehqjfjdh8gq
                                                                                  \n

                                                                                  5. GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent

                                                                                  \n
                                                                                  \n
                                                                                  \n 5\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"GOODRUN\n
                                                                                  Buy Now
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                                                                                  \n
                                                                                  \n
                                                                                  8.6/10 our score
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                                                                                  \n Buy Now\n
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                                                                                  • No drilling easy installation: strong double sided adhesive tape installation, comes with quick connect plugs and pre-applied adhesive tape for an easy installation. can be used in a variety of interior and exterior applications including all under car, truck beds, foot wells, rock lights,cars, suv,jeep boats, motorcycle, and many other decorative purposes.
                                                                                  • Multicolor & music sensor - the led strobe light kit can be changed into 8 colors by adjusting the light color and the lighting effect as you wish with the wireless music remote control, also sound-activated function allows you to enjoy the fun with the color changing following your car stereo music rhythm as well as your voice. 20key remote-8 color gradually varied 4 ...
                                                                                  • Higher quality materials: flexible rubber tubing and upgraded adhesive 3m backing used, super viscosity so that make installation easier. double layer copper, better quality than other single layer copper led strips to the same type. protected from road conditions and debris. high and low temperature resistant with a waterproof rate: ip67.
                                                                                  • Package include: 4 x smd5050 car led strip lights, 1 x music remote control, 1x music ir control box, 30 days money back guarantee and 90 days warranty, 100% high-quality & friendly service and 24-hours email response.
                                                                                  • High brightness:made of cree smd 5050 rgb light bulb, it can light up every side of your car underbody as decorative lighting. with the outstandingmerits: long service life, running at low temperature, low energy consumption, less heat radiation, it will be your first choice of wireless led light strip.
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                                                                                  6. OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync

                                                                                  \n
                                                                                  \n
                                                                                  \n 6\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"OPT7\n
                                                                                  Buy Now
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                                                                                  \n
                                                                                  \n
                                                                                  8.3/10 our score
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                                                                                  \n Buy Now\n
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                                                                                  • Access to a full spectrum of colors with advanced dimming controls.
                                                                                  • Exclusive soundsync bass-activated technology lets you visualize the beat.
                                                                                  • 4pc aluminum kit: (2) 48\" light bars with 72 leds each and (2) 36\" light bars with 54 leds each.
                                                                                  • Two ways to power: car charger adapter or plug-n-go fuse tap connectors (all included).
                                                                                  • Our smart-color leds mixes colors at the micro level that results in a brilliant colors true to the eye.
                                                                                  \n

                                                                                  7. Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody

                                                                                  \n
                                                                                  \n
                                                                                  \n 7\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"Xprite\n
                                                                                  Buy Now
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                                                                                  \n
                                                                                  \n
                                                                                  8.1/10 our score
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                                                                                  \n
                                                                                  \n
                                                                                  \n Buy Now\n
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                                                                                  • Multi-color function this undercar lighting kits has 4 preset static colors and over 358 lighting effects. with programmable diy mode and auto demo mode.
                                                                                  • High brightness led chiphigh intensity smd 5050 led chips. 3 in 1 rgb chips, brighter output ,uniform lighting. enhances the look and feel of your vehicle with brilliant eye-catching lighting effects.
                                                                                  • Easy installationquick connect plugs into any 12v cigarette adapter or positive/negative wiring. and pre-applied adhesive tape to apply the led strips under the car. can be used in a variety of interior and exterior applications including all cars, suv,jeep,vans,truck,boats,motorcycle decoration, etc,.(note: extension cable wire sold separately asin:b07r1x1mf1)
                                                                                  • High quality5050 led strips encased in flexible rubber tubing with quick-connectors and pre-applied adhesive tape. protected from road conditions and debris. waterproof rate: ip67note: it is very necessary to use the included fuse holder to install in order to proper protect your vehicle and equipment.
                                                                                  • New remote control design faster on/off response and more easy adjustable speed and brightness button, easy to operate. lightweight and easy to carry. remote range over 100ft.
                                                                                  \n

                                                                                  8. LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars

                                                                                  \n
                                                                                  \n
                                                                                  \n 8\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"LEDGlow\n
                                                                                  Buy Now
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                                                                                  \n
                                                                                  \n
                                                                                  7.8/10 our score
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                                                                                  \n
                                                                                  \n Buy Now\n
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                                                                                  • Add ultra-bright multi-color underglow accent lighting to your car with (2) 46\" & (2) 36\" water-resistant tubes that house 270 smd leds
                                                                                  • Free lifetime technical support & one year limited warranty included with each ledglow purchase
                                                                                  • Choose from 7 different solid color modes that include: blue, red, green, yellow, purple, teal & white
                                                                                  • Underbody tubes feature 6' of wire on each end to daisy chain each tube to the next
                                                                                  • Control box & wireless remote allows you to select from 7 solid colors, 6 flashing modes, 6 scanning modes, 2 chase modes, 2 color cycle modes & 2 sound activated modes that react to the music playing inside of your car!
                                                                                  \n

                                                                                  9. KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color

                                                                                  \n
                                                                                  \n
                                                                                  \n 9\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"KORJO\n
                                                                                  Buy Now
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                                                                                  \n
                                                                                  \n
                                                                                  7.5/10 our score
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                                                                                  \n
                                                                                  \n Buy Now\n
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                                                                                  • easy installation & universal compatibility you can power the strip lights by the 12v cigarette adapter or by positive/negative wiring. the features of flexible, strong self-adhesive make it easy to applied under the car. the 12v voltage means it can be used by almost all of the car. perfect for cars, suv, jeep, vans, trucks, boats decoration. and as a decorative ...
                                                                                  • app control car under lights download app, turn on bluetooth, then start the app to control the led lights, turn on/off, set brightness and chasing speed, everything controlled by your phone, very convenient and easy operation. https://youtu.be/ehqjfjdh8gq
                                                                                  • ip 68 waterproof & strong adhesive whole light kit waterproof. the 5050 led strips are protected by the flexible rubber tubing, the controller and all connectors are also made of waterproof material, well protected from road conditions and debris and rain day. strong pre-applied adhesive tape with spare mounting clips and cable tie.
                                                                                  • dream color led strips built-in 120kinds ofcolorchasingmodes looks like recurrently chasing, skipping, dancing, twinkling on the strip. you also can mark your favorite chasing effect. 16 million colors for you to diy a static color for lighting.
                                                                                  • all in 1 kit & 1 year warranty package includes: 6 pcs car led lights (2 * 61in+ 4 * 23.6in ) + 1 bluetooth controller + 1 cigarette power adapter + 1 bag of mounting clips + 1x user manual. our korjo car underglow kit comes with 1 year warranty & 90 days money backand life long technical support. if you have any questions, please don't hesitate to contact us....
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                                                                                  10. XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light

                                                                                  \n
                                                                                  \n
                                                                                  \n 10\n
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                                                                                  \n
                                                                                  \n \"XTAUTO\n
                                                                                  Buy Now
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                                                                                  \n
                                                                                  \n
                                                                                  7.2/10 our score
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                                                                                  \n
                                                                                  \n Buy Now\n
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                                                                                  • High brightness led chip -- with the wireless remote to control power switch,the led music light strip follows the beat and adjusts flashing to fit the strength of the sound, makes your car more attractive.
                                                                                  • Control mode--4 music control mode, 8 signal color control, 2 preset jump and 2 preset fading patterns, on/off.you can control it anywhere in the car and feel free to control the color change and brightness.
                                                                                  • Can be used in a variety of interior and exterior applications including all cars, suv,jeep,vans ,truck, boats, motorcycle, and many other decorative purposes.
                                                                                  • Easy to install - you don't even need the wiring. first plugged into car cigarette lighter, then peel the back 3m double-sided tape, paste the light strips on the car floor.
                                                                                  • Safe & energy saving--car led light strip has a working voltage of 12v and powered by car charger are equipped with short circuit protection and memory function.
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                                                                                  \n\n\n\n"} +{"content_list": [[{"type": "paragraph", "raw_content": "

                                                                                  Looking for the Best Underbody Underglow Kits that can last long and perform well? Read our review and buying guide on Best Underbody Underglow Kits below.

                                                                                  ", "content": [{"c": "Looking for the Best Underbody Underglow Kits that can last long and perform well? Read our review and buying guide on Best Underbody Underglow Kits below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

                                                                                  To help you in your search, we have completed this list of the Best Underbody Underglow Kits.

                                                                                  ", "content": [{"c": "To help you in your search, we have completed this list of the Best Underbody Underglow Kits.", "t": "text"}]}, {"type": "table", "raw_content": "
                                                                                  RankProduct NameScore
                                                                                  1
                                                                                  \"Car
                                                                                  Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color
                                                                                  9.7
                                                                                  2
                                                                                  \"4pcs
                                                                                  4pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent
                                                                                  9.5
                                                                                  3
                                                                                  \"Xprite
                                                                                  Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function
                                                                                  9.1
                                                                                  4
                                                                                  \"Car
                                                                                  Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof
                                                                                  8.8
                                                                                  5
                                                                                  \"GOODRUN
                                                                                  GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent
                                                                                  8.6
                                                                                  6
                                                                                  \"OPT7
                                                                                  OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync
                                                                                  8.3
                                                                                  7
                                                                                  \"Xprite
                                                                                  Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody
                                                                                  8.1
                                                                                  8
                                                                                  \"LEDGlow
                                                                                  LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars
                                                                                  7.8
                                                                                  9
                                                                                  \"KORJO
                                                                                  KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color
                                                                                  7.5
                                                                                  10
                                                                                  \"XTAUTO
                                                                                  XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light
                                                                                  7.2
                                                                                  ", "content": {"html": "
                                                                                  RankProduct NameScore
                                                                                  1Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color9.7
                                                                                  24pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent9.5
                                                                                  3Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function9.1
                                                                                  4Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof8.8
                                                                                  5GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent8.6
                                                                                  6OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync8.3
                                                                                  7Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody8.1
                                                                                  8LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars7.8
                                                                                  9KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color7.5
                                                                                  10XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light7.2
                                                                                  ", "is_complex": false, "table_nest_level": "1"}}]], "main_html": "

                                                                                  Looking for the Best Underbody Underglow Kits that can last long and perform well? Read our review and buying guide on Best Underbody Underglow Kits below.

                                                                                  To help you in your search, we have completed this list of the Best Underbody Underglow Kits.

                                                                                  RankProduct NameScore
                                                                                  1
                                                                                  \"Car
                                                                                  Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color
                                                                                  9.7
                                                                                  2
                                                                                  \"4pcs
                                                                                  4pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent
                                                                                  9.5
                                                                                  3
                                                                                  \"Xprite
                                                                                  Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function
                                                                                  9.1
                                                                                  4
                                                                                  \"Car
                                                                                  Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof
                                                                                  8.8
                                                                                  5
                                                                                  \"GOODRUN
                                                                                  GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent
                                                                                  8.6
                                                                                  6
                                                                                  \"OPT7
                                                                                  OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync
                                                                                  8.3
                                                                                  7
                                                                                  \"Xprite
                                                                                  Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody
                                                                                  8.1
                                                                                  8
                                                                                  \"LEDGlow
                                                                                  LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars
                                                                                  7.8
                                                                                  9
                                                                                  \"KORJO
                                                                                  KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color
                                                                                  7.5
                                                                                  10
                                                                                  \"XTAUTO
                                                                                  XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light
                                                                                  7.2
                                                                                  ", "statics": {"paragraph": 2, "paragraph.text": 2, "table": 1}, "url": "https://10bestreviewz.com/automotive/best-underbody-underglow-kits", "content": "Looking for the Best Underbody Underglow Kits that can last long and perform well? Read our review and buying guide on Best Underbody Underglow Kits below.\n\nTo help you in your search, we have completed this list of the Best Underbody Underglow Kits.\n\n| Rank | Product Name | Score |\n|---|---|---|\n| 1 | Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color | 9.7 |\n| 2 | 4pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent | 9.5 |\n| 3 | Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function | 9.1 |\n| 4 | Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof | 8.8 |\n| 5 | GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent | 8.6 |\n| 6 | OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync | 8.3 |\n| 7 | Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody | 8.1 |\n| 8 | LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars | 7.8 |\n| 9 | KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color | 7.5 |\n| 10 | XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light | 7.2 |\n", "html": "\n\n\n\n \n \n \n \n \n \n \n \n \n \n Top #10 Best Underbody Underglow Kits in 2025 | Reviews by Experts\n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n \n \n\n \n \n \n \n \n\n\n\n
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                                                                                  \n \n \"10BestReviewz\"\n \n
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                                                                                  \n

                                                                                  Best Underbody Underglow Kits In January 2025

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                                                                                  Looking for the Best Underbody Underglow Kits that can last long and perform well? Read our review and buying guide on Best Underbody Underglow Kits below.

                                                                                  \n

                                                                                  To help you in your search, we have completed this list of the Best Underbody Underglow Kits.

                                                                                  \n

                                                                                  Top Rated Best Underbody Underglow Kits Of 2025

                                                                                  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                                                  RankProduct NameScore
                                                                                  \n
                                                                                  \n
                                                                                  1
                                                                                  \n
                                                                                  \n \"Car\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  9.7
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  2
                                                                                  \n
                                                                                  \n \"4pcs\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  4pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  9.5
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  3
                                                                                  \n
                                                                                  \n \"Xprite\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  9.1
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  4
                                                                                  \n
                                                                                  \n \"Car\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  8.8
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
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                                                                                  \n
                                                                                  5
                                                                                  \n
                                                                                  \n \"GOODRUN\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  8.6
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
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                                                                                  \n
                                                                                  6
                                                                                  \n
                                                                                  \n \"OPT7\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  8.3
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  7
                                                                                  \n
                                                                                  \n \"Xprite\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  8.1
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  8
                                                                                  \n
                                                                                  \n \"LEDGlow\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  7.8
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
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                                                                                  \n
                                                                                  9
                                                                                  \n
                                                                                  \n \"KORJO\n
                                                                                  \n
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                                                                                  KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  7.5
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  10
                                                                                  \n
                                                                                  \n \"XTAUTO\n
                                                                                  \n
                                                                                  \n                                                                                   		\n
                                                                                  XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n                                                                                   		\n
                                                                                  \n
                                                                                  7.2
                                                                                  \n \n \n \n
                                                                                  \n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n

                                                                                  1. Car Underglow Lights,EJ's SUPER CAR Underglow Underbody System Neon Strip Lights Kit,8 Color

                                                                                  \n
                                                                                  \n
                                                                                  \n 1\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"Car\n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  9.7/10 our score
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n Buy Now\n
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                                                                                  • Sound active & music decorationthe car led neon underglow lights has also 8 color flashing modes with sound sensors that lets under car glow system synchronize with music beat.the led music light strip follows the beat and adjusts flashing to fit the strength of the sound,adjustable brightness.
                                                                                  • High quality led chipwireless remote control color changing, flashing mode and music mode, create colorful gradient mode. upgraded high intensity smd 5050 led chip, provide brighter light, add the look and feel of the vehicle with bright lighting effects.
                                                                                  • More color & energy savingthe led under car glow system lights up colorful colors,it can be bent, fold, with your imagination to create your ambience,give a stylishly stunning look to the areas you may want to put them on.even though they're really bright, they're great power savers so you don't have to worry energy consumption.
                                                                                  • Universal compatibilitycar underglow lights its voltage is compatible with 12v. led underbody lights is waterproof and comes with built-in overload protection, so it can be used safely.it can be used by almost all of the car. perfect for under wheel arches, around the front grille, under foot-wells, in the trunk, along the dash.
                                                                                  • Easy installationstrong adhesive led strip lights kit(with black fixed tie), flexible and easy to install.designed with flexible tubing, protected from terrible road conditions and debris. high quality material and low temperature resistance.
                                                                                  \n

                                                                                  2. 4pcs 8 Colors LED Strip Under Car Tube Underglow Underbody System Neon Accent

                                                                                  \n
                                                                                  \n
                                                                                  \n 2\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"4pcs\n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  9.5/10 our score
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n Buy Now\n
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                                                                                  \n

                                                                                  \n
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                                                                                  • High brightness led chip: high quality flexible smd 5050 led strip lights, waterproof, anti-collision, anti-corrosion, suitable for outdoor and indoor use. (remote works within 32ft/10m).
                                                                                  • Easy to install: with 3m sticker on the back of the light, soft and flexible, easy to install, no need to change the wire circuit in your car, can be used in a variety of interior and exterior applications including all cars, truck, jeep, suv, vans, motorcycle, boats and much more, installation area:car bumper, inlet grid, cainterior, car rear, car bottom, etc..
                                                                                  • Package included: 2 x 3ft (36 inch) & 2 x 4ft(48 inch) music control led strip lights, remote, ir/voice control box
                                                                                  • Sound functions and multi-color: contains 8 preset static colors. 4 lighting effects: 2 preset jump and 2 preset fading patterns. 4 sound active modes, the led under car glow system produces multi colors (blue, green, purple, red, teal, white, and yellow)
                                                                                  • Waterproof and protection: our underglow lights are entirely sealed by waterproof(ip68) pvc. made of flexible rubber tubing protecting from dust and high/low temperature. can be bent freely and used for more than 50000 hrs lifetime.
                                                                                  \n

                                                                                  3. Xprite Car Underglow Neon Accent Strip Lights Kit 8 Color Sound Active Function

                                                                                  \n
                                                                                  \n
                                                                                  \n 3\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"Xprite\n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  9.1/10 our score
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n Buy Now\n
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                                                                                  \n
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                                                                                  • Hight quality - made of flexible rubber tubing, protected from road conditions and debris. it can be bent freely and has the characteristic of high and low temperature resistance.waterproof rate: ip67 note: it is very necessary to use the included fuse holder to install in order to proper protect your vehicle and equipment.
                                                                                  • New remote control design - faster on/off response and more easy adjustable brightness/function button,easier to operate.lightweight and easy to carry.the remote range over 100ft.
                                                                                  • High brightness led chip - high intensity smd 5050 led chips. 3 in 1 rgb chips, brighter output ,uniform lighting. enhances the look and feel of your vehicle with brilliant lighting effects.
                                                                                  • Multi-color function - this undercar strips lighting kits have 8 preset static colors .4 lighting effects: 2 preset jump and 2 preset fading patterns. 4 sound active modes, leds change according to sound. pre-applied adhesive tape.
                                                                                  • Easy installation - with quick connect plugs and pre-applied adhesive tape to apply the led strips under the car. can be used in a variety of interior and exterior applications including all cars, suv,jeep,vans ,truck, boats, motorcycle decoration, etc,.
                                                                                  \n

                                                                                  4. Car Underglow Lights, Bluetooth Dream Color Chasing Strip Lights Kit, 6 PCS Waterproof

                                                                                  \n
                                                                                  \n
                                                                                  \n 4\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"Car\n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  8.8/10 our score
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n Buy Now\n
                                                                                  \n
                                                                                  \n

                                                                                  \n
                                                                                    \n
                                                                                  • Addressable led strip & chasing effect ----- special design by adding ic in strip lights, you can control a variety of colors, brightness and chasing speed as you want. built-in 120 kinds of color chasing modes looks like recurrently chasing, skipping, dancing, twinkling on the strip, give you an amazing visual experience. you also can mark your favorite chasing ...
                                                                                  • Easy installationlight strips with strong sturdy self-adhesive tape can be simply peeled off and pasted on any smooth surface (please make sure the surface you intend to stick the light is clean,flat and dry). waterproof rate: ip68
                                                                                  • Car underglow strip lights perfect application for all cars, suv, jeep, truck decoration, etc. low-power design neon lights strip : this car atmosphere lights kit designed to be simple, also provides elegance outside the car when driving at night or enjoying music.using for lighting up the entire under car, enhances the look and feel of your vehicle with brilliant eye-c...
                                                                                  • App controlldownload app in google play or app store, connect bluetooth, no setting or network management required. easily turn on/off, adjust color and brightness. everything controlled by your smartphone, very convenient and easy operation. note: app named: led hue. supports ios 10.0 or above / android 4.4 or above.
                                                                                  • Howtosetuptheappofthecarunderglowlights? https://youtu.be/ehqjfjdh8gq
                                                                                  \n

                                                                                  5. GOODRUN Underglow Underbody Lighting Kit, Multicolored LED Strip Light with Light Bulbs,Neon Accent

                                                                                  \n
                                                                                  \n
                                                                                  \n 5\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"GOODRUN\n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  8.6/10 our score
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n Buy Now\n
                                                                                  \n
                                                                                  \n

                                                                                  \n
                                                                                    \n
                                                                                  • No drilling easy installation: strong double sided adhesive tape installation, comes with quick connect plugs and pre-applied adhesive tape for an easy installation. can be used in a variety of interior and exterior applications including all under car, truck beds, foot wells, rock lights,cars, suv,jeep boats, motorcycle, and many other decorative purposes.
                                                                                  • Multicolor & music sensor - the led strobe light kit can be changed into 8 colors by adjusting the light color and the lighting effect as you wish with the wireless music remote control, also sound-activated function allows you to enjoy the fun with the color changing following your car stereo music rhythm as well as your voice. 20key remote-8 color gradually varied 4 ...
                                                                                  • Higher quality materials: flexible rubber tubing and upgraded adhesive 3m backing used, super viscosity so that make installation easier. double layer copper, better quality than other single layer copper led strips to the same type. protected from road conditions and debris. high and low temperature resistant with a waterproof rate: ip67.
                                                                                  • Package include: 4 x smd5050 car led strip lights, 1 x music remote control, 1x music ir control box, 30 days money back guarantee and 90 days warranty, 100% high-quality & friendly service and 24-hours email response.
                                                                                  • High brightness:made of cree smd 5050 rgb light bulb, it can light up every side of your car underbody as decorative lighting. with the outstandingmerits: long service life, running at low temperature, low energy consumption, less heat radiation, it will be your first choice of wireless led light strip.
                                                                                  \n
                                                                                  \n

                                                                                  \n\n

                                                                                  \n

                                                                                  6. OPT7 Aura 4pc Pickup Truck Underglow LED Lighting Kit w/remote - Soundsync

                                                                                  \n
                                                                                  \n
                                                                                  \n 6\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"OPT7\n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  8.3/10 our score
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n Buy Now\n
                                                                                  \n
                                                                                  \n

                                                                                  \n
                                                                                    \n
                                                                                  • Access to a full spectrum of colors with advanced dimming controls.
                                                                                  • Exclusive soundsync bass-activated technology lets you visualize the beat.
                                                                                  • 4pc aluminum kit: (2) 48\" light bars with 72 leds each and (2) 36\" light bars with 54 leds each.
                                                                                  • Two ways to power: car charger adapter or plug-n-go fuse tap connectors (all included).
                                                                                  • Our smart-color leds mixes colors at the micro level that results in a brilliant colors true to the eye.
                                                                                  \n

                                                                                  7. Xprite Car Underglow RGB Dancing Light Kit with Wireless Remote Control 6PC Underbody

                                                                                  \n
                                                                                  \n
                                                                                  \n 7\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"Xprite\n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  8.1/10 our score
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n Buy Now\n
                                                                                  \n
                                                                                  \n

                                                                                  \n
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                                                                                  • Multi-color function this undercar lighting kits has 4 preset static colors and over 358 lighting effects. with programmable diy mode and auto demo mode.
                                                                                  • High brightness led chiphigh intensity smd 5050 led chips. 3 in 1 rgb chips, brighter output ,uniform lighting. enhances the look and feel of your vehicle with brilliant eye-catching lighting effects.
                                                                                  • Easy installationquick connect plugs into any 12v cigarette adapter or positive/negative wiring. and pre-applied adhesive tape to apply the led strips under the car. can be used in a variety of interior and exterior applications including all cars, suv,jeep,vans,truck,boats,motorcycle decoration, etc,.(note: extension cable wire sold separately asin:b07r1x1mf1)
                                                                                  • High quality5050 led strips encased in flexible rubber tubing with quick-connectors and pre-applied adhesive tape. protected from road conditions and debris. waterproof rate: ip67note: it is very necessary to use the included fuse holder to install in order to proper protect your vehicle and equipment.
                                                                                  • New remote control design faster on/off response and more easy adjustable speed and brightness button, easy to operate. lightweight and easy to carry. remote range over 100ft.
                                                                                  \n

                                                                                  8. LEDGlow 4pc Multi-Color Slimline LED Underbody Underglow Accent Neon Lighting Kit for Cars

                                                                                  \n
                                                                                  \n
                                                                                  \n 8\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"LEDGlow\n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  7.8/10 our score
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n Buy Now\n
                                                                                  \n
                                                                                  \n

                                                                                  \n
                                                                                    \n
                                                                                  • Add ultra-bright multi-color underglow accent lighting to your car with (2) 46\" & (2) 36\" water-resistant tubes that house 270 smd leds
                                                                                  • Free lifetime technical support & one year limited warranty included with each ledglow purchase
                                                                                  • Choose from 7 different solid color modes that include: blue, red, green, yellow, purple, teal & white
                                                                                  • Underbody tubes feature 6' of wire on each end to daisy chain each tube to the next
                                                                                  • Control box & wireless remote allows you to select from 7 solid colors, 6 flashing modes, 6 scanning modes, 2 chase modes, 2 color cycle modes & 2 sound activated modes that react to the music playing inside of your car!
                                                                                  \n

                                                                                  9. KORJO Car Underglow Lights, 6 Pcs Bluetooth Led Strip Lights with Dream Color

                                                                                  \n
                                                                                  \n
                                                                                  \n 9\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"KORJO\n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  7.5/10 our score
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n Buy Now\n
                                                                                  \n
                                                                                  \n

                                                                                  \n
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                                                                                  • easy installation & universal compatibility you can power the strip lights by the 12v cigarette adapter or by positive/negative wiring. the features of flexible, strong self-adhesive make it easy to applied under the car. the 12v voltage means it can be used by almost all of the car. perfect for cars, suv, jeep, vans, trucks, boats decoration. and as a decorative ...
                                                                                  • app control car under lights download app, turn on bluetooth, then start the app to control the led lights, turn on/off, set brightness and chasing speed, everything controlled by your phone, very convenient and easy operation. https://youtu.be/ehqjfjdh8gq
                                                                                  • ip 68 waterproof & strong adhesive whole light kit waterproof. the 5050 led strips are protected by the flexible rubber tubing, the controller and all connectors are also made of waterproof material, well protected from road conditions and debris and rain day. strong pre-applied adhesive tape with spare mounting clips and cable tie.
                                                                                  • dream color led strips built-in 120kinds ofcolorchasingmodes looks like recurrently chasing, skipping, dancing, twinkling on the strip. you also can mark your favorite chasing effect. 16 million colors for you to diy a static color for lighting.
                                                                                  • all in 1 kit & 1 year warranty package includes: 6 pcs car led lights (2 * 61in+ 4 * 23.6in ) + 1 bluetooth controller + 1 cigarette power adapter + 1 bag of mounting clips + 1x user manual. our korjo car underglow kit comes with 1 year warranty & 90 days money backand life long technical support. if you have any questions, please don't hesitate to contact us....
                                                                                  \n

                                                                                  10. XTAUTO 4Pcs Car 72 LED Neon Undercar Underglow Glow Atmosphere Decorative Bar Light

                                                                                  \n
                                                                                  \n
                                                                                  \n 10\n
                                                                                  \n
                                                                                  \n
                                                                                  \n \"XTAUTO\n
                                                                                  Buy Now
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  7.2/10 our score
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n Buy Now\n
                                                                                  \n
                                                                                  \n

                                                                                  \n
                                                                                    \n
                                                                                  • High brightness led chip -- with the wireless remote to control power switch,the led music light strip follows the beat and adjusts flashing to fit the strength of the sound, makes your car more attractive.
                                                                                  • Control mode--4 music control mode, 8 signal color control, 2 preset jump and 2 preset fading patterns, on/off.you can control it anywhere in the car and feel free to control the color change and brightness.
                                                                                  • Can be used in a variety of interior and exterior applications including all cars, suv,jeep,vans ,truck, boats, motorcycle, and many other decorative purposes.
                                                                                  • Easy to install - you don't even need the wiring. first plugged into car cigarette lighter, then peel the back 3m double-sided tape, paste the light strips on the car floor.
                                                                                  • Safe & energy saving--car led light strip has a working voltage of 12v and powered by car charger are equipped with short circuit protection and memory function.
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                                                                                  \n\n\n\n"} diff --git a/bench/data/groundtruth/table_3.jsonl b/bench/data/groundtruth/table_3.jsonl index ca498ba3..129cac09 100644 --- a/bench/data/groundtruth/table_3.jsonl +++ b/bench/data/groundtruth/table_3.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "table", "raw_content": "
                                                                                  Mrs S Hindle
                                                                                  Show CC RCC
                                                                                  Driffield 5th October 2006 CH. Ricksbury Royal Hero CH. Keyingham Branwell
                                                                                  Manchester 16th January 2008 CH. Lochbuie Geordie Merryoth Maeve
                                                                                  Darlington 20th September 2009 CH. Maibee Make Believe CH. Loranka Just Like Heaven JW
                                                                                  Blackpool 22nd June 2012 CH. Loranka Sherrie Baby Dear Magic Touch De La Fi Au Songeur
                                                                                  Welsh Kennel Club 2014 Brymarden Carolina Sunrise Ch. Wandris Evan Elp Us
                                                                                  Welsh Kennel Club 2014 Ch. Charnell Clematis of Salegreen CH. Byermoor Queens Maid
                                                                                  ", "content": {"html": "
                                                                                  Mrs S Hindle
                                                                                  ShowCCRCC
                                                                                  Driffield 5th October 2006CH. Ricksbury Royal HeroCH. Keyingham Branwell
                                                                                  Manchester 16th January 2008CH. Lochbuie GeordieMerryoth Maeve
                                                                                  Darlington 20th September 2009CH. Maibee Make BelieveCH. Loranka Just Like Heaven JW
                                                                                  Blackpool 22nd June 2012CH. Loranka Sherrie BabyDear Magic Touch De La Fi Au Songeur
                                                                                  Welsh Kennel Club 2014Brymarden Carolina SunriseCh. Wandris Evan Elp Us
                                                                                  Welsh Kennel Club 2014Ch. Charnell Clematis of SalegreenCH. Byermoor Queens Maid
                                                                                  ", "is_complex": false}}]], "main_html": "
                                                                                  Mrs S Hindle
                                                                                  Show CC RCC
                                                                                  Driffield 5th October 2006 CH. Ricksbury Royal Hero CH. Keyingham Branwell
                                                                                  Manchester 16th January 2008 CH. Lochbuie Geordie Merryoth Maeve
                                                                                  Darlington 20th September 2009 CH. Maibee Make Believe CH. Loranka Just Like Heaven JW
                                                                                  Blackpool 22nd June 2012 CH. Loranka Sherrie Baby Dear Magic Touch De La Fi Au Songeur
                                                                                  Welsh Kennel Club 2014 Brymarden Carolina Sunrise Ch. Wandris Evan Elp Us
                                                                                  Welsh Kennel Club 2014 Ch. Charnell Clematis of Salegreen CH. Byermoor Queens Maid
                                                                                  ", "statics": {"table": 1}, "url": "https://cavaliers.co.uk/results/judges/Mrs%20S%20Hindle.htm", "content": "| Mrs S Hindle | | |\n|---|---|---|\n| Show | CC | RCC |\n| Driffield 5th October 2006 | CH. Ricksbury Royal Hero | CH. Keyingham Branwell |\n| Manchester 16th January 2008 | CH. Lochbuie Geordie | Merryoth Maeve |\n| Darlington 20th September 2009 | CH. Maibee Make Believe | CH. Loranka Just Like Heaven JW |\n| Blackpool 22nd June 2012 | CH. Loranka Sherrie Baby | Dear Magic Touch De La Fi Au Songeur |\n| Welsh Kennel Club 2014 | Brymarden Carolina Sunrise | Ch. Wandris Evan Elp Us |\n| Welsh Kennel Club 2014 | Ch. Charnell Clematis of Salegreen | CH. Byermoor Queens Maid |\n", "html": "\n\n Results by GAYHALO results program 21/01/2017\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                                                  Mrs S Hindle
                                                                                  Show CC RCC
                                                                                  Driffield 5th October 2006 CH. Ricksbury Royal Hero CH. Keyingham Branwell
                                                                                  Manchester 16th January 2008 CH. Lochbuie Geordie Merryoth Maeve
                                                                                  Darlington 20th September 2009 CH. Maibee Make Believe CH. Loranka Just Like Heaven JW
                                                                                  Blackpool 22nd June 2012 CH. Loranka Sherrie Baby Dear Magic Touch De La Fi Au Songeur
                                                                                  Welsh Kennel Club 2014 Brymarden Carolina Sunrise Ch. Wandris Evan Elp Us
                                                                                  Welsh Kennel Club 2014 Ch. Charnell Clematis of Salegreen CH. Byermoor Queens Maid
                                                                                  \n \n \n"} +{"content_list": [[{"type": "table", "raw_content": "
                                                                                  Mrs S Hindle
                                                                                  Show CC RCC
                                                                                  Driffield 5th October 2006 CH. Ricksbury Royal Hero CH. Keyingham Branwell
                                                                                  Manchester 16th January 2008 CH. Lochbuie Geordie Merryoth Maeve
                                                                                  Darlington 20th September 2009 CH. Maibee Make Believe CH. Loranka Just Like Heaven JW
                                                                                  Blackpool 22nd June 2012 CH. Loranka Sherrie Baby Dear Magic Touch De La Fi Au Songeur
                                                                                  Welsh Kennel Club 2014 Brymarden Carolina Sunrise Ch. Wandris Evan Elp Us
                                                                                  Welsh Kennel Club 2014 Ch. Charnell Clematis of Salegreen CH. Byermoor Queens Maid
                                                                                  ", "content": {"html": "
                                                                                  Mrs S Hindle
                                                                                  ShowCCRCC
                                                                                  Driffield 5th October 2006CH. Ricksbury Royal HeroCH. Keyingham Branwell
                                                                                  Manchester 16th January 2008CH. Lochbuie GeordieMerryoth Maeve
                                                                                  Darlington 20th September 2009CH. Maibee Make BelieveCH. Loranka Just Like Heaven JW
                                                                                  Blackpool 22nd June 2012CH. Loranka Sherrie BabyDear Magic Touch De La Fi Au Songeur
                                                                                  Welsh Kennel Club 2014Brymarden Carolina SunriseCh. Wandris Evan Elp Us
                                                                                  Welsh Kennel Club 2014Ch. Charnell Clematis of SalegreenCH. Byermoor Queens Maid
                                                                                  ", "is_complex": false, "table_nest_level": "1"}}]], "main_html": "
                                                                                  Mrs S Hindle
                                                                                  Show CC RCC
                                                                                  Driffield 5th October 2006 CH. Ricksbury Royal Hero CH. Keyingham Branwell
                                                                                  Manchester 16th January 2008 CH. Lochbuie Geordie Merryoth Maeve
                                                                                  Darlington 20th September 2009 CH. Maibee Make Believe CH. Loranka Just Like Heaven JW
                                                                                  Blackpool 22nd June 2012 CH. Loranka Sherrie Baby Dear Magic Touch De La Fi Au Songeur
                                                                                  Welsh Kennel Club 2014 Brymarden Carolina Sunrise Ch. Wandris Evan Elp Us
                                                                                  Welsh Kennel Club 2014 Ch. Charnell Clematis of Salegreen CH. Byermoor Queens Maid
                                                                                  ", "statics": {"table": 1}, "url": "https://cavaliers.co.uk/results/judges/Mrs%20S%20Hindle.htm", "content": "| Mrs S Hindle | | |\n|---|---|---|\n| Show | CC | RCC |\n| Driffield 5th October 2006 | CH. Ricksbury Royal Hero | CH. Keyingham Branwell |\n| Manchester 16th January 2008 | CH. Lochbuie Geordie | Merryoth Maeve |\n| Darlington 20th September 2009 | CH. Maibee Make Believe | CH. Loranka Just Like Heaven JW |\n| Blackpool 22nd June 2012 | CH. Loranka Sherrie Baby | Dear Magic Touch De La Fi Au Songeur |\n| Welsh Kennel Club 2014 | Brymarden Carolina Sunrise | Ch. Wandris Evan Elp Us |\n| Welsh Kennel Club 2014 | Ch. Charnell Clematis of Salegreen | CH. Byermoor Queens Maid |\n", "html": "\n\n Results by GAYHALO results program 21/01/2017\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
                                                                                  Mrs S Hindle
                                                                                  Show CC RCC
                                                                                  Driffield 5th October 2006 CH. Ricksbury Royal Hero CH. Keyingham Branwell
                                                                                  Manchester 16th January 2008 CH. Lochbuie Geordie Merryoth Maeve
                                                                                  Darlington 20th September 2009 CH. Maibee Make Believe CH. Loranka Just Like Heaven JW
                                                                                  Blackpool 22nd June 2012 CH. Loranka Sherrie Baby Dear Magic Touch De La Fi Au Songeur
                                                                                  Welsh Kennel Club 2014 Brymarden Carolina Sunrise Ch. Wandris Evan Elp Us
                                                                                  Welsh Kennel Club 2014 Ch. Charnell Clematis of Salegreen CH. Byermoor Queens Maid
                                                                                  \n \n \n"} diff --git a/bench/data/groundtruth/table_4.jsonl b/bench/data/groundtruth/table_4.jsonl index 7e1aba6a..28bd37a9 100644 --- a/bench/data/groundtruth/table_4.jsonl +++ b/bench/data/groundtruth/table_4.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "table", "raw_content": "
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \u9ad8\u7537\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u65b9\u9756\u7def
                                                                                  \u674e\u6631\u8ce2
                                                                                  \u5468\u6587\u6bc5
                                                                                  \u5468\u66c9\u98a8
                                                                                  \u6d2a\u53ef\u6d69
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                                                                                  \u8521\u53f3\u627f
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                                                                                  \u4faf\u5091\u6069
                                                                                  \u7d22\u5357\u55ac\u67cf
                                                                                  \u694a\u51f1\u6069
                                                                                  \u5289\u5b50\u8aa0
                                                                                  \u8cf4\u79b9\u821c
                                                                                  \u8607\u79b9\u4e1e
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u4f59\u51a0\u61b2
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                                                                                  \u5433\u6649\u5cb1
                                                                                  \u9673\u5b5f\u8b19
                                                                                  \u9ec3\u653f\u9594
                                                                                  \u8521\u548c\u8afa
                                                                                  \u8521\u627f\u7950
                                                                                  		\u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u738b\u7167\u9321
                                                                                  \u90b1\u9756\u748b
                                                                                  \u5510\u90e1
                                                                                  \u66f9\u9594\u7fd4
                                                                                  \u9673\u6cd3\u8aed
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                                                                                  \u6e6f\u71ca
                                                                                  		\u82b1\u84ee\u7e23\u82b1\u84ee\u9ad8\u4e2d
                                                                                  \u65b9\u5b8f\u6137
                                                                                  \u7530\u6fec\u744b
                                                                                  \u5442\u674e\u6b3d\u7fd4
                                                                                  \u8449\u4f73\u5049
                                                                                  \u6f58\u7acb\u8ed2
                                                                                  \u5c4f\u6771\u7e23\u5927\u540c\u9ad8\u4e2d
                                                                                  \u67ef\u535a\u9594
                                                                                  \u9ad8\u58eb\u51f1
                                                                                  \u90ed\u7d18\u777f
                                                                                  \u9673\u5764\u4f51
                                                                                  \u5289\u5f37\u68ee
                                                                                  \u912d\u4f0d\u6d0b
                                                                                  \u76e7\u5b87\u6f54
                                                                                  \u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u674e\u6649\u6e90
                                                                                  \u6797\u627f\u9032
                                                                                  \u5f35\u4e2d\u777f
                                                                                  \u9ec3\u97cb\u7db8
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                                                                                  \u5433\u6607\u6a3a
                                                                                  \u675c\u5764\u6649
                                                                                  \u9673\u4ee5\u8ed2
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                                                                                  \u5ed6\u6d69\u5ef7
                                                                                  \u6b50\u5091\u6587
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                                                                                  \u9ad8\u5973\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u674e\u4e88\u85b0
                                                                                  \u674e\u79d1\u7de3
                                                                                  \u6797\u82b3\u5b89
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                                                                                  \u842c\u5955\u5f63
                                                                                  		\u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u6e29\u4fde\u699b
                                                                                  \u5df4\u55ac\u767b
                                                                                  \u5442\u4f73\u7a4e
                                                                                  \u6797\u7acb\u5fc3
                                                                                  \u9ec3\u5955\u6674
                                                                                  \u5289\u5b5d\u6148
                                                                                  \u912d\u6f54\u682d
                                                                                  		\u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u9673\u6631\u5e0c
                                                                                  \u9673\u96c5\u541b
                                                                                  \u66fe\u7766\u6f54
                                                                                  \u9ec3\u6df5\u5100
                                                                                  \u9ec3\u9756\u6df3
                                                                                  \u5289\u598d\u5e0c
                                                                                  \u8b1d\u96e8\u6069
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u53e4\u9038\u5a77
                                                                                  \u674e\u6b23\u5a77
                                                                                  \u6797\u7acb\u7d43
                                                                                  \u9673\u744b\u55ac
                                                                                  \u66fe\u5b50\u7a4e
                                                                                  \u5289\u90c1\u82b3
                                                                                  \u8b1d\u4f73\u6069
                                                                                  		\u81fa\u5317\u5e02\u5927\u76f4\u9ad8\u4e2d
                                                                                  \u5353\u61ff\u7444
                                                                                  \u5f35\u5de7\u6db5
                                                                                  \u9ec3\u5a49\u59ae
                                                                                  \u9ec3\u59f5\u59ae
                                                                                  \u5289\u5955\u8431
                                                                                  \u8b1d\u5143\u85b0
                                                                                  \u8607\u739f\u6615
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u738b\u4f73\u5043
                                                                                  \u4f55\u5029\u4f69
                                                                                  \u675c\u4f9d\u7d14
                                                                                  \u9673\u5bd7\u81fb
                                                                                  \u9ec3\u7230\u84c1
                                                                                  \u8607\u6021\u5be7
                                                                                  \u81fa\u5317\u5e02\u5317\u4e00\u5973\u4e2d
                                                                                  \u738b\u5a55\u82b8
                                                                                  \u4f59\u5a55\u7433
                                                                                  \u674e\u5b9c\u84c1
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                                                                                  \u9673\u4e8e\u6db5
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                                                                                  \u5289\u66c8\u749f
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                                                                                  \u674e\u5b9c\u85b0
                                                                                  \u6797\u51a0\u59a4
                                                                                  \u9673\u601d\u9f4a
                                                                                  \u8521\u6085
                                                                                  \u8521\u6190\u61ab
                                                                                  \u8cf4\u73ee\u84c1
                                                                                  \u570b\u7537\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u6bdb\u51a0\u6dee
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                                                                                  \u6797\u90c1\u654f
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                                                                                  \u4f59\u6893\u52e4
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                                                                                  \u9673\u4ee5\u8ed2
                                                                                  \u9673\u6600\u744b
                                                                                  \u5ed6\u6d69\u5ef7
                                                                                  \u6b50\u5091\u6587
                                                                                  \u8b1d\u5fc3\u8afe
                                                                                  \u9ad8\u5973\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u674e\u4e88\u85b0
                                                                                  \u674e\u79d1\u7de3
                                                                                  \u6797\u82b3\u5b89
                                                                                  \u5f35\u5929\u99a8
                                                                                  \u6881\u5bb8\u5f0b
                                                                                  \u9673\u923a\u8553
                                                                                  \u842c\u5955\u5f63
                                                                                  		\u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u6e29\u4fde\u699b
                                                                                  \u5df4\u55ac\u767b
                                                                                  \u5442\u4f73\u7a4e
                                                                                  \u6797\u7acb\u5fc3
                                                                                  \u9ec3\u5955\u6674
                                                                                  \u5289\u5b5d\u6148
                                                                                  \u912d\u6f54\u682d
                                                                                  		\u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u9673\u6631\u5e0c
                                                                                  \u9673\u96c5\u541b
                                                                                  \u66fe\u7766\u6f54
                                                                                  \u9ec3\u6df5\u5100
                                                                                  \u9ec3\u9756\u6df3
                                                                                  \u5289\u598d\u5e0c
                                                                                  \u8b1d\u96e8\u6069
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u53e4\u9038\u5a77
                                                                                  \u674e\u6b23\u5a77
                                                                                  \u6797\u7acb\u7d43
                                                                                  \u9673\u744b\u55ac
                                                                                  \u66fe\u5b50\u7a4e
                                                                                  \u5289\u90c1\u82b3
                                                                                  \u8b1d\u4f73\u6069
                                                                                  		\u81fa\u5317\u5e02\u5927\u76f4\u9ad8\u4e2d
                                                                                  \u5353\u61ff\u7444
                                                                                  \u5f35\u5de7\u6db5
                                                                                  \u9ec3\u5a49\u59ae
                                                                                  \u9ec3\u59f5\u59ae
                                                                                  \u5289\u5955\u8431
                                                                                  \u8b1d\u5143\u85b0
                                                                                  \u8607\u739f\u6615
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u738b\u4f73\u5043
                                                                                  \u4f55\u5029\u4f69
                                                                                  \u675c\u4f9d\u7d14
                                                                                  \u9673\u5bd7\u81fb
                                                                                  \u9ec3\u7230\u84c1
                                                                                  \u8607\u6021\u5be7
                                                                                  \u81fa\u5317\u5e02\u5317\u4e00\u5973\u4e2d
                                                                                  \u738b\u5a55\u82b8
                                                                                  \u4f59\u5a55\u7433
                                                                                  \u674e\u5b9c\u84c1
                                                                                  \u6797\u80b2\u8431
                                                                                  \u9673\u4e8e\u6db5
                                                                                  \u9ec3\u90c1\u742a
                                                                                  \u5289\u66c8\u749f
                                                                                  \u65b0\u5317\u5e02\u65b0\u5317\u9ad8\u4e2d
                                                                                  \u738b\u7d2b\u82b8
                                                                                  \u674e\u5b9c\u85b0
                                                                                  \u6797\u51a0\u59a4
                                                                                  \u9673\u601d\u9f4a
                                                                                  \u8521\u6085
                                                                                  \u8521\u6190\u61ab
                                                                                  \u8cf4\u73ee\u84c1
                                                                                  \u570b\u7537\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u6bdb\u51a0\u6dee
                                                                                  \u674e\u5cfb\u4ea6
                                                                                  \u6797\u90c1\u654f
                                                                                  \u838a\u5b78\u6fc2
                                                                                  \u838a\u6def\u7fd4
                                                                                  \u6b50\u5b50\u6bc5
                                                                                  \u8b1d\u79c9\u5bb8
                                                                                  		\u65b0\u7af9\u5e02\u4e09\u6c11\u570b\u4e2d
                                                                                  \u674e\u5bb6\u7a4e
                                                                                  \u6797\u8c6a\u923a
                                                                                  \u5f90\u7693\u9706
                                                                                  \u8881\u4fca\u777f
                                                                                  \u694a\u6615\u7fa9
                                                                                  \u912d\u53cb\u6069
                                                                                  \u9f94\u6631\u627f
                                                                                  		\u81fa\u4e2d\u5e02\u9ece\u660e\u570b\u4e2d
                                                                                  \u6797\u6587\u8a70
                                                                                  \u6797\u7fbd\u8ed2
                                                                                  \u694a\u627f\u7ff0
                                                                                  \u8449\u9326\u4e1e
                                                                                  \u5ed6\u5927\u921e
                                                                                  \u8d99\u5149\u5ba5
                                                                                  \u6492\u7490\uff0e\u4f0a\u65af\u5361\u5361\u592b\u7279
                                                                                  		\u5c4f\u6771\u7e23\u81f3\u6b63\u570b\u4e2d
                                                                                  \u738b\u6689\u5ef7
                                                                                  \u4f59\u6893\u52e4
                                                                                  \u5433\u665f\u69d0
                                                                                  \u8a31\u5bb6\u7a0b
                                                                                  \u9673\u5b8f\u8ed2
                                                                                  \u9673\u6975\u8a9e
                                                                                  \u9ec3\u51a0\u744b
                                                                                  		\u81fa\u5317\u5e02\u65b0\u6c11\u570b\u4e2d
                                                                                  \u5433\u6587\u65ed
                                                                                  \u674e\u5149\u555f
                                                                                  \u5353\u6b63\u5091
                                                                                  \u6797\u8b1d\u5b89
                                                                                  \u8a31\u723e\u5a01
                                                                                  \u9ec3\u6587\u5ba5
                                                                                  \u8cf4\u6649\u5bec
                                                                                  \u82b1\u84ee\u7e23\u5b9c\u660c\u570b\u4e2d
                                                                                  \u738b\u67cf\u921e
                                                                                  \u80e1\u4fca\u7965
                                                                                  \u9ad8\u5b87\u8861
                                                                                  \u9ec3\u54c1\u582f
                                                                                  \u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u65b9\u97cb\u55ac
                                                                                  \u738b\u5f65\u73fa
                                                                                  \u738b\u555f\u5b89
                                                                                  \u6885\u5cea\u61f7
                                                                                  \u6b50\u967d\u4f2f\u5f65
                                                                                  \u912d\u4e0a\u99ad
                                                                                  \u912d\u7fd4\u4e91
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u738b\u5ba5\u921e
                                                                                  \u5433\u5cb3\u6fa4
                                                                                  \u9673\u5ba5\u9716
                                                                                  \u66fe\u5b50\u6df5
                                                                                  \u9ec3\u5b50\u5100
                                                                                  \u694a\u5ead\u79be
                                                                                  \u7c21\u5146\u5b8f
                                                                                  \u570b\u5973\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u81fa\u5317\u5e02\u81f3\u5584\u570b\u4e2d
                                                                                  \u82b1\u82e1\u6069
                                                                                  \u9ad8\u82b7\u6db5
                                                                                  \u8a31\u82b8\u8c9e
                                                                                  \u9673\u53c8\u7dad
                                                                                  \u9ec3\u6c38\u5609
                                                                                  \u9127\u774e\u55ac
                                                                                  \u8cf4\u5bb6\u965e
                                                                                  		\u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u91d1\u5de7\u502b
                                                                                  \u7d00\u84d3\u5ba3
                                                                                  \u66f9\u5609\u51cc
                                                                                  \u9ec3\u73c9\u81fb
                                                                                  \u9ec3\u59ff\u5609
                                                                                  \u859b\u5b50\u5100
                                                                                  \u8b1d\u79c9\u6a3a
                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u674e\u6c9b\u7df9
                                                                                  \u90b1\u923a\u55ac
                                                                                  \u67ef\u96e8\u5c91
                                                                                  \u5d14\u6085\u5bb9
                                                                                  \u838a\u598d\u9234
                                                                                  \u96f7\u5982\u5ba3
                                                                                  \u912d\u5b89\u6f54
                                                                                  		\u81fa\u5357\u5e02\u5fe0\u5b5d\u570b\u4e2d
                                                                                  \u6797\u5955\u5f64
                                                                                  \u9673\u5ba5\u9321
                                                                                  \u9ec3\u53ef\u6615
                                                                                  \u9ec3\u54c1\u745c
                                                                                  \u694a\u6e58\u82d3
                                                                                  \u6f58\u5de7\u6df3
                                                                                  \u8521\u8a60\u5e06
                                                                                  		\u65b0\u7af9\u5e02\u4e09\u6c11\u570b\u4e2d
                                                                                  \u6797\u5bb6\u7504
                                                                                  \u5f35\u5c91\u7433
                                                                                  \u90ed\u84c1
                                                                                  \u694a\u6b23\u4ead
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u738b\u54c1\u7440
                                                                                  \u9673\u6b46\u55ac
                                                                                  \u5289\u90c1\u9e97
                                                                                  \u8607\u5ba5\u84c1
                                                                                  \u81fa\u5317\u5e02\u5927\u76f4\u9ad8\u4e2d
                                                                                  \u9673\u76c8\u7a4e
                                                                                  \u9673\u52ad\u5ead
                                                                                  \u9ec3\u5b50\u82b8
                                                                                  \u8521\u5bb6\u745e
                                                                                  \u8521\u6c84\u84d2
                                                                                  \u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u9673\u82b7\u82b8
                                                                                  \u9673\u828a\u58ec
                                                                                  \u66fe\u598d\u84c9
                                                                                  \u8449\u4fd0\u4f36
                                                                                  ", "statics": {"table": 1}, "url": "https://112sport.hcc.edu.tw/Module/Score/FinalsReport_List.php?LID=203", "content": "| \u6bd4\u8cfd\u9805\u76ee | \u7b2c\u4e00\u540d | \u7b2c\u4e8c\u540d | \u7b2c\u4e09\u540d | \u7b2c\u56db\u540d | \u7b2c\u4e94\u540d | \u7b2c\u516d\u540d | \u7b2c\u4e03\u540d | \u7b2c\u516b\u540d |\n|---|---|---|---|---|---|---|---|---|\n| \u9ad8\u7537\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd | \u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u65b9\u9756\u7def
                                                                                  \u674e\u6631\u8ce2
                                                                                  \u5468\u6587\u6bc5
                                                                                  \u5468\u66c9\u98a8
                                                                                  \u6d2a\u53ef\u6d69
                                                                                  \u5f90\u6668\u63da
                                                                                  \u8521\u53f3\u627f | \u81fa\u5317\u5e02\u5efa\u570b\u4e2d\u5b78
                                                                                  \u5468\u9806\u5fb7
                                                                                  \u4faf\u5091\u6069
                                                                                  \u7d22\u5357\u55ac\u67cf
                                                                                  \u694a\u51f1\u6069
                                                                                  \u5289\u5b50\u8aa0
                                                                                  \u8cf4\u79b9\u821c
                                                                                  \u8607\u79b9\u4e1e | \u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u4f59\u51a0\u61b2
                                                                                  \u5433\u4f2f\u5b87
                                                                                  \u5433\u6649\u5cb1
                                                                                  \u9673\u5b5f\u8b19
                                                                                  \u9ec3\u653f\u9594
                                                                                  \u8521\u548c\u8afa
                                                                                  \u8521\u627f\u7950 | \u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u738b\u7167\u9321
                                                                                  \u90b1\u9756\u748b
                                                                                  \u5510\u90e1
                                                                                  \u66f9\u9594\u7fd4
                                                                                  \u9673\u6cd3\u8aed
                                                                                  \u9673\u8a9e\u7db8
                                                                                  \u6e6f\u71ca | \u82b1\u84ee\u7e23\u82b1\u84ee\u9ad8\u4e2d
                                                                                  \u65b9\u5b8f\u6137
                                                                                  \u7530\u6fec\u744b
                                                                                  \u5442\u674e\u6b3d\u7fd4
                                                                                  \u8449\u4f73\u5049
                                                                                  \u6f58\u7acb\u8ed2
                                                                                  \u5c4f\u6771\u7e23\u5927\u540c\u9ad8\u4e2d
                                                                                  \u67ef\u535a\u9594
                                                                                  \u9ad8\u58eb\u51f1
                                                                                  \u90ed\u7d18\u777f
                                                                                  \u9673\u5764\u4f51
                                                                                  \u5289\u5f37\u68ee
                                                                                  \u912d\u4f0d\u6d0b
                                                                                  \u76e7\u5b87\u6f54
                                                                                  \u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u674e\u6649\u6e90
                                                                                  \u6797\u627f\u9032
                                                                                  \u5f35\u4e2d\u777f
                                                                                  \u9ec3\u97cb\u7db8
                                                                                  \u8521\u4f73\u4fee
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u5433\u6607\u6a3a
                                                                                  \u675c\u5764\u6649
                                                                                  \u9673\u4ee5\u8ed2
                                                                                  \u9673\u6600\u744b
                                                                                  \u5ed6\u6d69\u5ef7
                                                                                  \u6b50\u5091\u6587
                                                                                  \u8b1d\u5fc3\u8afe | | | |\n| \u9ad8\u5973\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd | \u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u674e\u4e88\u85b0
                                                                                  \u674e\u79d1\u7de3
                                                                                  \u6797\u82b3\u5b89
                                                                                  \u5f35\u5929\u99a8
                                                                                  \u6881\u5bb8\u5f0b
                                                                                  \u9673\u923a\u8553
                                                                                  \u842c\u5955\u5f63 | \u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u6e29\u4fde\u699b
                                                                                  \u5df4\u55ac\u767b
                                                                                  \u5442\u4f73\u7a4e
                                                                                  \u6797\u7acb\u5fc3
                                                                                  \u9ec3\u5955\u6674
                                                                                  \u5289\u5b5d\u6148
                                                                                  \u912d\u6f54\u682d | \u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u9673\u6631\u5e0c
                                                                                  \u9673\u96c5\u541b
                                                                                  \u66fe\u7766\u6f54
                                                                                  \u9ec3\u6df5\u5100
                                                                                  \u9ec3\u9756\u6df3
                                                                                  \u5289\u598d\u5e0c
                                                                                  \u8b1d\u96e8\u6069 | \u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u53e4\u9038\u5a77
                                                                                  \u674e\u6b23\u5a77
                                                                                  \u6797\u7acb\u7d43
                                                                                  \u9673\u744b\u55ac
                                                                                  \u66fe\u5b50\u7a4e
                                                                                  \u5289\u90c1\u82b3
                                                                                  \u8b1d\u4f73\u6069 | \u81fa\u5317\u5e02\u5927\u76f4\u9ad8\u4e2d
                                                                                  \u5353\u61ff\u7444
                                                                                  \u5f35\u5de7\u6db5
                                                                                  \u9ec3\u5a49\u59ae
                                                                                  \u9ec3\u59f5\u59ae
                                                                                  \u5289\u5955\u8431
                                                                                  \u8b1d\u5143\u85b0
                                                                                  \u8607\u739f\u6615
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u738b\u4f73\u5043
                                                                                  \u4f55\u5029\u4f69
                                                                                  \u675c\u4f9d\u7d14
                                                                                  \u9673\u5bd7\u81fb
                                                                                  \u9ec3\u7230\u84c1
                                                                                  \u8607\u6021\u5be7
                                                                                  \u81fa\u5317\u5e02\u5317\u4e00\u5973\u4e2d
                                                                                  \u738b\u5a55\u82b8
                                                                                  \u4f59\u5a55\u7433
                                                                                  \u674e\u5b9c\u84c1
                                                                                  \u6797\u80b2\u8431
                                                                                  \u9673\u4e8e\u6db5
                                                                                  \u9ec3\u90c1\u742a
                                                                                  \u5289\u66c8\u749f
                                                                                  \u65b0\u5317\u5e02\u65b0\u5317\u9ad8\u4e2d
                                                                                  \u738b\u7d2b\u82b8
                                                                                  \u674e\u5b9c\u85b0
                                                                                  \u6797\u51a0\u59a4
                                                                                  \u9673\u601d\u9f4a
                                                                                  \u8521\u6085
                                                                                  \u8521\u6190\u61ab
                                                                                  \u8cf4\u73ee\u84c1 | | | |\n| \u570b\u7537\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd | \u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u6bdb\u51a0\u6dee
                                                                                  \u674e\u5cfb\u4ea6
                                                                                  \u6797\u90c1\u654f
                                                                                  \u838a\u5b78\u6fc2
                                                                                  \u838a\u6def\u7fd4
                                                                                  \u6b50\u5b50\u6bc5
                                                                                  \u8b1d\u79c9\u5bb8 | \u65b0\u7af9\u5e02\u4e09\u6c11\u570b\u4e2d
                                                                                  \u674e\u5bb6\u7a4e
                                                                                  \u6797\u8c6a\u923a
                                                                                  \u5f90\u7693\u9706
                                                                                  \u8881\u4fca\u777f
                                                                                  \u694a\u6615\u7fa9
                                                                                  \u912d\u53cb\u6069
                                                                                  \u9f94\u6631\u627f | \u81fa\u4e2d\u5e02\u9ece\u660e\u570b\u4e2d
                                                                                  \u6797\u6587\u8a70
                                                                                  \u6797\u7fbd\u8ed2
                                                                                  \u694a\u627f\u7ff0
                                                                                  \u8449\u9326\u4e1e
                                                                                  \u5ed6\u5927\u921e
                                                                                  \u8d99\u5149\u5ba5
                                                                                  \u6492\u7490\uff0e\u4f0a\u65af\u5361\u5361\u592b\u7279 | \u5c4f\u6771\u7e23\u81f3\u6b63\u570b\u4e2d
                                                                                  \u738b\u6689\u5ef7
                                                                                  \u4f59\u6893\u52e4
                                                                                  \u5433\u665f\u69d0
                                                                                  \u8a31\u5bb6\u7a0b
                                                                                  \u9673\u5b8f\u8ed2
                                                                                  \u9673\u6975\u8a9e
                                                                                  \u9ec3\u51a0\u744b | \u81fa\u5317\u5e02\u65b0\u6c11\u570b\u4e2d
                                                                                  \u5433\u6587\u65ed
                                                                                  \u674e\u5149\u555f
                                                                                  \u5353\u6b63\u5091
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                                                                                  \u7c21\u5146\u5b8f | | | |\n| \u570b\u5973\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd | \u81fa\u5317\u5e02\u81f3\u5584\u570b\u4e2d
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                                                                                  \u8b1d\u79c9\u6a3a | \u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u674e\u6c9b\u7df9
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                                                                                  \u912d\u5b89\u6f54 | \u81fa\u5357\u5e02\u5fe0\u5b5d\u570b\u4e2d
                                                                                  \u6797\u5955\u5f64
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                                                                                  \u694a\u6e58\u82d3
                                                                                  \u6f58\u5de7\u6df3
                                                                                  \u8521\u8a60\u5e06 | \u65b0\u7af9\u5e02\u4e09\u6c11\u570b\u4e2d
                                                                                  \u6797\u5bb6\u7504
                                                                                  \u5f35\u5c91\u7433
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                                                                                  \u8449\u4fd0\u4f36 | | | |\n", "html": "\n\n\n\n\n\u4e2d\u83ef\u6c11\u570b112\u5e74\u5168\u570b\u4e2d\u7b49\u5b78\u6821\u904b\u52d5\u6703-112\u5168\u4e2d\u904b\n\n\n\t\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n \n\n\n\t\n \n\n \n\n
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                                                                                  1. \n\t\t\t\t:::\n\t\t\t\t\u7db2\u7ad9\u5c0e\u89bd\n\t\t\t
                                                                                    2. \n\t\t\t
                                                                                  2. \u7cfb\u7d71\u767b\u5165
                                                                                    3. \n\t\t
                                                                                  \n\t

                                                                                  \"\u4e2d\u83ef\u6c11\u570b112\u5e74\u5168\u570b\u4e2d\u7b49\u5b78\u6821\u904b\u52d5\u6703\"

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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \"\"id=\"_WebBannerImg\"
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                                                                                  \u7af6\u8cfd\u6210\u7e3e
                                                                                  \n\t\t\t\n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t
                                                                                  \u8cfd\u7a0b\u6210\u7e3e
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                                                                                  \n\t\t\n                                                                                   		\n \n
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                                                                                  1. \n \"\"\n

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                                                                                  \n \t\t \t\t 4\u670815\u65e5\n \t\t \t\t 4\u670816\u65e5\n \t\t \t\t 4\u670818\u65e5\n \t\t \t\t 4\u670819\u65e5\n \t\t \t\t 4\u670820\u65e5\n \t\t
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                                                                                  \n\n\n\n\n\n\n \t\t \t\t \n\t\t\t
                                                                                  \"\u7db2\u7403\"
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                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
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                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
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                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
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                                                                                  \u9673\u5b8f\u8ed2
                                                                                  \u9673\u6975\u8a9e
                                                                                  \u9ec3\u51a0\u744b
                                                                                  		\u81fa\u5317\u5e02\u65b0\u6c11\u570b\u4e2d
                                                                                  \u5433\u6587\u65ed
                                                                                  \u674e\u5149\u555f
                                                                                  \u5353\u6b63\u5091
                                                                                  \u6797\u8b1d\u5b89
                                                                                  \u8a31\u723e\u5a01
                                                                                  \u9ec3\u6587\u5ba5
                                                                                  \u8cf4\u6649\u5bec
                                                                                  \u82b1\u84ee\u7e23\u5b9c\u660c\u570b\u4e2d
                                                                                  \u738b\u67cf\u921e
                                                                                  \u80e1\u4fca\u7965
                                                                                  \u9ad8\u5b87\u8861
                                                                                  \u9ec3\u54c1\u582f
                                                                                  \u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u65b9\u97cb\u55ac
                                                                                  \u738b\u5f65\u73fa
                                                                                  \u738b\u555f\u5b89
                                                                                  \u6885\u5cea\u61f7
                                                                                  \u6b50\u967d\u4f2f\u5f65
                                                                                  \u912d\u4e0a\u99ad
                                                                                  \u912d\u7fd4\u4e91
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u738b\u5ba5\u921e
                                                                                  \u5433\u5cb3\u6fa4
                                                                                  \u9673\u5ba5\u9716
                                                                                  \u66fe\u5b50\u6df5
                                                                                  \u9ec3\u5b50\u5100
                                                                                  \u694a\u5ead\u79be
                                                                                  \u7c21\u5146\u5b8f
                                                                                  \u570b\u5973\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u81fa\u5317\u5e02\u81f3\u5584\u570b\u4e2d
                                                                                  \u82b1\u82e1\u6069
                                                                                  \u9ad8\u82b7\u6db5
                                                                                  \u8a31\u82b8\u8c9e
                                                                                  \u9673\u53c8\u7dad
                                                                                  \u9ec3\u6c38\u5609
                                                                                  \u9127\u774e\u55ac
                                                                                  \u8cf4\u5bb6\u965e
                                                                                  		\u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u91d1\u5de7\u502b
                                                                                  \u7d00\u84d3\u5ba3
                                                                                  \u66f9\u5609\u51cc
                                                                                  \u9ec3\u73c9\u81fb
                                                                                  \u9ec3\u59ff\u5609
                                                                                  \u859b\u5b50\u5100
                                                                                  \u8b1d\u79c9\u6a3a
                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u674e\u6c9b\u7df9
                                                                                  \u90b1\u923a\u55ac
                                                                                  \u67ef\u96e8\u5c91
                                                                                  \u5d14\u6085\u5bb9
                                                                                  \u838a\u598d\u9234
                                                                                  \u96f7\u5982\u5ba3
                                                                                  \u912d\u5b89\u6f54
                                                                                  		\u81fa\u5357\u5e02\u5fe0\u5b5d\u570b\u4e2d
                                                                                  \u6797\u5955\u5f64
                                                                                  \u9673\u5ba5\u9321
                                                                                  \u9ec3\u53ef\u6615
                                                                                  \u9ec3\u54c1\u745c
                                                                                  \u694a\u6e58\u82d3
                                                                                  \u6f58\u5de7\u6df3
                                                                                  \u8521\u8a60\u5e06
                                                                                  		\u65b0\u7af9\u5e02\u4e09\u6c11\u570b\u4e2d
                                                                                  \u6797\u5bb6\u7504
                                                                                  \u5f35\u5c91\u7433
                                                                                  \u90ed\u84c1
                                                                                  \u694a\u6b23\u4ead
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u738b\u54c1\u7440
                                                                                  \u9673\u6b46\u55ac
                                                                                  \u5289\u90c1\u9e97
                                                                                  \u8607\u5ba5\u84c1
                                                                                  \u81fa\u5317\u5e02\u5927\u76f4\u9ad8\u4e2d
                                                                                  \u9673\u76c8\u7a4e
                                                                                  \u9673\u52ad\u5ead
                                                                                  \u9ec3\u5b50\u82b8
                                                                                  \u8521\u5bb6\u745e
                                                                                  \u8521\u6c84\u84d2
                                                                                  \u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u9673\u82b7\u82b8
                                                                                  \u9673\u828a\u58ec
                                                                                  \u66fe\u598d\u84c9
                                                                                  \u8449\u4fd0\u4f36
                                                                                  \t\t\t\t\n\t\t\t\t
                                                                                  4\u670816\u65e5
                                                                                  \n\n\t\t
                                                                                  \n\n\t\t \n\t\t \t\n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n \t\t \t\t\t
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \u9ad8\u7537\u7d44\u7db2\u7403
                                                                                  \u96d9\u6253\u8cfd                                                                                  		\u81fa\u5317\u5e02\u5efa\u570b\u4e2d\u5b78
                                                                                  \u4faf\u5091\u6069
                                                                                  \u8cf4\u79b9\u821c
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u5433\u4f2f\u5b87
                                                                                  \u9ec3\u653f\u9594
                                                                                  		\u81fa\u5357\u5e02\u9577\u69ae\u4e2d\u5b78
                                                                                  \u4f55\u5bb8\u8f1d
                                                                                  \u7f85\u839b\u51f1
                                                                                  		\u81fa\u4e2d\u5e02\u6771\u5c71\u9ad8\u4e2d
                                                                                  \u5b89\u96c5\u5229
                                                                                  \u694a\u8283\u51f1
                                                                                  		\u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u90b1\u9756\u748b
                                                                                  \u6e6f\u71ca
                                                                                  \u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u6731\u7ad1\u7444
                                                                                  \u6881\u93a7\u9e9f
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u9673\u4ee5\u8ed2
                                                                                  \u8b1d\u5fc3\u8afe
                                                                                  \u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u5f90\u6668\u63da
                                                                                  \u8521\u53f3\u627f
                                                                                  \u9ad8\u5973\u7d44\u7db2\u7403
                                                                                  \u96d9\u6253\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u6797\u82b3\u5b89
                                                                                  \u5f35\u5929\u99a8
                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u6881\u5bb8\u5f0b
                                                                                  \u842c\u5955\u5f63
                                                                                  		\u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u6797\u7acb\u5fc3
                                                                                  \u912d\u6f54\u682d
                                                                                  		\u81fa\u5317\u5e02\u5317\u4e00\u5973\u4e2d
                                                                                  \u674e\u5b9c\u84c1
                                                                                  \u5289\u66c8\u749f
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u9673\u744b\u55ac
                                                                                  \u8b1d\u4f73\u6069
                                                                                  \u81fa\u4e2d\u5e02\u6771\u5c71\u9ad8\u4e2d
                                                                                  \u5f90\u8473
                                                                                  \u9673\u540d\u99a5
                                                                                  \u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u674e\u6b23\u5a77
                                                                                  \u5289\u90c1\u82b3
                                                                                  \u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u9673\u96c5\u541b
                                                                                  \u5289\u598d\u5e0c
                                                                                  \t\t\t\t\n\t\t\t\t
                                                                                  4\u670818\u65e5
                                                                                  \n\n\t\t
                                                                                  \n\n\t\t \n\t\t \t\n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n \t\t \t\t\t
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \u9ad8\u7537\u7d44\u7db2\u7403
                                                                                  \u55ae\u6253\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u6d2a\u53ef\u6d69
                                                                                  		\u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u6b50\u5091\u6587
                                                                                  		\u81fa\u5317\u5e02\u967d\u660e\u9ad8\u4e2d
                                                                                  \u4faf\u9187\u8b19
                                                                                  		\u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u5468\u66c9\u98a8
                                                                                  		\u82b1\u84ee\u7e23\u82b1\u84ee\u9ad8\u4e2d
                                                                                  \u65b9\u5b8f\u6137
                                                                                  \u81fa\u5317\u5e02\u5efa\u570b\u4e2d\u5b78
                                                                                  \u5433\u93a7\u5b89
                                                                                  \u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u5510\u90e1
                                                                                  \u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u5468\u6587\u6bc5
                                                                                  \u9ad8\u5973\u7d44\u7db2\u7403
                                                                                  \u55ae\u6253\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u9673\u923a\u8553
                                                                                  		\u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u5442\u4f73\u7a4e
                                                                                  		\u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u66fe\u7766\u6f54
                                                                                  		\u65b0\u5317\u5e02\u65b0\u5317\u9ad8\u4e2d
                                                                                  \u6797\u51a0\u59a4
                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u5433\u7fca\u5be7
                                                                                  \u81fa\u4e2d\u5e02\u6771\u5c71\u9ad8\u4e2d
                                                                                  \u5f90\u8473
                                                                                  \u81fa\u5357\u5e02\u5927\u7063\u9ad8\u4e2d
                                                                                  \u9673\u4ead\u6c5d
                                                                                  \u65b0\u5317\u5e02\u65b0\u5317\u9ad8\u4e2d
                                                                                  \u8521\u6190\u61ab
                                                                                  \u9ad8\u4e2d\u7d44\u7db2\u7403
                                                                                  \u7537\u5973\u6df7\u5408\u96d9\u6253\u8cfd                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u5433\u6649\u5cb1
                                                                                  \u66fe\u5b50\u7a4e
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u4f59\u51a0\u61b2
                                                                                  \u53e4\u9038\u5a77
                                                                                  		\u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u5468\u5a01\u8fb0
                                                                                  \u9ec3\u9756\u6df3
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u8521\u548c\u8afa
                                                                                  \u6797\u7acb\u7d43
                                                                                  		\u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u674e\u6649\u6e90
                                                                                  \u9ec3\u5955\u6674
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u5ed6\u6d69\u5ef7
                                                                                  \u738b\u4f73\u5043
                                                                                  \u5c4f\u6771\u7e23\u5927\u540c\u9ad8\u4e2d
                                                                                  \u5289\u5f37\u68ee
                                                                                  \u8a31\u4f73\u742a
                                                                                  \u5357\u6295\u7e23\u5357\u6295\u9ad8\u4e2d
                                                                                  \u9673\u838a\u535a\u7fd4
                                                                                  \u8cf4\u5999\u6893
                                                                                  \u570b\u4e2d\u7d44\u7db2\u7403
                                                                                  \u7537\u5973\u6df7\u5408\u96d9\u6253\u8cfd                                                                                  		\u81fa\u5317\u5e02\u81f3\u5584\u570b\u4e2d
                                                                                  \u76e7\u5ba5\u5ef7
                                                                                  \u82b1\u82e1\u6069
                                                                                  		\u5f70\u5316\u7e23\u5927\u540c\u570b\u4e2d
                                                                                  \u674e\u7950\u6f84
                                                                                  \u5f35\u5ead\u7504
                                                                                  		\u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u9ec3\u667a\u5bec
                                                                                  \u66f9\u5609\u51cc
                                                                                  		\u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u738b\u555f\u5b89
                                                                                  \u9edb\u5922\u5a1c\u00b7\u963f\u53e4\u96c5\u7d0d
                                                                                  		\u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u9673\u51a0\u5ef7
                                                                                  \u9ec3\u73c9\u81fb
                                                                                  \u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u8b1d\u79c9\u5bb8
                                                                                  \u96f7\u5982\u5ba3
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u9673\u5ba5\u9716
                                                                                  \u738b\u54c1\u7440
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u738b\u5ba5\u921e
                                                                                  \u9673\u6b46\u55ac
                                                                                  \t\t\t\t\n\t\t\t\t
                                                                                  4\u670819\u65e5
                                                                                  \n\n\t\t
                                                                                  \n\n\t\t \n\t\t \t\n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n \t\t \t\t\t
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \u570b\u7537\u7d44\u7db2\u7403
                                                                                  \u96d9\u6253\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u6bdb\u51a0\u6dee
                                                                                  \u6b50\u5b50\u6bc5
                                                                                  		\u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u66fe\u5b50\u6df5
                                                                                  \u694a\u5ead\u79be
                                                                                  		\u65b0\u7af9\u5e02\u4e09\u6c11\u570b\u4e2d
                                                                                  \u674e\u5bb6\u7a4e
                                                                                  \u6797\u8c6a\u923a
                                                                                  		\u5c4f\u6771\u7e23\u81f3\u6b63\u570b\u4e2d
                                                                                  \u4f59\u6893\u52e4
                                                                                  \u9673\u6975\u8a9e
                                                                                  		\u81fa\u4e2d\u5e02\u9ece\u660e\u570b\u4e2d
                                                                                  \u694a\u627f\u7ff0
                                                                                  \u6492\u7490\uff0e\u4f0a\u65af\u5361\u5361\u592b\u7279
                                                                                  \u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u6e29\u4fde\u53e1
                                                                                  \u9673\u51a0\u52f3
                                                                                  \u82b1\u84ee\u7e23\u570b\u98a8\u570b\u4e2d
                                                                                  \u67ef\u8fb0\u6f94
                                                                                  \u694a\u8c50\u6e90
                                                                                  \u81fa\u5317\u5e02\u65b0\u6c11\u570b\u4e2d
                                                                                  \u5353\u6b63\u5091
                                                                                  \u8cf4\u6649\u5bec
                                                                                  \u570b\u5973\u7d44\u7db2\u7403
                                                                                  \u96d9\u6253\u8cfd                                                                                  		\u81fa\u5317\u5e02\u967d\u660e\u9ad8\u4e2d
                                                                                  \u9ec3\u7433\u9234
                                                                                  \u8521\u5b87\u752f
                                                                                  		\u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u8b1d\u8a9e\u6674
                                                                                  \u9edb\u5922\u5a1c\u00b7\u963f\u53e4\u96c5\u7d0d
                                                                                  		\u81fa\u5317\u5e02\u81f3\u5584\u570b\u4e2d
                                                                                  \u9ad8\u82b7\u6db5
                                                                                  \u9673\u53c8\u7dad
                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u67ef\u96e8\u5c91
                                                                                  \u838a\u598d\u9234
                                                                                  		\u5f70\u5316\u7e23\u5927\u540c\u570b\u4e2d
                                                                                  \u5f35\u5ead\u7504
                                                                                  \u8cf4\u7487
                                                                                  \u82b1\u84ee\u7e23\u5b9c\u660c\u570b\u4e2d
                                                                                  \u9ec3\u8a22\u5bb9
                                                                                  \u8a79\u5fc3\u79be
                                                                                  \u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u7d00\u84d3\u5ba3
                                                                                  \u859b\u5b50\u5100
                                                                                  \u6843\u5712\u5e02\u6148\u6587\u570b\u4e2d
                                                                                  \u5442\u5bb6\u752f
                                                                                  \u6797\u90c1\u6668
                                                                                  \t\t\t\t\n\t\t\t\t
                                                                                  4\u670820\u65e5
                                                                                  \n\n\t\t
                                                                                  \n\n\t\t \n\t\t \t\n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n \t
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \u570b\u7537\u7d44\u7db2\u7403
                                                                                  \u55ae\u6253\u8cfd                                                                                  		\u81fa\u5357\u5e02\u5927\u7063\u9ad8\u4e2d
                                                                                  \u9673\u51a0\u5b88
                                                                                  		\u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u738b\u5f65\u73fa
                                                                                  		\u81fa\u4e2d\u5e02\u9ece\u660e\u570b\u4e2d
                                                                                  \u6797\u6587\u8a70
                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u838a\u5b78\u6fc2
                                                                                  		\u82b1\u84ee\u7e23\u7389\u91cc\u570b\u4e2d
                                                                                  \u5f6d\u6977\u7a0b
                                                                                  \u65b0\u5317\u5e02\u65b0\u6cf0\u570b\u4e2d
                                                                                  \u6c88\u5eb7\u5ef7
                                                                                  \u5c4f\u6771\u7e23\u9577\u6cbb\u570b\u4e2d
                                                                                  \u6234\u5955\u5bcc
                                                                                  \u5f70\u5316\u7e23\u82b1\u58c7\u570b\u4e2d
                                                                                  \u8a31\u5fb7\u5ba5
                                                                                  \u570b\u5973\u7d44\u7db2\u7403
                                                                                  \u55ae\u6253\u8cfd                                                                                  		\u81fa\u5317\u5e02\u967d\u660e\u9ad8\u4e2d
                                                                                  \u8521\u5b87\u752f
                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u912d\u5b89\u6f54
                                                                                  		\u6843\u5712\u5e02\u6148\u6587\u570b\u4e2d
                                                                                  \u6797\u90c1\u6668
                                                                                  		\u5c4f\u6771\u7e23\u9577\u6cbb\u570b\u4e2d
                                                                                  \u912d\u82e5\u59a4
                                                                                  		\u81fa\u5317\u5e02\u897f\u6e56\u570b\u4e2d
                                                                                  \u6c6a\u4e88\u6674
                                                                                  \u81fa\u5317\u5e02\u967d\u660e\u9ad8\u4e2d
                                                                                  \u9ec3\u7433\u9234
                                                                                  \u5f70\u5316\u7e23\u5927\u540c\u570b\u4e2d
                                                                                  \u912d\u923a\u81fb
                                                                                  \u65b0\u5317\u5e02\u65b0\u6cf0\u570b\u4e2d
                                                                                  \u4f55\u73ee\u7dba
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \n\n
                                                                                  \n\n\n\n
                                                                                  \n \n
                                                                                  \n\n
                                                                                  \n

                                                                                  \n
                                                                                  \n
                                                                                  \n\n\n
                                                                                  \n
                                                                                  \n
                                                                                  \n\n
                                                                                  \n
                                                                                  \n
                                                                                  \n\t
                                                                                  \n\t
                                                                                    \n\t
                                                                                  • \n\t\t\t\u4e3b\u8fa6\u6a5f\u95dc\uff1a\n\t\t\t\n\t\t
                                                                                    • \n\t
                                                                                  • \n\t\t\t\u627f\u8fa6\u55ae\u4f4d\uff1a\n\t\t\t
                                                                                    • \n\t
                                                                                  \n\t
                                                                                    \n\t\t
                                                                                  • 112\u5168\u4e2d\u904b\u57f7\u59d4\u6703
                                                                                    • \n\t\t
                                                                                  • \u5730\u5740\uff1a30210\u65b0\u7af9\u7e23\u7af9\u5317\u5e02\u5149\u660e\u516d\u8def10\u865fB\u68df6\u6a13
                                                                                    • \n\t\t
                                                                                  • \u8fa6\u516c\u6642\u9593\uff1a\u9031\u4e00\u81f3\u9031\u4e94\u4e0a\u53488:00\u81f3\u4e0b\u534816:00
                                                                                    • \n\t\t
                                                                                  • \u96fb\u8a71\uff1a03-5518101\u5206\u6a5f2862
                                                                                    • \n\t\t
                                                                                  • \u4fe1\u7bb1\uff1a20058475@hchg.gov.tw
                                                                                    • \n\t\t
                                                                                  • \ufeff\u700f\u89bd\u4eba\u6578\uff1a3399541\n
                                                                                    • \n\t
                                                                                  \n\n\t\n\n\t
                                                                                  \n\t
                                                                                  \n
                                                                                  \n\n
                                                                                  \n
                                                                                  \n
                                                                                  \n\n\n\n \n\n"} +{"content_list": [[{"type": "table", "raw_content": "
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \u9ad8\u7537\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u65b9\u9756\u7def
                                                                                  \u674e\u6631\u8ce2
                                                                                  \u5468\u6587\u6bc5
                                                                                  \u5468\u66c9\u98a8
                                                                                  \u6d2a\u53ef\u6d69
                                                                                  \u5f90\u6668\u63da
                                                                                  \u8521\u53f3\u627f
                                                                                  		\u81fa\u5317\u5e02\u5efa\u570b\u4e2d\u5b78
                                                                                  \u5468\u9806\u5fb7
                                                                                  \u4faf\u5091\u6069
                                                                                  \u7d22\u5357\u55ac\u67cf
                                                                                  \u694a\u51f1\u6069
                                                                                  \u5289\u5b50\u8aa0
                                                                                  \u8cf4\u79b9\u821c
                                                                                  \u8607\u79b9\u4e1e
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u4f59\u51a0\u61b2
                                                                                  \u5433\u4f2f\u5b87
                                                                                  \u5433\u6649\u5cb1
                                                                                  \u9673\u5b5f\u8b19
                                                                                  \u9ec3\u653f\u9594
                                                                                  \u8521\u548c\u8afa
                                                                                  \u8521\u627f\u7950
                                                                                  		\u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u738b\u7167\u9321
                                                                                  \u90b1\u9756\u748b
                                                                                  \u5510\u90e1
                                                                                  \u66f9\u9594\u7fd4
                                                                                  \u9673\u6cd3\u8aed
                                                                                  \u9673\u8a9e\u7db8
                                                                                  \u6e6f\u71ca
                                                                                  		\u82b1\u84ee\u7e23\u82b1\u84ee\u9ad8\u4e2d
                                                                                  \u65b9\u5b8f\u6137
                                                                                  \u7530\u6fec\u744b
                                                                                  \u5442\u674e\u6b3d\u7fd4
                                                                                  \u8449\u4f73\u5049
                                                                                  \u6f58\u7acb\u8ed2
                                                                                  \u5c4f\u6771\u7e23\u5927\u540c\u9ad8\u4e2d
                                                                                  \u67ef\u535a\u9594
                                                                                  \u9ad8\u58eb\u51f1
                                                                                  \u90ed\u7d18\u777f
                                                                                  \u9673\u5764\u4f51
                                                                                  \u5289\u5f37\u68ee
                                                                                  \u912d\u4f0d\u6d0b
                                                                                  \u76e7\u5b87\u6f54
                                                                                  \u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u674e\u6649\u6e90
                                                                                  \u6797\u627f\u9032
                                                                                  \u5f35\u4e2d\u777f
                                                                                  \u9ec3\u97cb\u7db8
                                                                                  \u8521\u4f73\u4fee
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u5433\u6607\u6a3a
                                                                                  \u675c\u5764\u6649
                                                                                  \u9673\u4ee5\u8ed2
                                                                                  \u9673\u6600\u744b
                                                                                  \u5ed6\u6d69\u5ef7
                                                                                  \u6b50\u5091\u6587
                                                                                  \u8b1d\u5fc3\u8afe
                                                                                  \u9ad8\u5973\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u674e\u4e88\u85b0
                                                                                  \u674e\u79d1\u7de3
                                                                                  \u6797\u82b3\u5b89
                                                                                  \u5f35\u5929\u99a8
                                                                                  \u6881\u5bb8\u5f0b
                                                                                  \u9673\u923a\u8553
                                                                                  \u842c\u5955\u5f63
                                                                                  		\u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u6e29\u4fde\u699b
                                                                                  \u5df4\u55ac\u767b
                                                                                  \u5442\u4f73\u7a4e
                                                                                  \u6797\u7acb\u5fc3
                                                                                  \u9ec3\u5955\u6674
                                                                                  \u5289\u5b5d\u6148
                                                                                  \u912d\u6f54\u682d
                                                                                  		\u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u9673\u6631\u5e0c
                                                                                  \u9673\u96c5\u541b
                                                                                  \u66fe\u7766\u6f54
                                                                                  \u9ec3\u6df5\u5100
                                                                                  \u9ec3\u9756\u6df3
                                                                                  \u5289\u598d\u5e0c
                                                                                  \u8b1d\u96e8\u6069
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u53e4\u9038\u5a77
                                                                                  \u674e\u6b23\u5a77
                                                                                  \u6797\u7acb\u7d43
                                                                                  \u9673\u744b\u55ac
                                                                                  \u66fe\u5b50\u7a4e
                                                                                  \u5289\u90c1\u82b3
                                                                                  \u8b1d\u4f73\u6069
                                                                                  		\u81fa\u5317\u5e02\u5927\u76f4\u9ad8\u4e2d
                                                                                  \u5353\u61ff\u7444
                                                                                  \u5f35\u5de7\u6db5
                                                                                  \u9ec3\u5a49\u59ae
                                                                                  \u9ec3\u59f5\u59ae
                                                                                  \u5289\u5955\u8431
                                                                                  \u8b1d\u5143\u85b0
                                                                                  \u8607\u739f\u6615
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u738b\u4f73\u5043
                                                                                  \u4f55\u5029\u4f69
                                                                                  \u675c\u4f9d\u7d14
                                                                                  \u9673\u5bd7\u81fb
                                                                                  \u9ec3\u7230\u84c1
                                                                                  \u8607\u6021\u5be7
                                                                                  \u81fa\u5317\u5e02\u5317\u4e00\u5973\u4e2d
                                                                                  \u738b\u5a55\u82b8
                                                                                  \u4f59\u5a55\u7433
                                                                                  \u674e\u5b9c\u84c1
                                                                                  \u6797\u80b2\u8431
                                                                                  \u9673\u4e8e\u6db5
                                                                                  \u9ec3\u90c1\u742a
                                                                                  \u5289\u66c8\u749f
                                                                                  \u65b0\u5317\u5e02\u65b0\u5317\u9ad8\u4e2d
                                                                                  \u738b\u7d2b\u82b8
                                                                                  \u674e\u5b9c\u85b0
                                                                                  \u6797\u51a0\u59a4
                                                                                  \u9673\u601d\u9f4a
                                                                                  \u8521\u6085
                                                                                  \u8521\u6190\u61ab
                                                                                  \u8cf4\u73ee\u84c1
                                                                                  \u570b\u7537\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u6bdb\u51a0\u6dee
                                                                                  \u674e\u5cfb\u4ea6
                                                                                  \u6797\u90c1\u654f
                                                                                  \u838a\u5b78\u6fc2
                                                                                  \u838a\u6def\u7fd4
                                                                                  \u6b50\u5b50\u6bc5
                                                                                  \u8b1d\u79c9\u5bb8
                                                                                  		\u65b0\u7af9\u5e02\u4e09\u6c11\u570b\u4e2d
                                                                                  \u674e\u5bb6\u7a4e
                                                                                  \u6797\u8c6a\u923a
                                                                                  \u5f90\u7693\u9706
                                                                                  \u8881\u4fca\u777f
                                                                                  \u694a\u6615\u7fa9
                                                                                  \u912d\u53cb\u6069
                                                                                  \u9f94\u6631\u627f
                                                                                  		\u81fa\u4e2d\u5e02\u9ece\u660e\u570b\u4e2d
                                                                                  \u6797\u6587\u8a70
                                                                                  \u6797\u7fbd\u8ed2
                                                                                  \u694a\u627f\u7ff0
                                                                                  \u8449\u9326\u4e1e
                                                                                  \u5ed6\u5927\u921e
                                                                                  \u8d99\u5149\u5ba5
                                                                                  \u6492\u7490\uff0e\u4f0a\u65af\u5361\u5361\u592b\u7279
                                                                                  		\u5c4f\u6771\u7e23\u81f3\u6b63\u570b\u4e2d
                                                                                  \u738b\u6689\u5ef7
                                                                                  \u4f59\u6893\u52e4
                                                                                  \u5433\u665f\u69d0
                                                                                  \u8a31\u5bb6\u7a0b
                                                                                  \u9673\u5b8f\u8ed2
                                                                                  \u9673\u6975\u8a9e
                                                                                  \u9ec3\u51a0\u744b
                                                                                  		\u81fa\u5317\u5e02\u65b0\u6c11\u570b\u4e2d
                                                                                  \u5433\u6587\u65ed
                                                                                  \u674e\u5149\u555f
                                                                                  \u5353\u6b63\u5091
                                                                                  \u6797\u8b1d\u5b89
                                                                                  \u8a31\u723e\u5a01
                                                                                  \u9ec3\u6587\u5ba5
                                                                                  \u8cf4\u6649\u5bec
                                                                                  \u82b1\u84ee\u7e23\u5b9c\u660c\u570b\u4e2d
                                                                                  \u738b\u67cf\u921e
                                                                                  \u80e1\u4fca\u7965
                                                                                  \u9ad8\u5b87\u8861
                                                                                  \u9ec3\u54c1\u582f
                                                                                  \u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u65b9\u97cb\u55ac
                                                                                  \u738b\u5f65\u73fa
                                                                                  \u738b\u555f\u5b89
                                                                                  \u6885\u5cea\u61f7
                                                                                  \u6b50\u967d\u4f2f\u5f65
                                                                                  \u912d\u4e0a\u99ad
                                                                                  \u912d\u7fd4\u4e91
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u738b\u5ba5\u921e
                                                                                  \u5433\u5cb3\u6fa4
                                                                                  \u9673\u5ba5\u9716
                                                                                  \u66fe\u5b50\u6df5
                                                                                  \u9ec3\u5b50\u5100
                                                                                  \u694a\u5ead\u79be
                                                                                  \u7c21\u5146\u5b8f
                                                                                  \u570b\u5973\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u81fa\u5317\u5e02\u81f3\u5584\u570b\u4e2d
                                                                                  \u82b1\u82e1\u6069
                                                                                  \u9ad8\u82b7\u6db5
                                                                                  \u8a31\u82b8\u8c9e
                                                                                  \u9673\u53c8\u7dad
                                                                                  \u9ec3\u6c38\u5609
                                                                                  \u9127\u774e\u55ac
                                                                                  \u8cf4\u5bb6\u965e
                                                                                  		\u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u91d1\u5de7\u502b
                                                                                  \u7d00\u84d3\u5ba3
                                                                                  \u66f9\u5609\u51cc
                                                                                  \u9ec3\u73c9\u81fb
                                                                                  \u9ec3\u59ff\u5609
                                                                                  \u859b\u5b50\u5100
                                                                                  \u8b1d\u79c9\u6a3a
                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u674e\u6c9b\u7df9
                                                                                  \u90b1\u923a\u55ac
                                                                                  \u67ef\u96e8\u5c91
                                                                                  \u5d14\u6085\u5bb9
                                                                                  \u838a\u598d\u9234
                                                                                  \u96f7\u5982\u5ba3
                                                                                  \u912d\u5b89\u6f54
                                                                                  		\u81fa\u5357\u5e02\u5fe0\u5b5d\u570b\u4e2d
                                                                                  \u6797\u5955\u5f64
                                                                                  \u9673\u5ba5\u9321
                                                                                  \u9ec3\u53ef\u6615
                                                                                  \u9ec3\u54c1\u745c
                                                                                  \u694a\u6e58\u82d3
                                                                                  \u6f58\u5de7\u6df3
                                                                                  \u8521\u8a60\u5e06
                                                                                  		\u65b0\u7af9\u5e02\u4e09\u6c11\u570b\u4e2d
                                                                                  \u6797\u5bb6\u7504
                                                                                  \u5f35\u5c91\u7433
                                                                                  \u90ed\u84c1
                                                                                  \u694a\u6b23\u4ead
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u738b\u54c1\u7440
                                                                                  \u9673\u6b46\u55ac
                                                                                  \u5289\u90c1\u9e97
                                                                                  \u8607\u5ba5\u84c1
                                                                                  \u81fa\u5317\u5e02\u5927\u76f4\u9ad8\u4e2d
                                                                                  \u9673\u76c8\u7a4e
                                                                                  \u9673\u52ad\u5ead
                                                                                  \u9ec3\u5b50\u82b8
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                                                                                  \u8df3\u5230\u4e3b\u8981\u5167\u5bb9\n\t\t
                                                                                    \n\t\t\t
                                                                                  1. \n\t\t\t\t:::\n\t\t\t\t\u7db2\u7ad9\u5c0e\u89bd\n\t\t\t
                                                                                    2. \n\t\t\t
                                                                                  2. \u7cfb\u7d71\u767b\u5165
                                                                                    3. \n\t\t
                                                                                  \n\t

                                                                                  \"\u4e2d\u83ef\u6c11\u570b112\u5e74\u5168\u570b\u4e2d\u7b49\u5b78\u6821\u904b\u52d5\u6703\"

                                                                                  \n\n\n\t
                                                                                  \n\n
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n\n\n\n\n
                                                                                  \n\t\t\t
                                                                                  \n\t\t\t\t
                                                                                  \n\n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n\n
                                                                                  \n\t\t\t
                                                                                  \n\t\t\t
                                                                                  \n\t\t\t\t
                                                                                  \n\n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n
                                                                                  \"\"id=\"_WebBannerImg\"
                                                                                  \n\n\n
                                                                                  \n\t\t\t
                                                                                  \n\t
                                                                                  \n
                                                                                  \n\t\n\t\n\t
                                                                                  \n
                                                                                  \n
                                                                                  \n\t
                                                                                  \n\t \n \n \n \n \n
                                                                                  \n \n
                                                                                  \n\t\t\t
                                                                                  \u7af6\u8cfd\u6210\u7e3e
                                                                                  \n\t\t\t\n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \t\t\t
                                                                                  \u8cfd\u7a0b\u6210\u7e3e
                                                                                  \u6c7a\u8cfd\u6210\u7e3e\u7d9c\u5408\u67e5\u8a62
                                                                                  \u6210\u7e3e\u8b49\u660e\u5217\u5370
                                                                                  \u6c7a\u8cfd\u6210\u7e3e\u5831\u544a\u8868
                                                                                  \u61c9(\u5df2)\u9812\u734e\u724c
                                                                                  \u734e\u724c\u7d71\u8a08
                                                                                  \u7834\u7d00\uf93f\u67e5\u8a62
                                                                                  \u7af6\u8cfd\u7a2e\u985e\u9326\u6a19
                                                                                  \u6b77\u5e74\u6210\u7e3e\u67e5\u8a62
                                                                                  \n\t\t
                                                                                  \n\t\t\n                                                                                   		\n \n
                                                                                  \n
                                                                                    \n\n
                                                                                  1. \n \"\"\n

                                                                                    \u6c7a\u8cfd\u6210\u7e3e\u5831\u544a\u8868

                                                                                    \n
                                                                                    2. \n
                                                                                  2. \n
                                                                                  \n
                                                                                  \n
                                                                                  \n
                                                                                  \n \n \n \n \n
                                                                                  \n\n
                                                                                  \n
                                                                                  \n \n \n \n\n\t
                                                                                  \n \t\t \t\t 4\u670815\u65e5\n \t\t \t\t 4\u670816\u65e5\n \t\t \t\t 4\u670818\u65e5\n \t\t \t\t 4\u670819\u65e5\n \t\t \t\t 4\u670820\u65e5\n \t\t
                                                                                  \n \t\t
                                                                                  \n
                                                                                  \n\n\n\n\n\n\n \t\t \t\t \n\t\t\t
                                                                                  \"\u7db2\u7403\"
                                                                                  \u7db2\u7403
                                                                                  \n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t
                                                                                  4\u670815\u65e5
                                                                                  \n\n\t\t
                                                                                  \n\n\t\t \n\t\t \t\n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n \t\t \t\t\t
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \u9ad8\u7537\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u65b9\u9756\u7def
                                                                                  \u674e\u6631\u8ce2
                                                                                  \u5468\u6587\u6bc5
                                                                                  \u5468\u66c9\u98a8
                                                                                  \u6d2a\u53ef\u6d69
                                                                                  \u5f90\u6668\u63da
                                                                                  \u8521\u53f3\u627f
                                                                                  		\u81fa\u5317\u5e02\u5efa\u570b\u4e2d\u5b78
                                                                                  \u5468\u9806\u5fb7
                                                                                  \u4faf\u5091\u6069
                                                                                  \u7d22\u5357\u55ac\u67cf
                                                                                  \u694a\u51f1\u6069
                                                                                  \u5289\u5b50\u8aa0
                                                                                  \u8cf4\u79b9\u821c
                                                                                  \u8607\u79b9\u4e1e
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u4f59\u51a0\u61b2
                                                                                  \u5433\u4f2f\u5b87
                                                                                  \u5433\u6649\u5cb1
                                                                                  \u9673\u5b5f\u8b19
                                                                                  \u9ec3\u653f\u9594
                                                                                  \u8521\u548c\u8afa
                                                                                  \u8521\u627f\u7950
                                                                                  		\u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u738b\u7167\u9321
                                                                                  \u90b1\u9756\u748b
                                                                                  \u5510\u90e1
                                                                                  \u66f9\u9594\u7fd4
                                                                                  \u9673\u6cd3\u8aed
                                                                                  \u9673\u8a9e\u7db8
                                                                                  \u6e6f\u71ca
                                                                                  		\u82b1\u84ee\u7e23\u82b1\u84ee\u9ad8\u4e2d
                                                                                  \u65b9\u5b8f\u6137
                                                                                  \u7530\u6fec\u744b
                                                                                  \u5442\u674e\u6b3d\u7fd4
                                                                                  \u8449\u4f73\u5049
                                                                                  \u6f58\u7acb\u8ed2
                                                                                  \u5c4f\u6771\u7e23\u5927\u540c\u9ad8\u4e2d
                                                                                  \u67ef\u535a\u9594
                                                                                  \u9ad8\u58eb\u51f1
                                                                                  \u90ed\u7d18\u777f
                                                                                  \u9673\u5764\u4f51
                                                                                  \u5289\u5f37\u68ee
                                                                                  \u912d\u4f0d\u6d0b
                                                                                  \u76e7\u5b87\u6f54
                                                                                  \u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u674e\u6649\u6e90
                                                                                  \u6797\u627f\u9032
                                                                                  \u5f35\u4e2d\u777f
                                                                                  \u9ec3\u97cb\u7db8
                                                                                  \u8521\u4f73\u4fee
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u5433\u6607\u6a3a
                                                                                  \u675c\u5764\u6649
                                                                                  \u9673\u4ee5\u8ed2
                                                                                  \u9673\u6600\u744b
                                                                                  \u5ed6\u6d69\u5ef7
                                                                                  \u6b50\u5091\u6587
                                                                                  \u8b1d\u5fc3\u8afe
                                                                                  \u9ad8\u5973\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u674e\u4e88\u85b0
                                                                                  \u674e\u79d1\u7de3
                                                                                  \u6797\u82b3\u5b89
                                                                                  \u5f35\u5929\u99a8
                                                                                  \u6881\u5bb8\u5f0b
                                                                                  \u9673\u923a\u8553
                                                                                  \u842c\u5955\u5f63
                                                                                  		\u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u6e29\u4fde\u699b
                                                                                  \u5df4\u55ac\u767b
                                                                                  \u5442\u4f73\u7a4e
                                                                                  \u6797\u7acb\u5fc3
                                                                                  \u9ec3\u5955\u6674
                                                                                  \u5289\u5b5d\u6148
                                                                                  \u912d\u6f54\u682d
                                                                                  		\u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u9673\u6631\u5e0c
                                                                                  \u9673\u96c5\u541b
                                                                                  \u66fe\u7766\u6f54
                                                                                  \u9ec3\u6df5\u5100
                                                                                  \u9ec3\u9756\u6df3
                                                                                  \u5289\u598d\u5e0c
                                                                                  \u8b1d\u96e8\u6069
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u53e4\u9038\u5a77
                                                                                  \u674e\u6b23\u5a77
                                                                                  \u6797\u7acb\u7d43
                                                                                  \u9673\u744b\u55ac
                                                                                  \u66fe\u5b50\u7a4e
                                                                                  \u5289\u90c1\u82b3
                                                                                  \u8b1d\u4f73\u6069
                                                                                  		\u81fa\u5317\u5e02\u5927\u76f4\u9ad8\u4e2d
                                                                                  \u5353\u61ff\u7444
                                                                                  \u5f35\u5de7\u6db5
                                                                                  \u9ec3\u5a49\u59ae
                                                                                  \u9ec3\u59f5\u59ae
                                                                                  \u5289\u5955\u8431
                                                                                  \u8b1d\u5143\u85b0
                                                                                  \u8607\u739f\u6615
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u738b\u4f73\u5043
                                                                                  \u4f55\u5029\u4f69
                                                                                  \u675c\u4f9d\u7d14
                                                                                  \u9673\u5bd7\u81fb
                                                                                  \u9ec3\u7230\u84c1
                                                                                  \u8607\u6021\u5be7
                                                                                  \u81fa\u5317\u5e02\u5317\u4e00\u5973\u4e2d
                                                                                  \u738b\u5a55\u82b8
                                                                                  \u4f59\u5a55\u7433
                                                                                  \u674e\u5b9c\u84c1
                                                                                  \u6797\u80b2\u8431
                                                                                  \u9673\u4e8e\u6db5
                                                                                  \u9ec3\u90c1\u742a
                                                                                  \u5289\u66c8\u749f
                                                                                  \u65b0\u5317\u5e02\u65b0\u5317\u9ad8\u4e2d
                                                                                  \u738b\u7d2b\u82b8
                                                                                  \u674e\u5b9c\u85b0
                                                                                  \u6797\u51a0\u59a4
                                                                                  \u9673\u601d\u9f4a
                                                                                  \u8521\u6085
                                                                                  \u8521\u6190\u61ab
                                                                                  \u8cf4\u73ee\u84c1
                                                                                  \u570b\u7537\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u6bdb\u51a0\u6dee
                                                                                  \u674e\u5cfb\u4ea6
                                                                                  \u6797\u90c1\u654f
                                                                                  \u838a\u5b78\u6fc2
                                                                                  \u838a\u6def\u7fd4
                                                                                  \u6b50\u5b50\u6bc5
                                                                                  \u8b1d\u79c9\u5bb8
                                                                                  		\u65b0\u7af9\u5e02\u4e09\u6c11\u570b\u4e2d
                                                                                  \u674e\u5bb6\u7a4e
                                                                                  \u6797\u8c6a\u923a
                                                                                  \u5f90\u7693\u9706
                                                                                  \u8881\u4fca\u777f
                                                                                  \u694a\u6615\u7fa9
                                                                                  \u912d\u53cb\u6069
                                                                                  \u9f94\u6631\u627f
                                                                                  		\u81fa\u4e2d\u5e02\u9ece\u660e\u570b\u4e2d
                                                                                  \u6797\u6587\u8a70
                                                                                  \u6797\u7fbd\u8ed2
                                                                                  \u694a\u627f\u7ff0
                                                                                  \u8449\u9326\u4e1e
                                                                                  \u5ed6\u5927\u921e
                                                                                  \u8d99\u5149\u5ba5
                                                                                  \u6492\u7490\uff0e\u4f0a\u65af\u5361\u5361\u592b\u7279
                                                                                  		\u5c4f\u6771\u7e23\u81f3\u6b63\u570b\u4e2d
                                                                                  \u738b\u6689\u5ef7
                                                                                  \u4f59\u6893\u52e4
                                                                                  \u5433\u665f\u69d0
                                                                                  \u8a31\u5bb6\u7a0b
                                                                                  \u9673\u5b8f\u8ed2
                                                                                  \u9673\u6975\u8a9e
                                                                                  \u9ec3\u51a0\u744b
                                                                                  		\u81fa\u5317\u5e02\u65b0\u6c11\u570b\u4e2d
                                                                                  \u5433\u6587\u65ed
                                                                                  \u674e\u5149\u555f
                                                                                  \u5353\u6b63\u5091
                                                                                  \u6797\u8b1d\u5b89
                                                                                  \u8a31\u723e\u5a01
                                                                                  \u9ec3\u6587\u5ba5
                                                                                  \u8cf4\u6649\u5bec
                                                                                  \u82b1\u84ee\u7e23\u5b9c\u660c\u570b\u4e2d
                                                                                  \u738b\u67cf\u921e
                                                                                  \u80e1\u4fca\u7965
                                                                                  \u9ad8\u5b87\u8861
                                                                                  \u9ec3\u54c1\u582f
                                                                                  \u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u65b9\u97cb\u55ac
                                                                                  \u738b\u5f65\u73fa
                                                                                  \u738b\u555f\u5b89
                                                                                  \u6885\u5cea\u61f7
                                                                                  \u6b50\u967d\u4f2f\u5f65
                                                                                  \u912d\u4e0a\u99ad
                                                                                  \u912d\u7fd4\u4e91
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u738b\u5ba5\u921e
                                                                                  \u5433\u5cb3\u6fa4
                                                                                  \u9673\u5ba5\u9716
                                                                                  \u66fe\u5b50\u6df5
                                                                                  \u9ec3\u5b50\u5100
                                                                                  \u694a\u5ead\u79be
                                                                                  \u7c21\u5146\u5b8f
                                                                                  \u570b\u5973\u7d44\u7db2\u7403
                                                                                  \u5718\u9ad4\u8cfd                                                                                  		\u81fa\u5317\u5e02\u81f3\u5584\u570b\u4e2d
                                                                                  \u82b1\u82e1\u6069
                                                                                  \u9ad8\u82b7\u6db5
                                                                                  \u8a31\u82b8\u8c9e
                                                                                  \u9673\u53c8\u7dad
                                                                                  \u9ec3\u6c38\u5609
                                                                                  \u9127\u774e\u55ac
                                                                                  \u8cf4\u5bb6\u965e
                                                                                  		\u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u91d1\u5de7\u502b
                                                                                  \u7d00\u84d3\u5ba3
                                                                                  \u66f9\u5609\u51cc
                                                                                  \u9ec3\u73c9\u81fb
                                                                                  \u9ec3\u59ff\u5609
                                                                                  \u859b\u5b50\u5100
                                                                                  \u8b1d\u79c9\u6a3a
                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u674e\u6c9b\u7df9
                                                                                  \u90b1\u923a\u55ac
                                                                                  \u67ef\u96e8\u5c91
                                                                                  \u5d14\u6085\u5bb9
                                                                                  \u838a\u598d\u9234
                                                                                  \u96f7\u5982\u5ba3
                                                                                  \u912d\u5b89\u6f54
                                                                                  		\u81fa\u5357\u5e02\u5fe0\u5b5d\u570b\u4e2d
                                                                                  \u6797\u5955\u5f64
                                                                                  \u9673\u5ba5\u9321
                                                                                  \u9ec3\u53ef\u6615
                                                                                  \u9ec3\u54c1\u745c
                                                                                  \u694a\u6e58\u82d3
                                                                                  \u6f58\u5de7\u6df3
                                                                                  \u8521\u8a60\u5e06
                                                                                  		\u65b0\u7af9\u5e02\u4e09\u6c11\u570b\u4e2d
                                                                                  \u6797\u5bb6\u7504
                                                                                  \u5f35\u5c91\u7433
                                                                                  \u90ed\u84c1
                                                                                  \u694a\u6b23\u4ead
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u738b\u54c1\u7440
                                                                                  \u9673\u6b46\u55ac
                                                                                  \u5289\u90c1\u9e97
                                                                                  \u8607\u5ba5\u84c1
                                                                                  \u81fa\u5317\u5e02\u5927\u76f4\u9ad8\u4e2d
                                                                                  \u9673\u76c8\u7a4e
                                                                                  \u9673\u52ad\u5ead
                                                                                  \u9ec3\u5b50\u82b8
                                                                                  \u8521\u5bb6\u745e
                                                                                  \u8521\u6c84\u84d2
                                                                                  \u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u9673\u82b7\u82b8
                                                                                  \u9673\u828a\u58ec
                                                                                  \u66fe\u598d\u84c9
                                                                                  \u8449\u4fd0\u4f36
                                                                                  \t\t\t\t\n\t\t\t\t
                                                                                  4\u670816\u65e5
                                                                                  \n\n\t\t
                                                                                  \n\n\t\t \n\t\t \t\n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n \t\t \t\t\t
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \u9ad8\u7537\u7d44\u7db2\u7403
                                                                                  \u96d9\u6253\u8cfd                                                                                  		\u81fa\u5317\u5e02\u5efa\u570b\u4e2d\u5b78
                                                                                  \u4faf\u5091\u6069
                                                                                  \u8cf4\u79b9\u821c
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u5433\u4f2f\u5b87
                                                                                  \u9ec3\u653f\u9594
                                                                                  		\u81fa\u5357\u5e02\u9577\u69ae\u4e2d\u5b78
                                                                                  \u4f55\u5bb8\u8f1d
                                                                                  \u7f85\u839b\u51f1
                                                                                  		\u81fa\u4e2d\u5e02\u6771\u5c71\u9ad8\u4e2d
                                                                                  \u5b89\u96c5\u5229
                                                                                  \u694a\u8283\u51f1
                                                                                  		\u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u90b1\u9756\u748b
                                                                                  \u6e6f\u71ca
                                                                                  \u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u6731\u7ad1\u7444
                                                                                  \u6881\u93a7\u9e9f
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u9673\u4ee5\u8ed2
                                                                                  \u8b1d\u5fc3\u8afe
                                                                                  \u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u5f90\u6668\u63da
                                                                                  \u8521\u53f3\u627f
                                                                                  \u9ad8\u5973\u7d44\u7db2\u7403
                                                                                  \u96d9\u6253\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u6797\u82b3\u5b89
                                                                                  \u5f35\u5929\u99a8
                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u6881\u5bb8\u5f0b
                                                                                  \u842c\u5955\u5f63
                                                                                  		\u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u6797\u7acb\u5fc3
                                                                                  \u912d\u6f54\u682d
                                                                                  		\u81fa\u5317\u5e02\u5317\u4e00\u5973\u4e2d
                                                                                  \u674e\u5b9c\u84c1
                                                                                  \u5289\u66c8\u749f
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u9673\u744b\u55ac
                                                                                  \u8b1d\u4f73\u6069
                                                                                  \u81fa\u4e2d\u5e02\u6771\u5c71\u9ad8\u4e2d
                                                                                  \u5f90\u8473
                                                                                  \u9673\u540d\u99a5
                                                                                  \u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u674e\u6b23\u5a77
                                                                                  \u5289\u90c1\u82b3
                                                                                  \u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u9673\u96c5\u541b
                                                                                  \u5289\u598d\u5e0c
                                                                                  \t\t\t\t\n\t\t\t\t
                                                                                  4\u670818\u65e5
                                                                                  \n\n\t\t
                                                                                  \n\n\t\t \n\t\t \t\n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n \t\t \t\t\t
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \u9ad8\u7537\u7d44\u7db2\u7403
                                                                                  \u55ae\u6253\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u6d2a\u53ef\u6d69
                                                                                  		\u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u6b50\u5091\u6587
                                                                                  		\u81fa\u5317\u5e02\u967d\u660e\u9ad8\u4e2d
                                                                                  \u4faf\u9187\u8b19
                                                                                  		\u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u5468\u66c9\u98a8
                                                                                  		\u82b1\u84ee\u7e23\u82b1\u84ee\u9ad8\u4e2d
                                                                                  \u65b9\u5b8f\u6137
                                                                                  \u81fa\u5317\u5e02\u5efa\u570b\u4e2d\u5b78
                                                                                  \u5433\u93a7\u5b89
                                                                                  \u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u5510\u90e1
                                                                                  \u9ad8\u96c4\u5e02\u65b0\u838a\u9ad8\u4e2d
                                                                                  \u5468\u6587\u6bc5
                                                                                  \u9ad8\u5973\u7d44\u7db2\u7403
                                                                                  \u55ae\u6253\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u9673\u923a\u8553
                                                                                  		\u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u5442\u4f73\u7a4e
                                                                                  		\u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u66fe\u7766\u6f54
                                                                                  		\u65b0\u5317\u5e02\u65b0\u5317\u9ad8\u4e2d
                                                                                  \u6797\u51a0\u59a4
                                                                                  		\u9ad8\u96c4\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u5433\u7fca\u5be7
                                                                                  \u81fa\u4e2d\u5e02\u6771\u5c71\u9ad8\u4e2d
                                                                                  \u5f90\u8473
                                                                                  \u81fa\u5357\u5e02\u5927\u7063\u9ad8\u4e2d
                                                                                  \u9673\u4ead\u6c5d
                                                                                  \u65b0\u5317\u5e02\u65b0\u5317\u9ad8\u4e2d
                                                                                  \u8521\u6190\u61ab
                                                                                  \u9ad8\u4e2d\u7d44\u7db2\u7403
                                                                                  \u7537\u5973\u6df7\u5408\u96d9\u6253\u8cfd                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u5433\u6649\u5cb1
                                                                                  \u66fe\u5b50\u7a4e
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u4f59\u51a0\u61b2
                                                                                  \u53e4\u9038\u5a77
                                                                                  		\u65b0\u5317\u5e02\u4e09\u91cd\u9ad8\u4e2d
                                                                                  \u5468\u5a01\u8fb0
                                                                                  \u9ec3\u9756\u6df3
                                                                                  		\u81fa\u5357\u5e02\u5584\u5316\u9ad8\u4e2d
                                                                                  \u8521\u548c\u8afa
                                                                                  \u6797\u7acb\u7d43
                                                                                  		\u81fa\u5317\u5e02\u5357\u6e2f\u9ad8\u5de5
                                                                                  \u674e\u6649\u6e90
                                                                                  \u9ec3\u5955\u6674
                                                                                  \u65b0\u5317\u5e02\u4e09\u6c11\u9ad8\u4e2d
                                                                                  \u5ed6\u6d69\u5ef7
                                                                                  \u738b\u4f73\u5043
                                                                                  \u5c4f\u6771\u7e23\u5927\u540c\u9ad8\u4e2d
                                                                                  \u5289\u5f37\u68ee
                                                                                  \u8a31\u4f73\u742a
                                                                                  \u5357\u6295\u7e23\u5357\u6295\u9ad8\u4e2d
                                                                                  \u9673\u838a\u535a\u7fd4
                                                                                  \u8cf4\u5999\u6893
                                                                                  \u570b\u4e2d\u7d44\u7db2\u7403
                                                                                  \u7537\u5973\u6df7\u5408\u96d9\u6253\u8cfd                                                                                  		\u81fa\u5317\u5e02\u81f3\u5584\u570b\u4e2d
                                                                                  \u76e7\u5ba5\u5ef7
                                                                                  \u82b1\u82e1\u6069
                                                                                  		\u5f70\u5316\u7e23\u5927\u540c\u570b\u4e2d
                                                                                  \u674e\u7950\u6f84
                                                                                  \u5f35\u5ead\u7504
                                                                                  		\u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u9ec3\u667a\u5bec
                                                                                  \u66f9\u5609\u51cc
                                                                                  		\u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u738b\u555f\u5b89
                                                                                  \u9edb\u5922\u5a1c\u00b7\u963f\u53e4\u96c5\u7d0d
                                                                                  		\u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u9673\u51a0\u5ef7
                                                                                  \u9ec3\u73c9\u81fb
                                                                                  \u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u8b1d\u79c9\u5bb8
                                                                                  \u96f7\u5982\u5ba3
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u9673\u5ba5\u9716
                                                                                  \u738b\u54c1\u7440
                                                                                  \u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u738b\u5ba5\u921e
                                                                                  \u9673\u6b46\u55ac
                                                                                  \t\t\t\t\n\t\t\t\t
                                                                                  4\u670819\u65e5
                                                                                  \n\n\t\t
                                                                                  \n\n\t\t \n\t\t \t\n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n \t\t \t\t\t
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \u570b\u7537\u7d44\u7db2\u7403
                                                                                  \u96d9\u6253\u8cfd                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u6bdb\u51a0\u6dee
                                                                                  \u6b50\u5b50\u6bc5
                                                                                  		\u81fa\u5357\u5e02\u5b89\u5b9a\u570b\u4e2d
                                                                                  \u66fe\u5b50\u6df5
                                                                                  \u694a\u5ead\u79be
                                                                                  		\u65b0\u7af9\u5e02\u4e09\u6c11\u570b\u4e2d
                                                                                  \u674e\u5bb6\u7a4e
                                                                                  \u6797\u8c6a\u923a
                                                                                  		\u5c4f\u6771\u7e23\u81f3\u6b63\u570b\u4e2d
                                                                                  \u4f59\u6893\u52e4
                                                                                  \u9673\u6975\u8a9e
                                                                                  		\u81fa\u4e2d\u5e02\u9ece\u660e\u570b\u4e2d
                                                                                  \u694a\u627f\u7ff0
                                                                                  \u6492\u7490\uff0e\u4f0a\u65af\u5361\u5361\u592b\u7279
                                                                                  \u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u6e29\u4fde\u53e1
                                                                                  \u9673\u51a0\u52f3
                                                                                  \u82b1\u84ee\u7e23\u570b\u98a8\u570b\u4e2d
                                                                                  \u67ef\u8fb0\u6f94
                                                                                  \u694a\u8c50\u6e90
                                                                                  \u81fa\u5317\u5e02\u65b0\u6c11\u570b\u4e2d
                                                                                  \u5353\u6b63\u5091
                                                                                  \u8cf4\u6649\u5bec
                                                                                  \u570b\u5973\u7d44\u7db2\u7403
                                                                                  \u96d9\u6253\u8cfd                                                                                  		\u81fa\u5317\u5e02\u967d\u660e\u9ad8\u4e2d
                                                                                  \u9ec3\u7433\u9234
                                                                                  \u8521\u5b87\u752f
                                                                                  		\u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u8b1d\u8a9e\u6674
                                                                                  \u9edb\u5922\u5a1c\u00b7\u963f\u53e4\u96c5\u7d0d
                                                                                  		\u81fa\u5317\u5e02\u81f3\u5584\u570b\u4e2d
                                                                                  \u9ad8\u82b7\u6db5
                                                                                  \u9673\u53c8\u7dad
                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u67ef\u96e8\u5c91
                                                                                  \u838a\u598d\u9234
                                                                                  		\u5f70\u5316\u7e23\u5927\u540c\u570b\u4e2d
                                                                                  \u5f35\u5ead\u7504
                                                                                  \u8cf4\u7487
                                                                                  \u82b1\u84ee\u7e23\u5b9c\u660c\u570b\u4e2d
                                                                                  \u9ec3\u8a22\u5bb9
                                                                                  \u8a79\u5fc3\u79be
                                                                                  \u81fa\u5317\u5e02\u65b0\u8208\u570b\u4e2d
                                                                                  \u7d00\u84d3\u5ba3
                                                                                  \u859b\u5b50\u5100
                                                                                  \u6843\u5712\u5e02\u6148\u6587\u570b\u4e2d
                                                                                  \u5442\u5bb6\u752f
                                                                                  \u6797\u90c1\u6668
                                                                                  \t\t\t\t\n\t\t\t\t
                                                                                  4\u670820\u65e5
                                                                                  \n\n\t\t
                                                                                  \n\n\t\t \n\t\t \t\n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n\n \t
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \u570b\u7537\u7d44\u7db2\u7403
                                                                                  \u55ae\u6253\u8cfd                                                                                  		\u81fa\u5357\u5e02\u5927\u7063\u9ad8\u4e2d
                                                                                  \u9673\u51a0\u5b88
                                                                                  		\u81fa\u5357\u5e02\u5357\u79d1\u9ad8\u4e2d
                                                                                  \u738b\u5f65\u73fa
                                                                                  		\u81fa\u4e2d\u5e02\u9ece\u660e\u570b\u4e2d
                                                                                  \u6797\u6587\u8a70
                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u838a\u5b78\u6fc2
                                                                                  		\u82b1\u84ee\u7e23\u7389\u91cc\u570b\u4e2d
                                                                                  \u5f6d\u6977\u7a0b
                                                                                  \u65b0\u5317\u5e02\u65b0\u6cf0\u570b\u4e2d
                                                                                  \u6c88\u5eb7\u5ef7
                                                                                  \u5c4f\u6771\u7e23\u9577\u6cbb\u570b\u4e2d
                                                                                  \u6234\u5955\u5bcc
                                                                                  \u5f70\u5316\u7e23\u82b1\u58c7\u570b\u4e2d
                                                                                  \u8a31\u5fb7\u5ba5
                                                                                  \u570b\u5973\u7d44\u7db2\u7403
                                                                                  \u55ae\u6253\u8cfd                                                                                  		\u81fa\u5317\u5e02\u967d\u660e\u9ad8\u4e2d
                                                                                  \u8521\u5b87\u752f
                                                                                  		\u9ad8\u96c4\u5e02\u6b63\u8208\u570b\u4e2d
                                                                                  \u912d\u5b89\u6f54
                                                                                  		\u6843\u5712\u5e02\u6148\u6587\u570b\u4e2d
                                                                                  \u6797\u90c1\u6668
                                                                                  		\u5c4f\u6771\u7e23\u9577\u6cbb\u570b\u4e2d
                                                                                  \u912d\u82e5\u59a4
                                                                                  		\u81fa\u5317\u5e02\u897f\u6e56\u570b\u4e2d
                                                                                  \u6c6a\u4e88\u6674
                                                                                  \u81fa\u5317\u5e02\u967d\u660e\u9ad8\u4e2d
                                                                                  \u9ec3\u7433\u9234
                                                                                  \u5f70\u5316\u7e23\u5927\u540c\u570b\u4e2d
                                                                                  \u912d\u923a\u81fb
                                                                                  \u65b0\u5317\u5e02\u65b0\u6cf0\u570b\u4e2d
                                                                                  \u4f55\u73ee\u7dba
                                                                                  \u6bd4\u8cfd\u9805\u76ee\u7b2c\u4e00\u540d\u7b2c\u4e8c\u540d\u7b2c\u4e09\u540d\u7b2c\u56db\u540d\u7b2c\u4e94\u540d\u7b2c\u516d\u540d\u7b2c\u4e03\u540d\u7b2c\u516b\u540d
                                                                                  \n\n
                                                                                  \n\n\n\n
                                                                                  \n \n
                                                                                  \n\n
                                                                                  \n

                                                                                  \n
                                                                                  \n
                                                                                  \n\n\n
                                                                                  \n
                                                                                  \n
                                                                                  \n\n
                                                                                  \n
                                                                                  \n
                                                                                  \n\t
                                                                                  \n\t
                                                                                    \n\t
                                                                                  • \n\t\t\t\u4e3b\u8fa6\u6a5f\u95dc\uff1a\n\t\t\t\n\t\t
                                                                                    • \n\t
                                                                                  • \n\t\t\t\u627f\u8fa6\u55ae\u4f4d\uff1a\n\t\t\t
                                                                                    • \n\t
                                                                                  \n\t
                                                                                    \n\t\t
                                                                                  • 112\u5168\u4e2d\u904b\u57f7\u59d4\u6703
                                                                                    • \n\t\t
                                                                                  • \u5730\u5740\uff1a30210\u65b0\u7af9\u7e23\u7af9\u5317\u5e02\u5149\u660e\u516d\u8def10\u865fB\u68df6\u6a13
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                                                                                  • \u8fa6\u516c\u6642\u9593\uff1a\u9031\u4e00\u81f3\u9031\u4e94\u4e0a\u53488:00\u81f3\u4e0b\u534816:00
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                                                                                  • \u96fb\u8a71\uff1a03-5518101\u5206\u6a5f2862
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                                                                                  • \u4fe1\u7bb1\uff1a20058475@hchg.gov.tw
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                                                                                  • \ufeff\u700f\u89bd\u4eba\u6578\uff1a3399541\n
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                                                                                  \n\n\t\n\n\t
                                                                                  \n\t
                                                                                  \n
                                                                                  \n\n
                                                                                  \n
                                                                                  \n
                                                                                  \n\n\n\n \n\n"} diff --git a/bench/run.py b/bench/run.py index e82a5346..f7067d79 100644 --- a/bench/run.py +++ b/bench/run.py @@ -61,6 +61,8 @@ def main(): url = files[fileName]['url'] origin_filepath = files[fileName]['origin_filepath'] groundtruth_filepath = files[fileName]['groundtruth_filepath'] + layout_type = files[fileName].get('layout_type', '') + print(layout_type) html = reader.read(f'{root}/data/{origin_filepath}').decode('utf-8') groundtruth = json.loads(reader.read(f'{root}/data/{groundtruth_filepath}').decode('utf-8')) statics_gt.merge_statics(groundtruth.get('statics', {})) @@ -77,7 +79,7 @@ def main(): print(pipelineConfigPath) print(pipeline_data_path) print(f'{root}/data/{origin_filepath}') - output, content_list, main_html, statics = eval_ours_extract_html(pipelineConfigPath, pipeline_data_path, f'{root}/data/{origin_filepath}') + output, content_list, main_html, statics = eval_ours_extract_html(pipelineConfigPath, pipeline_data_path, f'{root}/data/{origin_filepath}', layout_type) out['content_list'] = content_list out['main_html'] = main_html out['statics'] = statics diff --git a/docs/llm_web_kit/model/lang_id.md b/docs/llm_web_kit/model/lang_id.md index 5459bb86..f8492313 100644 --- a/docs/llm_web_kit/model/lang_id.md +++ b/docs/llm_web_kit/model/lang_id.md @@ -47,17 +47,15 @@ from llm_web_kit.model.lang_id import * text = 'hello world, this is a test. the language is english' print(update_language_by_str(text)) #{'language': 'en','language_details': 'eng_Latn'} -print(decide_lang_by_str(text)) -#en -print(decide_lang_by_str_v218(text)) -#eng_Latn ``` ## 运行时间 -总共有 2099 条数据 +使用单cpu进行推理 +共有 2099 条数据 总 token 数: 379375 平均 token 数: 180.74 -载入数据时间: 0.02 秒 -处理函数时间: 0.02 秒 -总时间: 0.04 秒 +载入数据时间: 0.0214 秒 +语言识别时间: 2.4313 秒 +总时间: 2.4527 秒 +处理速度: 863.33 条/秒 diff --git a/llm_web_kit/config/cfg_reader.py b/llm_web_kit/config/cfg_reader.py index 652eaad0..5b648f76 100644 --- a/llm_web_kit/config/cfg_reader.py +++ b/llm_web_kit/config/cfg_reader.py @@ -2,15 +2,22 @@ import commentjson as json +from llm_web_kit.exception.exception import ModelResourceException + def load_config() -> dict: - """_summary_ + """Load the configuration file for the web kit. First try to read the + configuration file from the environment variable LLM_WEB_KIT_CFG_PATH. If + the environment variable is not set, use the default configuration file + path ~/.llm-web-kit.jsonc. If the configuration file does not exist, raise + an exception. - Args: - config_file (_type_): _description_ + Raises: + ModelResourceException: LLM_WEB_KIT_CFG_PATH points to a non-exist file + ModelResourceException: cfg_path does not exist Returns: - _type_: _description_ + config(dict): The configuration dictionary """ # 首先从环境变量LLM_WEB_KIT_CFG_PATH 读取配置文件的位置 # 如果没有配置，就使用默认的配置文件位置 @@ -19,12 +26,15 @@ def load_config() -> dict: if env_cfg_path: cfg_path = env_cfg_path if not os.path.exists(cfg_path): - raise FileNotFoundError(f'environment variable LLM_WEB_KIT_CFG_PATH points to a non-exist file: {cfg_path}') + raise ModelResourceException( + f'environment variable LLM_WEB_KIT_CFG_PATH points to a non-exist file: {cfg_path}' + ) else: cfg_path = os.path.expanduser('~/.llm-web-kit.jsonc') if not os.path.exists(cfg_path): - raise FileNotFoundError( - f'{cfg_path} does not exist, please create one or set environment variable LLM_WEB_KIT_CFG_PATH to a valid file path') + raise ModelResourceException( + f'{cfg_path} does not exist, please create one or set environment variable LLM_WEB_KIT_CFG_PATH to a valid file path' + ) # 读取配置文件 with open(cfg_path, 'r', encoding='utf-8') as f: diff --git a/llm_web_kit/extractor/html/magic_html/config.py b/llm_web_kit/extractor/html/magic_html/config.py index aef945ba..010726d2 100644 --- a/llm_web_kit/extractor/html/magic_html/config.py +++ b/llm_web_kit/extractor/html/magic_html/config.py @@ -109,6 +109,7 @@ '//div[starts-with(@class, "advert") or starts-with(@name, "advert") or starts-with(@id, "advert")]', '//div[contains(@style, "display: none")]', '//div[contains(@style, "display:none")]', + '//*[contains(@class, "d-none")]', # 任意标签，class包含d-none，如这类网站https://stats.stackexchange.com/questions/11544/testing-for-stability-in-a-time-series/11750 # '//head//script' ] diff --git a/llm_web_kit/extractor/html/recognizer/cc_math/common.py b/llm_web_kit/extractor/html/recognizer/cc_math/common.py index 87a5c1ca..f98e4cd8 100644 --- a/llm_web_kit/extractor/html/recognizer/cc_math/common.py +++ b/llm_web_kit/extractor/html/recognizer/cc_math/common.py @@ -85,11 +85,13 @@ class MATH_TYPE_PATTERN: latex_config = { MATH_TYPE_PATTERN.INLINEMATH: [ ['$', '$'], - ['\\(', '\\)'] + ['\\(', '\\)'], + ['[itex]', '[/itex]'], # 这个网站自定义的分割，https://www.physicsforums.com/threads/turning-to-a-single-logarithm-then-simply.269419/ ], MATH_TYPE_PATTERN.DISPLAYMATH: [ ['\\[', '\\]'], ['$$', '$$'], + ['[tex]', '[/tex]'], # 这个网站自定义的分割，https://www.physicsforums.com/threads/turning-to-a-single-logarithm-then-simply.269419/ ['\\begin{equation}', '\\end{equation}'], ['\\begin{align}', '\\end{align}'], ['\\begin{alignat}', '\\end{alignat}'], @@ -420,6 +422,9 @@ def build_cc_exception_tag(self, text, math_type, math_render) -> str: print(cm.get_equation_type('a≠0')) print(cm.get_equation_type('a≠0')) print(cm.get_equation_type('

                                                                                  这是p的text

                                                                                  ')) + # print(cm.get_equation_type(r'

                                                                                  [tex]\frac{1}{4} Log(x-1)=Log((x-1)^{1\over{4}})= Log(\sqrt[4]{x-1})[/tex]

                                                                                  ')) + # print(cm.get_equation_type(r'

                                                                                  abc [itex]x^2[/itex] abc

                                                                                  ')) + # print(cm.get_equation_type(r'

                                                                                  abc [itex]x^2 abc

                                                                                  ')) print(cm.get_equation_type(r'

                                                                                  \begin{align} a^2+b=c\end{align}

                                                                                  ')) print(cm.get_equation_type(r'

                                                                                  \begin{abc} a^2+b=c\end{abc}

                                                                                  ')) print(cm.wrap_math(r'{\displaystyle \operatorname {Var} (X)=\operatorname {E} \left[(X-\mu)^{2}\right].}')) diff --git a/llm_web_kit/model/lang_id.py b/llm_web_kit/model/lang_id.py index 2f8c4d9f..a2e898ce 100644 --- a/llm_web_kit/model/lang_id.py +++ b/llm_web_kit/model/lang_id.py @@ -1,6 +1,6 @@ import os import re -from typing import Tuple +from typing import Dict, Tuple import fasttext @@ -196,7 +196,7 @@ def detect_latex_env(content_str: str) -> bool: return latex_env_pattern.search(content_str) is not None -def decide_language_func(content_str: str, lang_detect: LanguageIdentification) -> str: +def decide_language_func(content_str: str, lang_detect: LanguageIdentification) -> Dict[str, str]: """Decide language based on the content string. This function will truncate the content string if it is too long. This function will return "empty" if the content string is empty. @@ -216,7 +216,7 @@ def decide_language_func(content_str: str, lang_detect: LanguageIdentification) lang_detect (LanguageIdentification): The language identification model Returns: - str: The final language label + dict: Dictionary containing 'language' and 'language_details' keys """ # truncate the content string if it is too long @@ -236,40 +236,35 @@ def decide_language_func(content_str: str, lang_detect: LanguageIdentification) # return "empty" if the content string is empty if len(content_str.strip()) == 0: - return 'empty' + return {'language': 'empty', 'language_details': None} - if lang_detect.version in ['176.bin', '218.bin']: - predictions, probabilities = lang_detect.predict(content_str) - result = decide_language_by_prob_v176(predictions, probabilities) - else: + if lang_detect.version not in LANG_ID_SUPPORTED_VERSIONS: raise ValueError(f'Unsupported version: {lang_detect.version}. Supported versions: {LANG_ID_SUPPORTED_VERSIONS}') - return result + predictions, probabilities = lang_detect.predict(content_str) + language = decide_language_by_prob_v176(predictions, probabilities) -def decide_lang_by_str(content_str: str, model_path: str = None) -> str: - """Decide language based on the content string, based on - decide_language_func.""" - lang_detect = get_singleton_lang_detect(model_path) + language_details = None + if lang_detect.version == '218.bin': + first_pred = predictions[0] + match = re.match(r'^__label__([a-z]+)_[A-Za-z]+$', first_pred) + if match: + lang_code = match.group(1) + else: + lang_code = first_pred.replace('__label__', '').split('_')[0] + language_details = lang_code - return decide_language_func(content_str, lang_detect) + return { + 'language': language, + 'language_details': language_details + } -def decide_lang_by_str_v218(content_str: str, model_path: str = None) -> str: - """Decide language based on the content string, displayed in the format of - the fasttext218 model.""" +def update_language_by_str(content_str: str, model_path: str = None) -> Dict[str, str]: + """Decide language based on the content string and return a dictionary with + language and details.""" lang_detect = get_singleton_lang_detect(model_path) - if lang_detect.version == '176.bin': - return None - else: - return lang_detect.predict(content_str)[0][0].replace('__label__', '') - - -def update_language_by_str(content_str: str, model_path: str = None) -> str: - """Decide language based on the content string.""" - return { - 'language': decide_lang_by_str(content_str, model_path), - 'language_details': decide_lang_by_str_v218(content_str, model_path) - } + return decide_language_func(content_str, lang_detect) if __name__ == '__main__': diff --git a/llm_web_kit/model/resource_utils/download_assets.py b/llm_web_kit/model/resource_utils/download_assets.py index efa07619..5e5db168 100644 --- a/llm_web_kit/model/resource_utils/download_assets.py +++ b/llm_web_kit/model/resource_utils/download_assets.py @@ -10,7 +10,7 @@ from tqdm import tqdm from llm_web_kit.config.cfg_reader import load_config -from llm_web_kit.exception.exception import ModelInputException +from llm_web_kit.exception.exception import ModelResourceException from llm_web_kit.libs.logger import mylogger as logger from llm_web_kit.model.resource_utils.boto3_ext import (get_s3_client, is_s3_path, @@ -117,14 +117,18 @@ def __del__(self): class FileLock: """基于文件锁的上下文管理器（跨平台兼容版）""" - def __init__(self, lock_path: str, timeout: float = 300): + def __init__(self, lock_path: str, check_callback=None, timeout: float = 300): self.lock_path = lock_path + self.check_callback = check_callback self.timeout = timeout self._fd = None def __enter__(self): start_time = time.time() while True: + if self.check_callback: + if self.check_callback(): + return True try: # 原子性创建锁文件（O_EXCL标志是关键） self._fd = os.open( @@ -166,7 +170,7 @@ def verify_file_checksum( ) -> bool: """校验文件哈希值.""" if not sum([bool(md5_sum), bool(sha256_sum)]) == 1: - raise ModelInputException( + raise ModelResourceException( 'Exactly one of md5_sum or sha256_sum must be provided' ) @@ -254,20 +258,29 @@ def download_auto_file( """线程安全的文件下载函数""" lock_path = f'{target_path}.lock' - with FileLock(lock_path, timeout=lock_timeout): - # 二次检查（其他进程可能已经完成下载） - if os.path.exists(target_path): - if verify_file_checksum(target_path, md5_sum, sha256_sum): - logger.info(f'File already exists with valid checksum: {target_path}') - return target_path + def check_callback(): + return verify_file_checksum(target_path, md5_sum, sha256_sum) - if not exist_ok: - raise FileExistsError( - f'File exists with invalid checksum: {target_path}' - ) + if os.path.exists(target_path): + if not exist_ok: + raise ModelResourceException( + f'File exists with invalid checksum: {target_path}' + ) + + if verify_file_checksum(target_path, md5_sum, sha256_sum): + logger.info(f'File already exists with valid checksum: {target_path}') + return target_path + else: logger.warning(f'Removing invalid file: {target_path}') try_remove(target_path) + with FileLock(lock_path, check_callback, timeout=lock_timeout) as lock: + if lock is True: + logger.info( + f'File already exists with valid checksum: {target_path} while waiting' + ) + return target_path + # 创建连接 conn_cls = S3Connection if is_s3_path(resource_path) else HttpConnection conn = conn_cls(resource_path) diff --git a/llm_web_kit/model/resource_utils/unzip_ext.py b/llm_web_kit/model/resource_utils/unzip_ext.py index ca1a70b2..1819c358 100644 --- a/llm_web_kit/model/resource_utils/unzip_ext.py +++ b/llm_web_kit/model/resource_utils/unzip_ext.py @@ -4,7 +4,10 @@ import zipfile from typing import Optional -from llm_web_kit.model.resource_utils.download_assets import FileLock +from llm_web_kit.exception.exception import ModelResourceException +from llm_web_kit.libs.logger import mylogger as logger +from llm_web_kit.model.resource_utils.download_assets import (FileLock, + try_remove) def get_unzip_dir(zip_path: str) -> str: @@ -23,11 +26,32 @@ def get_unzip_dir(zip_path: str) -> str: return os.path.join(zip_dir, base_name + '_unzip') +def check_zip_file(zip_ref: zipfile.ZipFile, target_dir: str) -> bool: + """Check if the zip file is correctly unzipped to the target directory. + + Args: + zip_ref (zipfile.ZipFile): The zip file object. + target_dir (str): The target directory. + + Returns: + bool: True if the zip file is correctly unzipped to the target directory, False otherwise. + """ + + zip_info_list = [info for info in zip_ref.infolist() if not info.is_dir()] + for info in zip_info_list: + file_path = os.path.join(target_dir, info.filename) + if not os.path.exists(file_path): + return False + if os.path.getsize(file_path) != info.file_size: + return False + return True + + def unzip_local_file( zip_path: str, target_dir: str, password: Optional[str] = None, - exist_ok: bool = False, + exist_ok: bool = True, lock_timeout: float = 300, ) -> str: """Unzip a zip file to a target directory. @@ -40,20 +64,45 @@ def unzip_local_file( If False, raise an exception if the target directory already exists. Defaults to False. Raises: - Exception: If the target directory already exists and exist_ok is False. + ModelResourceException: If the zip file does not exist. + ModelResourceException: If the target directory already exists and exist_ok is False Returns: str: The path to the target directory. """ lock_path = f'{zip_path}.lock' - with FileLock(lock_path, timeout=lock_timeout): + + if not os.path.exists(zip_path): + logger.error(f'zip file {zip_path} does not exist') + raise ModelResourceException(f'zip file {zip_path} does not exist') + + def check_zip(): + with zipfile.ZipFile(zip_path, 'r') as zip_ref: + if password: + zip_ref.setpassword(password.encode()) + return check_zip_file(zip_ref, target_dir) + + if os.path.exists(target_dir): + if not exist_ok: + raise ModelResourceException(f'Target directory {target_dir} already exists') + + if check_zip(): + logger.info(f'zip file {zip_path} is already unzipped to {target_dir}') + return target_dir + else: + logger.warning( + f'zip file {zip_path} is not correctly unzipped to {target_dir}, retry to unzip' + ) + try_remove(target_dir) + + with FileLock(lock_path, check_zip, timeout=lock_timeout) as lock: + if lock is True: + logger.info( + f'zip file {zip_path} is already unzipped to {target_dir} while waiting' + ) + return target_dir # ensure target directory not exists - if os.path.exists(target_dir): - if exist_ok: - shutil.rmtree(target_dir) - else: - raise Exception(f'Target directory {target_dir} already exists') # 创建临时解压目录 with tempfile.TemporaryDirectory() as temp_dir: diff --git a/tests/llm_web_kit/extractor/html/recognizer/test_math.py b/tests/llm_web_kit/extractor/html/recognizer/test_math.py index f9499832..985306ee 100644 --- a/tests/llm_web_kit/extractor/html/recognizer/test_math.py +++ b/tests/llm_web_kit/extractor/html/recognizer/test_math.py @@ -219,6 +219,14 @@ { 'input': r'

                                                                                  $$x=5$$,$x=6$

                                                                                  ', 'expected': [('ccmath-interline', 'latex'), ('ccmath-inline', 'latex')] + }, + { + 'input': r'

                                                                                  [tex]\frac{1}{4} Log(x-1)=Log((x-1)^{1\over{4}})= Log(\sqrt[4]{x-1})[/tex]

                                                                                  ', + 'expected': [('ccmath-interline', 'latex')] + }, + { + 'input': r'

                                                                                  abc [itex]x^2[/itex] abc

                                                                                  ', + 'expected': [('ccmath-inline', 'latex')] } ] diff --git a/tests/llm_web_kit/model/resource_utils/test_download_assets.py b/tests/llm_web_kit/model/resource_utils/test_download_assets.py index a3c40d5c..f686ae43 100644 --- a/tests/llm_web_kit/model/resource_utils/test_download_assets.py +++ b/tests/llm_web_kit/model/resource_utils/test_download_assets.py @@ -6,7 +6,7 @@ from typing import Tuple from unittest.mock import MagicMock, call, mock_open, patch -from llm_web_kit.exception.exception import ModelInputException +from llm_web_kit.exception.exception import ModelResourceException from llm_web_kit.model.resource_utils.download_assets import ( FileLock, HttpConnection, S3Connection, calc_file_md5, calc_file_sha256, decide_cache_dir, download_auto_file, download_to_temp, move_to_target, @@ -448,7 +448,7 @@ def test_file_not_exists_download_http( # ) -> bool: # """校验文件哈希值.""" # if not sum([bool(md5_sum), bool(sha256_sum)]) == 1: -# raise ModelInputException('Exactly one of md5_sum or sha256_sum must be provided') +# raise ModelResourceException('Exactly one of md5_sum or sha256_sum must be provided') # if md5_sum: # actual = calc_file_md5(file_path) @@ -482,8 +482,8 @@ def test_pass_two_value(self, mock_calc_file_sha256, mock_calc_file_md5): sha256_sum = 'sha256_sum' mock_calc_file_md5.return_value = md5_sum mock_calc_file_sha256.return_value = sha256_sum - # will raise ModelInputException - with self.assertRaises(ModelInputException): + # will raise ModelResourceException + with self.assertRaises(ModelResourceException): verify_file_checksum(file_path, md5_sum, sha256_sum) @patch('llm_web_kit.model.resource_utils.download_assets.calc_file_md5') @@ -492,8 +492,8 @@ def test_pass_two_None(self, mock_calc_file_sha256, mock_calc_file_md5): file_path = 'file_path' md5_sum = None sha256_sum = None - # will raise ModelInputException - with self.assertRaises(ModelInputException): + # will raise ModelResourceException + with self.assertRaises(ModelResourceException): verify_file_checksum(file_path, md5_sum, sha256_sum) @patch('llm_web_kit.model.resource_utils.download_assets.calc_file_md5') diff --git a/tests/llm_web_kit/model/resource_utils/test_unzip_ext.py b/tests/llm_web_kit/model/resource_utils/test_unzip_ext.py index 39267ab1..6d8d28bb 100644 --- a/tests/llm_web_kit/model/resource_utils/test_unzip_ext.py +++ b/tests/llm_web_kit/model/resource_utils/test_unzip_ext.py @@ -2,6 +2,7 @@ import tempfile import zipfile +from llm_web_kit.exception.exception import ModelResourceException from llm_web_kit.model.resource_utils.unzip_ext import (get_unzip_dir, unzip_local_file) @@ -35,5 +36,5 @@ def test_unzip_local_file(): assert f.read() == 'This is another test file' try: unzip_local_file(zip_path, target_dir, exist_ok=False) - except Exception as e: - assert str(e) == f'Target directory {target_dir} already exists' + except ModelResourceException as e: + assert e.custom_message == f'Target directory {target_dir} already exists' diff --git a/tests/llm_web_kit/model/test_lang_id.py b/tests/llm_web_kit/model/test_lang_id.py index 4542c9dd..3ff5ea5d 100644 --- a/tests/llm_web_kit/model/test_lang_id.py +++ b/tests/llm_web_kit/model/test_lang_id.py @@ -2,8 +2,6 @@ from unittest.mock import MagicMock, patch from llm_web_kit.model.lang_id import (LanguageIdentification, - decide_lang_by_str, - decide_lang_by_str_v218, decide_language_by_prob_v176, decide_language_func, detect_code_block, detect_inline_equation, @@ -126,25 +124,27 @@ def test_decide_language_func(): lang_detect = MagicMock() lang_detect.version = '176.bin' lang_detect.predict.return_value = (['__label__en', '__label__zh'], [0.6, 0.4]) - assert decide_language_func('test text', lang_detect) == 'en' + result = decide_language_func('test text', lang_detect) + assert result == {'language': 'en', 'language_details': None} + # Test for 218.bin version + lang_detect.version = '218.bin' + lang_detect.predict.return_value = (['__label__eng_Latn', '__label__zho_Hans'], [0.6, 0.4]) + result = decide_language_func('test text', lang_detect) + assert result == {'language': 'en', 'language_details': 'eng'} -def test_decide_lang_by_str(): - with patch('llm_web_kit.model.lang_id.get_singleton_lang_detect') as mock_get_singleton_lang_detect, patch( - 'llm_web_kit.model.lang_id.decide_language_func') as mock_decide_language_func: - mock_get_singleton_lang_detect.return_value = MagicMock() - mock_decide_language_func.return_value = 'en' - assert decide_lang_by_str('test text') == 'en' + # Test for empty string + result = decide_language_func('', lang_detect) + assert result == {'language': 'empty', 'language_details': None} def test_update_language_by_str(): - # 模拟 decide_lang_by_str 和 decide_lang_by_str_v218 的行为 - with patch('llm_web_kit.model.lang_id.decide_lang_by_str') as mock_decide_lang_by_str, \ - patch('llm_web_kit.model.lang_id.decide_lang_by_str_v218') as mock_decide_lang_by_str_v218: + with patch('llm_web_kit.model.lang_id.get_singleton_lang_detect') as mock_get_singleton_lang_detect, \ + patch('llm_web_kit.model.lang_id.decide_language_func') as mock_decide_language_func: # 设置模拟函数的返回值 - mock_decide_lang_by_str.return_value = 'en' - mock_decide_lang_by_str_v218.return_value = 'en_v218' + mock_get_singleton_lang_detect.return_value = MagicMock() + mock_decide_language_func.return_value = {'language': 'en', 'language_details': 'eng'} # 调用被测函数 result = update_language_by_str('test text') @@ -152,30 +152,7 @@ def test_update_language_by_str(): # 验证返回结果 expected_result = { 'language': 'en', - 'language_details': 'en_v218' + 'language_details': 'eng' } assert result == expected_result, f'Expected {expected_result}, but got {result}' print('Test passed!') - - -class TestDecideLangByStrV218(unittest.TestCase): - - @patch('llm_web_kit.model.lang_id.get_singleton_lang_detect') - def test_decide_lang_by_str_v218(self, mock_get_singleton_lang_detect): - mock_lang_detect = MagicMock() - mock_lang_detect.predict.return_value = [('__label__en', 0.8), ('__label__fr', 0.2)] - mock_get_singleton_lang_detect.return_value = mock_lang_detect - - content_str = 'This is an English text.' - result = decide_lang_by_str_v218(content_str, 'model_path') - self.assertEqual(result, 'en') - - @patch('llm_web_kit.model.lang_id.get_singleton_lang_detect') - def test_decide_lang_by_str_v218_custom_model_path(self, mock_get_singleton_lang_detect): - mock_lang_detect = MagicMock() - mock_lang_detect.predict.return_value = [('__label__es', 0.9), ('__label__de', 0.1)] - mock_get_singleton_lang_detect.return_value = mock_lang_detect - - content_str = 'Este es un texto en español.' - result = decide_lang_by_str_v218(content_str, 'custom_model_path') - self.assertEqual(result, 'es') diff --git a/tests/st/test_st.py b/tests/st/test_st.py index 54b07a77..74e8d2cd 100644 --- a/tests/st/test_st.py +++ b/tests/st/test_st.py @@ -79,14 +79,19 @@ def test_st_bench(self): # files结构是{'filename': {'url': '', 'filepath': ''}}，获取filepath for fileName in files: filepath = files[fileName]['origin_filepath'] + groundtruth_filepath = os.path.join(self.root, 'bench/data/', files[fileName]['groundtruth_filepath']) page_layout_type = files[fileName]['layout_type'] summary.total += 1 print(f'开始抽取:{filepath}...') - # TODO: code_5.html当前因代码有bug，导致抽取失败，先跳过 - if 'code_5.html' in filepath: - continue try: output, content_list, main_html, statics = eval_ours_extract_html(self.chainConfig, self.pipeline_data_path, f'{self.root}/bench/data/{filepath}', page_layout_type) + # 断言statics中的元素数量和groundtruth_filepath中的元素数量一致 + with open(groundtruth_filepath, 'r') as f: + groundtruth = json.loads(f.readline().strip()) + # 断言equation-interline, paragraph.equation-inline和list.equation-inline元素数一致 + self.assertEqual(statics.get('equation-interline'), groundtruth.get('statics', {}).get('equation-interline'), msg=f'{filepath}抽取equation-interline数量和groundtruth:{groundtruth_filepath}不一致') + self.assertEqual(statics.get('paragraph.equation-inline'), groundtruth.get('statics', {}).get('paragraph.equation-inline'), msg=f'{filepath}抽取paragraph.equation-inline数量和groundtruth:{groundtruth_filepath}不一致') + self.assertEqual(statics.get('list.equation-inline'), groundtruth.get('statics', {}).get('list.equation-inline'), msg=f'{filepath}抽取list.equation-inline数量和groundtruth:{groundtruth_filepath}不一致') except Exception as e: summary.error_summary['count'] += 1 detail.result_detail['error_result'].append(Error_Item( From 4efe5d437ed12a053805ab189ff14a6c66e3f8e5 Mon Sep 17 00:00:00 2001 From: qiujiantao Date: Thu, 6 Mar 2025 11:41:49 +0800 Subject: [PATCH 2/8] =?UTF-8?q?feat:=20=E5=A2=9E=E5=BC=BA=20S3=20=E8=B7=AF?= =?UTF-8?q?=E5=BE=84=E5=A4=84=E7=90=86=E5=92=8C=E9=94=99=E8=AF=AF=E5=A4=84?= =?UTF-8?q?=E7=90=86=EF=BC=8C=E6=B7=BB=E5=8A=A0=E8=AF=A6=E7=BB=86=E6=96=87?= =?UTF-8?q?=E6=A1=A3=E6=B3=A8=E9=87=8A?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- llm_web_kit/model/resource_utils/boto3_ext.py | 79 ++++++++++++++++--- .../model/resource_utils/test_boto3_ext.py | 50 ++++++------ 2 files changed, 91 insertions(+), 38 deletions(-) diff --git a/llm_web_kit/model/resource_utils/boto3_ext.py b/llm_web_kit/model/resource_utils/boto3_ext.py index 11341568..ac75fb8b 100644 --- a/llm_web_kit/model/resource_utils/boto3_ext.py +++ b/llm_web_kit/model/resource_utils/boto3_ext.py @@ -1,20 +1,37 @@ import re -from typing import Dict, List, Union +from typing import Dict, List, Tuple, Union import boto3 from botocore.config import Config from botocore.exceptions import ClientError from llm_web_kit.config.cfg_reader import load_config +from llm_web_kit.exception.exception import ModelResourceException __re_s3_path = re.compile('^s3://([^/]+)(?:/(.*))?$') def is_s3_path(path: str) -> bool: + """check a path is s3 path or not. + + Args: + path (str): path + + Returns: + bool: is s3 path or not + """ return path.startswith('s3://') -def is_s3_404_error(e: Exception): +def is_s3_404_error(e: Exception) -> bool: + """check if an exception is 404 error. + + Args: + e (Exception): exception + + Returns: + bool: is 404 error or not + """ if not isinstance(e, ClientError): return False flag_1 = e.response.get('Error', {}).get('Code') in ['404', 'NoSuchKey'] @@ -23,22 +40,35 @@ def is_s3_404_error(e: Exception): return any([flag_1, flag_2, flag_3]) -def split_s3_path(path: str): - """split bucket and key from path.""" +def split_s3_path(path: str) -> Tuple[str, str]: + """split bucket and key from path. + + Args: + path (str): s3 path + + Returns: + Tuple[str, str]: bucket and key + + Raises: + ModelResourceException: if path is not s3 path + """ + if not is_s3_path(path): + raise ModelResourceException(f'{path} is not a s3 path') m = __re_s3_path.match(path) if m is None: return '', '' return m.group(1), (m.group(2) or '') -def get_s3_config(path: str): +def get_s3_config(path: str) -> Dict: """Get s3 config for a given path by its bucket name from the config file. Args: path (str): s3 path Raises: - ValueError: if bucket not found in config + ModelResourceException: if bucket not in config + ModelResourceException: if path is not s3 path Returns: dict: s3 config @@ -48,10 +78,23 @@ def get_s3_config(path: str): if bucket in config_dict['s3']: return config_dict['s3'][bucket] else: - raise ValueError(f'bucket {bucket} not in config') + raise ModelResourceException(f'bucket {bucket} not in config') + + +def get_s3_client(path: Union[str, List[str]]) -> boto3.client: + """Get s3 client for a given path. + + Args: + path (Union[str, List[str]]): s3 path + + Returns: + boto3.client: s3 client + Raises: + ModelResourceException: if bucket not in config + ModelResourceException: if path is not s3 path + """ -def get_s3_client(path: Union[str, List[str]]): s3_config = get_s3_config(path) try: return boto3.client( @@ -61,10 +104,7 @@ def get_s3_client(path: Union[str, List[str]]): endpoint_url=s3_config['endpoint'], config=Config( s3={'addressing_style': s3_config.get('addressing_style', 'path')}, - retries={ - 'max_attempts': 8, - 'mode': 'standard' - }, + retries={'max_attempts': 8, 'mode': 'standard'}, connect_timeout=600, read_timeout=600, ), @@ -84,6 +124,21 @@ def get_s3_client(path: Union[str, List[str]]): def head_s3_object(client, path: str, raise_404=False) -> Union[Dict, None]: + """Get s3 object metadata. + + Args: + client (boto3.client): the s3 client + path (str): the s3 path + raise_404 (bool, optional): raise 404 error or not. Defaults to False. + + Returns: + Union[Dict, None]: s3 object metadata or None if not found + + Raises: + ClientError: if raise_404 is True and object not + ModelResourceException: if path is not s3 path + ModelResourceException: if bucket not in config + """ bucket, key = split_s3_path(path) try: resp = client.head_object(Bucket=bucket, Key=key) diff --git a/tests/llm_web_kit/model/resource_utils/test_boto3_ext.py b/tests/llm_web_kit/model/resource_utils/test_boto3_ext.py index 5be52f4a..5510dec0 100644 --- a/tests/llm_web_kit/model/resource_utils/test_boto3_ext.py +++ b/tests/llm_web_kit/model/resource_utils/test_boto3_ext.py @@ -3,6 +3,7 @@ import pytest from botocore.exceptions import ClientError +from llm_web_kit.exception.exception import ModelResourceException from llm_web_kit.model.resource_utils.boto3_ext import (get_s3_client, get_s3_config, head_s3_object, @@ -19,13 +20,8 @@ def test_is_s3_path(): def test_is_s3_404_error(): not_found_error = ClientError( error_response={ - 'Error': { - 'Code': '404', - 'Message': 'Not Found' - }, - 'ResponseMetadata': { - 'HTTPStatusCode': 404 - }, + 'Error': {'Code': '404', 'Message': 'Not Found'}, + 'ResponseMetadata': {'HTTPStatusCode': 404}, }, operation_name='test', ) @@ -33,13 +29,8 @@ def test_is_s3_404_error(): not_404_error = ClientError( error_response={ - 'Error': { - 'Code': '403', - 'Message': 'Forbidden' - }, - 'ResponseMetadata': { - 'HTTPStatusCode': 403 - }, + 'Error': {'Code': '403', 'Message': 'Forbidden'}, + 'ResponseMetadata': {'HTTPStatusCode': 403}, }, operation_name='test', ) @@ -54,20 +45,28 @@ def test_split_s3_path(): @patch('llm_web_kit.model.resource_utils.boto3_ext.load_config') def test_get_s3_config(get_config_mock): - get_config_mock.return_value = {'s3': {'bucket': {'ak': 'test_ak', 'sk': 'test_sk', 'endpoint': 'test_endpoint'}}} + get_config_mock.return_value = { + 's3': { + 'bucket': {'ak': 'test_ak', 'sk': 'test_sk', 'endpoint': 'test_endpoint'} + } + } assert get_s3_config('s3://bucket/key') == { 'ak': 'test_ak', 'sk': 'test_sk', 'endpoint': 'test_endpoint', } - with pytest.raises(ValueError): + with pytest.raises(ModelResourceException): get_s3_config('s3://nonexistent_bucket/key') @patch('llm_web_kit.model.resource_utils.boto3_ext.load_config') @patch('llm_web_kit.model.resource_utils.boto3_ext.boto3.client') def test_get_s3_client(boto3_client_mock, get_config_mock): - get_config_mock.return_value = {'s3': {'bucket': {'ak': 'test_ak', 'sk': 'test_sk', 'endpoint': 'test_endpoint'}}} + get_config_mock.return_value = { + 's3': { + 'bucket': {'ak': 'test_ak', 'sk': 'test_sk', 'endpoint': 'test_endpoint'} + } + } mock_client = MagicMock() boto3_client_mock.return_value = mock_client assert get_s3_client('s3://bucket/key') == mock_client @@ -78,19 +77,18 @@ def test_get_s3_client(boto3_client_mock, get_config_mock): def test_head_s3_object(boto3_client_mock, is_s3_404_error_mock): s3_client_mock = MagicMock() boto3_client_mock.return_value = s3_client_mock - s3_client_mock.head_object.return_value = {'ResponseMetadata': {'HTTPStatusCode': 200}} + s3_client_mock.head_object.return_value = { + 'ResponseMetadata': {'HTTPStatusCode': 200} + } - assert head_s3_object(s3_client_mock, 's3://bucket/key') == {'ResponseMetadata': {'HTTPStatusCode': 200}} + assert head_s3_object(s3_client_mock, 's3://bucket/key') == { + 'ResponseMetadata': {'HTTPStatusCode': 200} + } s3_client_mock.head_object.side_effect = ClientError( error_response={ - 'Error': { - 'Code': '404', - 'Message': 'Not Found' - }, - 'ResponseMetadata': { - 'HTTPStatusCode': 404 - }, + 'Error': {'Code': '404', 'Message': 'Not Found'}, + 'ResponseMetadata': {'HTTPStatusCode': 404}, }, operation_name='test', ) From 189e5a980a516664c6d82902b3a3ad7678387bf6 Mon Sep 17 00:00:00 2001 From: qiujiantao Date: Thu, 6 Mar 2025 13:27:29 +0800 Subject: [PATCH 3/8] =?UTF-8?q?feat:=20=E6=B7=BB=E5=8A=A0=20unzip=5Flocal?= =?UTF-8?q?=5Ffile=20=E5=87=BD=E6=95=B0=E7=9A=84=E9=94=99=E8=AF=AF?= =?UTF-8?q?=E5=A4=84=E7=90=86=EF=BC=8C=E5=A2=9E=E5=8A=A0=20zip=20=E6=96=87?= =?UTF-8?q?=E4=BB=B6=E6=A3=80=E6=9F=A5=E7=9A=84=E5=8D=95=E5=85=83=E6=B5=8B?= =?UTF-8?q?=E8=AF=95?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- llm_web_kit/model/resource_utils/unzip_ext.py | 4 +- tests/llm_web_kit/model/assets/zip_demo.zip | Bin 0 -> 848 bytes .../model/resource_utils/test_unzip_ext.py | 91 +++++++++++++++++- 3 files changed, 90 insertions(+), 5 deletions(-) create mode 100644 tests/llm_web_kit/model/assets/zip_demo.zip diff --git a/llm_web_kit/model/resource_utils/unzip_ext.py b/llm_web_kit/model/resource_utils/unzip_ext.py index 1819c358..7b48f967 100644 --- a/llm_web_kit/model/resource_utils/unzip_ext.py +++ b/llm_web_kit/model/resource_utils/unzip_ext.py @@ -84,7 +84,9 @@ def check_zip(): if os.path.exists(target_dir): if not exist_ok: - raise ModelResourceException(f'Target directory {target_dir} already exists') + raise ModelResourceException( + f'Target directory {target_dir} already exists' + ) if check_zip(): logger.info(f'zip file {zip_path} is already unzipped to {target_dir}') diff --git a/tests/llm_web_kit/model/assets/zip_demo.zip b/tests/llm_web_kit/model/assets/zip_demo.zip new file mode 100644 index 0000000000000000000000000000000000000000..62be30491dc895932d099bc84b365a3e9a6a0a3b GIT binary patch literal 848 zcmWIWW@h1H0D*(CX;EMXl;C8LVJJy0E{U(oEYJ@P;bdU0lscIX!lf1542&!C$r zL;#8r(&mvhY(OJGSP0n&kXeR$B^4$3%mORoLNyWDQG&=Orse0Pq!t;G;5Y(C8IoYs zjg*pbm_Ma3U1g{b@`@2L{xt&nmyt=18CNt&KwT}s@YWH;M2ZwvNTgsy2&!?YQ35fJ z5opkoMrUN>kRpgUb3u-Q0YL_2XHEqQ!<~tUCQ6NcfN3l!%7`--HNucXK^qttXs$(x WK3wLrg5sZn6$rNiO=txLDFXnsez8OV literal 0 HcmV?d00001 diff --git a/tests/llm_web_kit/model/resource_utils/test_unzip_ext.py b/tests/llm_web_kit/model/resource_utils/test_unzip_ext.py index 6d8d28bb..bf514d14 100644 --- a/tests/llm_web_kit/model/resource_utils/test_unzip_ext.py +++ b/tests/llm_web_kit/model/resource_utils/test_unzip_ext.py @@ -1,15 +1,98 @@ import os import tempfile import zipfile +from unittest import TestCase from llm_web_kit.exception.exception import ModelResourceException -from llm_web_kit.model.resource_utils.unzip_ext import (get_unzip_dir, +from llm_web_kit.model.resource_utils.unzip_ext import (check_zip_file, + get_unzip_dir, unzip_local_file) -def test_get_unzip_dir(): - assert get_unzip_dir('/path/to/test.zip') == '/path/to/test_unzip' - assert get_unzip_dir('/path/to/test') == '/path/to/test_unzip' +def get_assert_dir(): + file_path = os.path.abspath(__file__) + assert_dir = os.path.join(os.path.dirname(os.path.dirname(file_path)), 'assets') + return assert_dir + + +class TestGetUnzipDir(TestCase): + + def test_get_unzip_dir_case1(self): + assert get_unzip_dir('/path/to/test.zip') == '/path/to/test_unzip' + + def test_get_unzip_dir_case2(self): + assert get_unzip_dir('/path/to/test') == '/path/to/test_unzip' + + +class TestCheckZipFile(TestCase): + # # test_zip/ + # # ├── test1.txt "test1\n" + # # ├── folder1 + # # │ └── test2.txt "test2\n" + # # └── folder2 + + def get_zipfile(self): + # 创建一个临时文件夹 + zip_path = os.path.join(get_assert_dir(), 'zip_demo.zip') + zip_file = zipfile.ZipFile(zip_path, 'r') + return zip_file + + def test_check_zip_file_cese1(self): + zip_file = self.get_zipfile() + # # test_zip/ + # # ├── test1.txt + # # ├── folder1 + # # │ └── test2.txt + # # └── folder2 + + with tempfile.TemporaryDirectory() as temp_dir: + root_dir = os.path.join(temp_dir, 'test_zip') + os.makedirs(os.path.join(root_dir, 'test_zip')) + os.makedirs(os.path.join(root_dir, 'folder1')) + os.makedirs(os.path.join(root_dir, 'folder2')) + with open(os.path.join(root_dir, 'test1.txt'), 'w') as f: + f.write('test1\n') + with open(os.path.join(root_dir, 'folder1', 'test2.txt'), 'w') as f: + f.write('test2\n') + + assert check_zip_file(zip_file, temp_dir) is True + + def test_check_zip_file_cese2(self): + zip_file = self.get_zipfile() + with tempfile.TemporaryDirectory() as temp_dir: + root_dir = os.path.join(temp_dir, 'test_zip') + os.makedirs(os.path.join(root_dir, 'test_zip')) + os.makedirs(os.path.join(root_dir, 'folder1')) + with open(os.path.join(root_dir, 'test1.txt'), 'w') as f: + f.write('test1\n') + with open(os.path.join(root_dir, 'folder1', 'test2.txt'), 'w') as f: + f.write('test2\n') + + assert check_zip_file(zip_file, temp_dir) is True + + def test_check_zip_file_cese3(self): + zip_file = self.get_zipfile() + with tempfile.TemporaryDirectory() as temp_dir: + root_dir = os.path.join(temp_dir, 'test_zip') + os.makedirs(os.path.join(root_dir, 'test_zip')) + os.makedirs(os.path.join(root_dir, 'folder1')) + with open(os.path.join(root_dir, 'folder1', 'test2.txt'), 'w') as f: + f.write('test2\n') + + assert check_zip_file(zip_file, temp_dir) is False + + def test_check_zip_file_cese4(self): + zip_file = self.get_zipfile() + with tempfile.TemporaryDirectory() as temp_dir: + root_dir = os.path.join(temp_dir, 'test_zip') + os.makedirs(os.path.join(root_dir, 'test_zip')) + os.makedirs(os.path.join(root_dir, 'folder1')) + with open(os.path.join(root_dir, 'test1.txt'), 'w') as f: + f.write('test1\n') + with open(os.path.join(root_dir, 'folder1', 'test2.txt'), 'w') as f: + f.write('test123\n') + + assert check_zip_file(zip_file, temp_dir) is False def test_unzip_local_file(): From 9996a68b93b7e5734665456d85f3dcc15f1d34a9 Mon Sep 17 00:00:00 2001 From: qiujiantao Date: Thu, 6 Mar 2025 13:29:56 +0800 Subject: [PATCH 4/8] =?UTF-8?q?feat:=20=E4=BD=BF=E7=94=A8=20ModelResourceE?= =?UTF-8?q?xception=20=E6=9B=BF=E6=8D=A2=E5=BC=82=E5=B8=B8=E5=A4=84?= =?UTF-8?q?=E7=90=86=EF=BC=8C=E5=A2=9E=E5=BC=BA=E8=B5=84=E6=BA=90=E7=AE=A1?= =?UTF-8?q?=E7=90=86=E5=99=A8=E7=9A=84=E9=94=99=E8=AF=AF=E6=8F=90=E7=A4=BA?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../resource_utils/singleton_resource_manager.py | 11 ++++++++--- .../resource_utils/test_singleton_resource_manager.py | 11 ++++++----- 2 files changed, 14 insertions(+), 8 deletions(-) diff --git a/llm_web_kit/model/resource_utils/singleton_resource_manager.py b/llm_web_kit/model/resource_utils/singleton_resource_manager.py index 24849131..84ddcc53 100644 --- a/llm_web_kit/model/resource_utils/singleton_resource_manager.py +++ b/llm_web_kit/model/resource_utils/singleton_resource_manager.py @@ -1,3 +1,6 @@ +from llm_web_kit.exception.exception import ModelResourceException + + class SingletonResourceManager: def __init__(self): @@ -8,9 +11,11 @@ def has_name(self, name): def set_resource(self, name: str, resource): if not isinstance(name, str): - raise TypeError('name should be a string') + raise ModelResourceException( + f'Name should be a string, but got {type(name)}' + ) if name in self.resources: - raise AssertionError(f'Resource {name} already exists') + raise ModelResourceException(f'Resource {name} already exists') self.resources[name] = resource @@ -18,7 +23,7 @@ def get_resource(self, name): if name in self.resources: return self.resources[name] else: - raise Exception(f'Resource {name} does not exist') + raise ModelResourceException(f'Resource {name} does not exist') def release_resource(self, name): if name in self.resources: diff --git a/tests/llm_web_kit/model/resource_utils/test_singleton_resource_manager.py b/tests/llm_web_kit/model/resource_utils/test_singleton_resource_manager.py index 8e345495..22a9282f 100644 --- a/tests/llm_web_kit/model/resource_utils/test_singleton_resource_manager.py +++ b/tests/llm_web_kit/model/resource_utils/test_singleton_resource_manager.py @@ -1,5 +1,6 @@ import pytest +from llm_web_kit.exception.exception import ModelResourceException from llm_web_kit.model.resource_utils.singleton_resource_manager import \ SingletonResourceManager @@ -27,15 +28,15 @@ def test_set_resource(self): assert self.manager.get_resource('test') == 'resource' # "test" should not be set again - with pytest.raises(AssertionError): + with pytest.raises(ModelResourceException): self.manager.set_resource('test', 'resource') # name should be a string - with pytest.raises(TypeError): + with pytest.raises(ModelResourceException): self.manager.set_resource(1, 'resource') # resource should not be None - with pytest.raises(TypeError): + with pytest.raises(ModelResourceException): self.manager.set_resource(None, 'resource') def test_get_resource(self): @@ -43,7 +44,7 @@ def test_get_resource(self): # "test" should exist after setting and the resource should be "resource" assert self.manager.get_resource('test') == 'resource' # Exception should be raised if the resource does not exist - with pytest.raises(Exception): + with pytest.raises(ModelResourceException): self.manager.get_resource('test1') def test_release_resource(self): @@ -52,7 +53,7 @@ def test_release_resource(self): # "test" should not exist after releasing assert not self.manager.has_name('test') # Exception should be raised if the resource does not exist - with pytest.raises(Exception): + with pytest.raises(ModelResourceException): self.manager.get_resource('test') # Should not raise exception if the resource does not exist self.manager.release_resource('test') From 91ac58223750aa7a6a93bb6bcaaa30aba5d872ad Mon Sep 17 00:00:00 2001 From: qiujiantao Date: Thu, 6 Mar 2025 13:57:36 +0800 Subject: [PATCH 5/8] =?UTF-8?q?feat:=20=E6=B7=BB=E5=8A=A0=E8=B7=A8?= =?UTF-8?q?=E5=B9=B3=E5=8F=B0=E6=96=87=E4=BB=B6=E9=94=81=E7=9A=84=E4=B8=8A?= =?UTF-8?q?=E4=B8=8B=E6=96=87=E7=AE=A1=E7=90=86=E5=99=A8=EF=BC=8C=E5=A2=9E?= =?UTF-8?q?=E5=BC=BA=E6=96=87=E4=BB=B6=E6=93=8D=E4=BD=9C=E7=9A=84=E5=AE=89?= =?UTF-8?q?=E5=85=A8=E6=80=A7=E5=92=8C=E7=A8=B3=E5=AE=9A=E6=80=A7?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../model/resource_utils/download_assets.py | 62 +------- llm_web_kit/model/resource_utils/unzip_ext.py | 3 +- llm_web_kit/model/resource_utils/utils.py | 62 ++++++++ .../resource_utils/test_download_assets.py | 141 +---------------- .../model/resource_utils/test_utils.py | 142 ++++++++++++++++++ 5 files changed, 208 insertions(+), 202 deletions(-) create mode 100644 llm_web_kit/model/resource_utils/utils.py create mode 100644 tests/llm_web_kit/model/resource_utils/test_utils.py diff --git a/llm_web_kit/model/resource_utils/download_assets.py b/llm_web_kit/model/resource_utils/download_assets.py index 5e5db168..2821b35a 100644 --- a/llm_web_kit/model/resource_utils/download_assets.py +++ b/llm_web_kit/model/resource_utils/download_assets.py @@ -1,9 +1,7 @@ -import errno import hashlib import os import shutil import tempfile -import time from typing import Iterable, Optional import requests @@ -15,14 +13,7 @@ from llm_web_kit.model.resource_utils.boto3_ext import (get_s3_client, is_s3_path, split_s3_path) - - -def try_remove(path: str): - """Attempt to remove a file, but ignore any exceptions that occur.""" - try: - os.remove(path) - except Exception: - pass +from llm_web_kit.model.resource_utils.utils import FileLock, try_remove def decide_cache_dir(): @@ -114,57 +105,6 @@ def __del__(self): self.response.close() -class FileLock: - """基于文件锁的上下文管理器（跨平台兼容版）""" - - def __init__(self, lock_path: str, check_callback=None, timeout: float = 300): - self.lock_path = lock_path - self.check_callback = check_callback - self.timeout = timeout - self._fd = None - - def __enter__(self): - start_time = time.time() - while True: - if self.check_callback: - if self.check_callback(): - return True - try: - # 原子性创建锁文件（O_EXCL标志是关键） - self._fd = os.open( - self.lock_path, os.O_CREAT | os.O_EXCL | os.O_WRONLY, 0o644 - ) - # 写入进程信息和时间戳 - with os.fdopen(self._fd, 'w') as f: - f.write(f'{os.getpid()}\n{time.time()}') - return self - except OSError as e: - if e.errno != errno.EEXIST: - raise - - # 检查锁是否过期 - try: - with open(self.lock_path, 'r') as f: - pid, timestamp = f.read().split('\n')[:2] - if time.time() - float(timestamp) > self.timeout: - os.remove(self.lock_path) - except (FileNotFoundError, ValueError): - pass - - if time.time() - start_time > self.timeout: - raise TimeoutError(f'Could not acquire lock after {self.timeout}s') - time.sleep(0.1) - - def __exit__(self, exc_type, exc_val, exc_tb): - try: - if self._fd: - os.close(self._fd) - except OSError: - pass - finally: - try_remove(self.lock_path) - - def verify_file_checksum( file_path: str, md5_sum: Optional[str] = None, sha256_sum: Optional[str] = None ) -> bool: diff --git a/llm_web_kit/model/resource_utils/unzip_ext.py b/llm_web_kit/model/resource_utils/unzip_ext.py index 7b48f967..8b884a85 100644 --- a/llm_web_kit/model/resource_utils/unzip_ext.py +++ b/llm_web_kit/model/resource_utils/unzip_ext.py @@ -6,8 +6,7 @@ from llm_web_kit.exception.exception import ModelResourceException from llm_web_kit.libs.logger import mylogger as logger -from llm_web_kit.model.resource_utils.download_assets import (FileLock, - try_remove) +from llm_web_kit.model.resource_utils.utils import FileLock, try_remove def get_unzip_dir(zip_path: str) -> str: diff --git a/llm_web_kit/model/resource_utils/utils.py b/llm_web_kit/model/resource_utils/utils.py new file mode 100644 index 00000000..f2e142e9 --- /dev/null +++ b/llm_web_kit/model/resource_utils/utils.py @@ -0,0 +1,62 @@ +import errno +import os +import time + + +def try_remove(path: str): + """Attempt to remove a file, but ignore any exceptions that occur.""" + try: + os.remove(path) + except Exception: + pass + + +class FileLock: + """基于文件锁的上下文管理器（跨平台兼容版）""" + + def __init__(self, lock_path: str, check_callback=None, timeout: float = 300): + self.lock_path = lock_path + self.check_callback = check_callback + self.timeout = timeout + self._fd = None + + def __enter__(self): + start_time = time.time() + while True: + if self.check_callback: + if self.check_callback(): + return True + try: + # 原子性创建锁文件（O_EXCL标志是关键） + self._fd = os.open( + self.lock_path, os.O_CREAT | os.O_EXCL | os.O_WRONLY, 0o644 + ) + # 写入进程信息和时间戳 + with os.fdopen(self._fd, 'w') as f: + f.write(f'{os.getpid()}\n{time.time()}') + return self + except OSError as e: + if e.errno != errno.EEXIST: + raise + + # 检查锁是否过期 + try: + with open(self.lock_path, 'r') as f: + pid, timestamp = f.read().split('\n')[:2] + if time.time() - float(timestamp) > self.timeout: + os.remove(self.lock_path) + except (FileNotFoundError, ValueError): + pass + + if time.time() - start_time > self.timeout: + raise TimeoutError(f'Could not acquire lock after {self.timeout}s') + time.sleep(0.1) + + def __exit__(self, exc_type, exc_val, exc_tb): + try: + if self._fd: + os.close(self._fd) + except OSError: + pass + finally: + try_remove(self.lock_path) diff --git a/tests/llm_web_kit/model/resource_utils/test_download_assets.py b/tests/llm_web_kit/model/resource_utils/test_download_assets.py index f686ae43..7b4276c8 100644 --- a/tests/llm_web_kit/model/resource_utils/test_download_assets.py +++ b/tests/llm_web_kit/model/resource_utils/test_download_assets.py @@ -1,4 +1,3 @@ -import errno import io import os import tempfile @@ -8,23 +7,9 @@ from llm_web_kit.exception.exception import ModelResourceException from llm_web_kit.model.resource_utils.download_assets import ( - FileLock, HttpConnection, S3Connection, calc_file_md5, calc_file_sha256, + HttpConnection, S3Connection, calc_file_md5, calc_file_sha256, decide_cache_dir, download_auto_file, download_to_temp, move_to_target, - try_remove, verify_file_checksum) - - -class Test_try_remove: - - @patch('os.remove') - def test_remove(self, removeMock): - try_remove('path') - removeMock.assert_called_once_with('path') - - @patch('os.remove') - def test_remove_exception(self, removeMock): - removeMock.side_effect = Exception - try_remove('path') - removeMock.assert_called_once_with('path') + verify_file_checksum) class Test_decide_cache_dir: @@ -141,128 +126,6 @@ def test_HttpConnection(requests_get_mock): assert b''.join(conn.read_stream()) == test_data -class TestFileLock(unittest.TestCase): - - def setUp(self): - self.lock_path = 'test.lock' - - @patch('os.fdopen') - @patch('os.open') - @patch('os.close') - @patch('os.remove') - def test_acquire_and_release_lock( - self, mock_remove, mock_close, mock_open, mock_os_fdopen - ): - # 模拟成功获取锁 - mock_open.return_value = 123 # 假设文件描述符为123 - # 模拟文件描述符 - mock_fd = MagicMock() - mock_fd.__enter__.return_value = mock_fd - mock_fd.write.return_value = None - mock_os_fdopen.return_value = mock_fd - - with FileLock(self.lock_path): - mock_open.assert_called_once_with( - self.lock_path, os.O_CREAT | os.O_EXCL | os.O_WRONLY, 0o644 - ) - mock_close.assert_called_once_with(123) - mock_remove.assert_called_once_with(self.lock_path) - - @patch('os.fdopen') - @patch('os.open') - @patch('builtins.open', new_callable=mock_open, read_data='1234\n100') - @patch('time.time') - @patch('os.remove') - def test_remove_stale_lock( - self, mock_remove, mock_time, mock_file_open, mock_os_open, mock_os_fdopen - ): - # 第一次尝试创建锁文件失败（锁已存在） - mock_os_open.side_effect = [ - OSError(errno.EEXIST, 'File exists'), - 123, # 第二次成功 - ] - - # 模拟文件描述符 - mock_fd = MagicMock() - mock_fd.__enter__.return_value = mock_fd - mock_fd.write.return_value = None - mock_os_fdopen.return_value = mock_fd - - # 当前时间设置为超过超时时间（timeout=300） - mock_time.return_value = 401 # 100 + 300 + 1 - - with FileLock(self.lock_path, timeout=300): - mock_remove.assert_called_once_with(self.lock_path) - mock_os_open.assert_any_call( - self.lock_path, os.O_CREAT | os.O_EXCL | os.O_WRONLY, 0o644 - ) - - @patch('os.open') - @patch('time.time') - def test_timeout_acquiring_lock(self, mock_time, mock_os_open): - # 总是返回EEXIST错误 - mock_os_open.side_effect = OSError(errno.EEXIST, 'File exists') - # 时间累计超过超时时间 - start_time = 1000 - mock_time.side_effect = [ - start_time, - start_time + 301, - start_time + 302, - start_time + 303, - ] - - with self.assertRaises(TimeoutError): - with FileLock(self.lock_path, timeout=300): - pass - - @patch('os.open') - def test_other_os_error(self, mock_os_open): - # 模拟其他OS错误（如权限不足） - mock_os_open.side_effect = OSError(errno.EACCES, 'Permission denied') - with self.assertRaises(OSError): - with FileLock(self.lock_path): - pass - - @patch('os.close') - @patch('os.remove') - def test_cleanup_on_exit(self, mock_remove, mock_close): - - mock_close.side_effect = None - # 确保退出上下文时执行清理 - lock_path = 'test.lock' - lock = FileLock(lock_path) - lock._fd = 123 # 模拟已打开的文件描述符 - lock.__exit__('!!!!!!!!!!!!!!!!!!!!!!!!!!!!!', None, None) - mock_remove.assert_called_once_with(self.lock_path) - - @patch('os.remove') - def test_cleanup_failure_handled(self, mock_remove): - # 模拟删除锁文件时失败 - mock_remove.side_effect = OSError - lock = FileLock(self.lock_path) - lock._fd = 123 - # 不应抛出异常 - lock.__exit__(None, None, None) - - @patch('os.getpid') - @patch('time.time') - def test_lock_file_content(self, mock_time, mock_pid): - # 验证锁文件内容格式 - mock_pid.return_value = 9999 - mock_time.return_value = 123456.789 - - with patch('os.open') as mock_os_open: - mock_os_open.return_value = 123 - with patch('os.fdopen') as mock_fdopen: - # 模拟写入文件描述符 - mock_file = MagicMock() - mock_fdopen.return_value.__enter__.return_value = mock_file - - with FileLock(self.lock_path): - mock_fdopen.assert_called_once_with(123, 'w') - mock_file.write.assert_called_once_with('9999\n123456.789') - - class TestDownloadAutoFile(unittest.TestCase): @patch('llm_web_kit.model.resource_utils.download_assets.os.path.exists') diff --git a/tests/llm_web_kit/model/resource_utils/test_utils.py b/tests/llm_web_kit/model/resource_utils/test_utils.py new file mode 100644 index 00000000..542b4904 --- /dev/null +++ b/tests/llm_web_kit/model/resource_utils/test_utils.py @@ -0,0 +1,142 @@ +import errno +import os +import unittest +from unittest.mock import MagicMock, mock_open, patch + +from llm_web_kit.model.resource_utils.utils import FileLock, try_remove + + +class Test_try_remove: + + @patch('os.remove') + def test_remove(self, removeMock): + try_remove('path') + removeMock.assert_called_once_with('path') + + @patch('os.remove') + def test_remove_exception(self, removeMock): + removeMock.side_effect = Exception + try_remove('path') + removeMock.assert_called_once_with('path') + + +class TestFileLock(unittest.TestCase): + + def setUp(self): + self.lock_path = 'test.lock' + + @patch('os.fdopen') + @patch('os.open') + @patch('os.close') + @patch('os.remove') + def test_acquire_and_release_lock( + self, mock_remove, mock_close, mock_open, mock_os_fdopen + ): + # 模拟成功获取锁 + mock_open.return_value = 123 # 假设文件描述符为123 + # 模拟文件描述符 + mock_fd = MagicMock() + mock_fd.__enter__.return_value = mock_fd + mock_fd.write.return_value = None + mock_os_fdopen.return_value = mock_fd + + with FileLock(self.lock_path): + mock_open.assert_called_once_with( + self.lock_path, os.O_CREAT | os.O_EXCL | os.O_WRONLY, 0o644 + ) + mock_close.assert_called_once_with(123) + mock_remove.assert_called_once_with(self.lock_path) + + @patch('os.fdopen') + @patch('os.open') + @patch('builtins.open', new_callable=mock_open, read_data='1234\n100') + @patch('time.time') + @patch('os.remove') + def test_remove_stale_lock( + self, mock_remove, mock_time, mock_file_open, mock_os_open, mock_os_fdopen + ): + # 第一次尝试创建锁文件失败（锁已存在） + mock_os_open.side_effect = [ + OSError(errno.EEXIST, 'File exists'), + 123, # 第二次成功 + ] + + # 模拟文件描述符 + mock_fd = MagicMock() + mock_fd.__enter__.return_value = mock_fd + mock_fd.write.return_value = None + mock_os_fdopen.return_value = mock_fd + + # 当前时间设置为超过超时时间（timeout=300） + mock_time.return_value = 401 # 100 + 300 + 1 + + with FileLock(self.lock_path, timeout=300): + mock_remove.assert_called_once_with(self.lock_path) + mock_os_open.assert_any_call( + self.lock_path, os.O_CREAT | os.O_EXCL | os.O_WRONLY, 0o644 + ) + + @patch('os.open') + @patch('time.time') + def test_timeout_acquiring_lock(self, mock_time, mock_os_open): + # 总是返回EEXIST错误 + mock_os_open.side_effect = OSError(errno.EEXIST, 'File exists') + # 时间累计超过超时时间 + start_time = 1000 + mock_time.side_effect = [ + start_time, + start_time + 301, + start_time + 302, + start_time + 303, + ] + + with self.assertRaises(TimeoutError): + with FileLock(self.lock_path, timeout=300): + pass + + @patch('os.open') + def test_other_os_error(self, mock_os_open): + # 模拟其他OS错误（如权限不足） + mock_os_open.side_effect = OSError(errno.EACCES, 'Permission denied') + with self.assertRaises(OSError): + with FileLock(self.lock_path): + pass + + @patch('os.close') + @patch('os.remove') + def test_cleanup_on_exit(self, mock_remove, mock_close): + + mock_close.side_effect = None + # 确保退出上下文时执行清理 + lock_path = 'test.lock' + lock = FileLock(lock_path) + lock._fd = 123 # 模拟已打开的文件描述符 + lock.__exit__('!!!!!!!!!!!!!!!!!!!!!!!!!!!!!', None, None) + mock_remove.assert_called_once_with(self.lock_path) + + @patch('os.remove') + def test_cleanup_failure_handled(self, mock_remove): + # 模拟删除锁文件时失败 + mock_remove.side_effect = OSError + lock = FileLock(self.lock_path) + lock._fd = 123 + # 不应抛出异常 + lock.__exit__(None, None, None) + + @patch('os.getpid') + @patch('time.time') + def test_lock_file_content(self, mock_time, mock_pid): + # 验证锁文件内容格式 + mock_pid.return_value = 9999 + mock_time.return_value = 123456.789 + + with patch('os.open') as mock_os_open: + mock_os_open.return_value = 123 + with patch('os.fdopen') as mock_fdopen: + # 模拟写入文件描述符 + mock_file = MagicMock() + mock_fdopen.return_value.__enter__.return_value = mock_file + + with FileLock(self.lock_path): + mock_fdopen.assert_called_once_with(123, 'w') + mock_file.write.assert_called_once_with('9999\n123456.789') From 860c8d6ded2c0e291de18105d24204f559f760f1 Mon Sep 17 00:00:00 2001 From: qiujiantao Date: Thu, 6 Mar 2025 14:12:17 +0800 Subject: [PATCH 6/8] =?UTF-8?q?feat:=20=E5=B0=86=20FileLock=20=E7=B1=BB?= =?UTF-8?q?=E9=87=8D=E5=91=BD=E5=90=8D=E4=B8=BA=20FileLockContext=EF=BC=8C?= =?UTF-8?q?=E6=9B=B4=E6=96=B0=E7=9B=B8=E5=85=B3=E5=BC=95=E7=94=A8=E4=BB=A5?= =?UTF-8?q?=E5=A2=9E=E5=BC=BA=E4=BB=A3=E7=A0=81=E5=8F=AF=E8=AF=BB=E6=80=A7?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../model/resource_utils/download_assets.py | 4 ++-- llm_web_kit/model/resource_utils/unzip_ext.py | 4 ++-- llm_web_kit/model/resource_utils/utils.py | 2 +- .../model/resource_utils/test_utils.py | 16 ++++++++-------- 4 files changed, 13 insertions(+), 13 deletions(-) diff --git a/llm_web_kit/model/resource_utils/download_assets.py b/llm_web_kit/model/resource_utils/download_assets.py index 2821b35a..0452d16e 100644 --- a/llm_web_kit/model/resource_utils/download_assets.py +++ b/llm_web_kit/model/resource_utils/download_assets.py @@ -13,7 +13,7 @@ from llm_web_kit.model.resource_utils.boto3_ext import (get_s3_client, is_s3_path, split_s3_path) -from llm_web_kit.model.resource_utils.utils import FileLock, try_remove +from llm_web_kit.model.resource_utils.utils import FileLockContext, try_remove def decide_cache_dir(): @@ -214,7 +214,7 @@ def check_callback(): logger.warning(f'Removing invalid file: {target_path}') try_remove(target_path) - with FileLock(lock_path, check_callback, timeout=lock_timeout) as lock: + with FileLockContext(lock_path, check_callback, timeout=lock_timeout) as lock: if lock is True: logger.info( f'File already exists with valid checksum: {target_path} while waiting' diff --git a/llm_web_kit/model/resource_utils/unzip_ext.py b/llm_web_kit/model/resource_utils/unzip_ext.py index 8b884a85..6579fd62 100644 --- a/llm_web_kit/model/resource_utils/unzip_ext.py +++ b/llm_web_kit/model/resource_utils/unzip_ext.py @@ -6,7 +6,7 @@ from llm_web_kit.exception.exception import ModelResourceException from llm_web_kit.libs.logger import mylogger as logger -from llm_web_kit.model.resource_utils.utils import FileLock, try_remove +from llm_web_kit.model.resource_utils.utils import FileLockContext, try_remove def get_unzip_dir(zip_path: str) -> str: @@ -96,7 +96,7 @@ def check_zip(): ) try_remove(target_dir) - with FileLock(lock_path, check_zip, timeout=lock_timeout) as lock: + with FileLockContext(lock_path, check_zip, timeout=lock_timeout) as lock: if lock is True: logger.info( f'zip file {zip_path} is already unzipped to {target_dir} while waiting' diff --git a/llm_web_kit/model/resource_utils/utils.py b/llm_web_kit/model/resource_utils/utils.py index f2e142e9..b80a35d0 100644 --- a/llm_web_kit/model/resource_utils/utils.py +++ b/llm_web_kit/model/resource_utils/utils.py @@ -11,7 +11,7 @@ def try_remove(path: str): pass -class FileLock: +class FileLockContext: """基于文件锁的上下文管理器（跨平台兼容版）""" def __init__(self, lock_path: str, check_callback=None, timeout: float = 300): diff --git a/tests/llm_web_kit/model/resource_utils/test_utils.py b/tests/llm_web_kit/model/resource_utils/test_utils.py index 542b4904..15448bcf 100644 --- a/tests/llm_web_kit/model/resource_utils/test_utils.py +++ b/tests/llm_web_kit/model/resource_utils/test_utils.py @@ -3,7 +3,7 @@ import unittest from unittest.mock import MagicMock, mock_open, patch -from llm_web_kit.model.resource_utils.utils import FileLock, try_remove +from llm_web_kit.model.resource_utils.utils import FileLockContext, try_remove class Test_try_remove: @@ -40,7 +40,7 @@ def test_acquire_and_release_lock( mock_fd.write.return_value = None mock_os_fdopen.return_value = mock_fd - with FileLock(self.lock_path): + with FileLockContext(self.lock_path): mock_open.assert_called_once_with( self.lock_path, os.O_CREAT | os.O_EXCL | os.O_WRONLY, 0o644 ) @@ -70,7 +70,7 @@ def test_remove_stale_lock( # 当前时间设置为超过超时时间（timeout=300） mock_time.return_value = 401 # 100 + 300 + 1 - with FileLock(self.lock_path, timeout=300): + with FileLockContext(self.lock_path, timeout=300): mock_remove.assert_called_once_with(self.lock_path) mock_os_open.assert_any_call( self.lock_path, os.O_CREAT | os.O_EXCL | os.O_WRONLY, 0o644 @@ -91,7 +91,7 @@ def test_timeout_acquiring_lock(self, mock_time, mock_os_open): ] with self.assertRaises(TimeoutError): - with FileLock(self.lock_path, timeout=300): + with FileLockContext(self.lock_path, timeout=300): pass @patch('os.open') @@ -99,7 +99,7 @@ def test_other_os_error(self, mock_os_open): # 模拟其他OS错误（如权限不足） mock_os_open.side_effect = OSError(errno.EACCES, 'Permission denied') with self.assertRaises(OSError): - with FileLock(self.lock_path): + with FileLockContext(self.lock_path): pass @patch('os.close') @@ -109,7 +109,7 @@ def test_cleanup_on_exit(self, mock_remove, mock_close): mock_close.side_effect = None # 确保退出上下文时执行清理 lock_path = 'test.lock' - lock = FileLock(lock_path) + lock = FileLockContext(lock_path) lock._fd = 123 # 模拟已打开的文件描述符 lock.__exit__('!!!!!!!!!!!!!!!!!!!!!!!!!!!!!', None, None) mock_remove.assert_called_once_with(self.lock_path) @@ -118,7 +118,7 @@ def test_cleanup_on_exit(self, mock_remove, mock_close): def test_cleanup_failure_handled(self, mock_remove): # 模拟删除锁文件时失败 mock_remove.side_effect = OSError - lock = FileLock(self.lock_path) + lock = FileLockContext(self.lock_path) lock._fd = 123 # 不应抛出异常 lock.__exit__(None, None, None) @@ -137,6 +137,6 @@ def test_lock_file_content(self, mock_time, mock_pid): mock_file = MagicMock() mock_fdopen.return_value.__enter__.return_value = mock_file - with FileLock(self.lock_path): + with FileLockContext(self.lock_path): mock_fdopen.assert_called_once_with(123, 'w') mock_file.write.assert_called_once_with('9999\n123456.789') From a666754eafd9036de62e7b79a5401fca97f1db35 Mon Sep 17 00:00:00 2001 From: qiujiantao Date: Thu, 6 Mar 2025 14:18:01 +0800 Subject: [PATCH 7/8] =?UTF-8?q?feat:=20=E4=BD=BF=E7=94=A8=20ModelResourceE?= =?UTF-8?q?xception=20=E6=9B=BF=E6=8D=A2=20move=5Fto=5Ftarget=20=E5=87=BD?= =?UTF-8?q?=E6=95=B0=E4=B8=AD=E7=9A=84=E5=BC=82=E5=B8=B8=E5=A4=84=E7=90=86?= =?UTF-8?q?=EF=BC=8C=E5=A2=9E=E5=BC=BA=E9=94=99=E8=AF=AF=E6=8F=90=E7=A4=BA?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- llm_web_kit/model/resource_utils/download_assets.py | 4 ++-- .../llm_web_kit/model/resource_utils/test_download_assets.py | 2 +- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/llm_web_kit/model/resource_utils/download_assets.py b/llm_web_kit/model/resource_utils/download_assets.py index 0452d16e..ab7cbead 100644 --- a/llm_web_kit/model/resource_utils/download_assets.py +++ b/llm_web_kit/model/resource_utils/download_assets.py @@ -154,7 +154,7 @@ def download_to_temp(conn, progress_bar) -> str: def move_to_target(tmp_path: str, target_path: str, expected_size: int): """移动文件并验证.""" if os.path.getsize(tmp_path) != expected_size: - raise ValueError( + raise ModelResourceException( f'File size mismatch: {os.path.getsize(tmp_path)} vs {expected_size}' ) @@ -162,7 +162,7 @@ def move_to_target(tmp_path: str, target_path: str, expected_size: int): shutil.move(tmp_path, target_path) # 原子操作替换 if not os.path.exists(target_path): - raise RuntimeError(f'Move failed: {tmp_path} -> {target_path}') + raise ModelResourceException(f'Move failed: {tmp_path} -> {target_path}') def download_auto_file( diff --git a/tests/llm_web_kit/model/resource_utils/test_download_assets.py b/tests/llm_web_kit/model/resource_utils/test_download_assets.py index 7b4276c8..b3e55e74 100644 --- a/tests/llm_web_kit/model/resource_utils/test_download_assets.py +++ b/tests/llm_web_kit/model/resource_utils/test_download_assets.py @@ -464,7 +464,7 @@ def test_size_mismatch(self): with open(tmp_path, 'wb') as f: f.write(b'short') - with self.assertRaisesRegex(ValueError, 'size mismatch'): + with self.assertRaisesRegex(ModelResourceException, 'size mismatch'): move_to_target(tmp_path, self.target_path, 100) def test_directory_creation(self): From 923ad0487291af402034033a55754edd455d73ea Mon Sep 17 00:00:00 2001 From: qiujiantao Date: Thu, 6 Mar 2025 14:22:50 +0800 Subject: [PATCH 8/8] feat: move test_utils.py to test_resource_utils.py --- .../resource_utils/{test_utils.py => test_resource_utils.py} | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename tests/llm_web_kit/model/resource_utils/{test_utils.py => test_resource_utils.py} (100%) diff --git a/tests/llm_web_kit/model/resource_utils/test_utils.py b/tests/llm_web_kit/model/resource_utils/test_resource_utils.py similarity index 100% rename from tests/llm_web_kit/model/resource_utils/test_utils.py rename to tests/llm_web_kit/model/resource_utils/test_resource_utils.py