From 32a3b06968115b82eac1697b8a5208b77e4376f7 Mon Sep 17 00:00:00 2001 From: veecue Date: Thu, 23 Jan 2020 17:31:47 +0100 Subject: [PATCH 1/6] edits for ws19/20 --- Analysis-3.tex | 10 ++++++++-- 1 file changed, 8 insertions(+), 2 deletions(-) diff --git a/Analysis-3.tex b/Analysis-3.tex index cd4e3bb..5ce3b83 100644 --- a/Analysis-3.tex +++ b/Analysis-3.tex @@ -257,6 +257,11 @@ \section{Fouriertransformation \quad $f(t) \ra F(\omega)$} 0, & \abs{t-a} > T \end{cases}$ & \multicolumn{3}{l}{\kern-2em $\FT 2ATe^{-\i\omega a} \mathrm{si}(\omega T)$} \end{tabular} + \textbf{Spezialfälle:} + \begin{itemize} + \item $f$ gerade $\Leftrightarrow \hat{f}(\omega)=2\int_0^\infty f(t)\cos(\omega t)dt$ + \item $f$ ungerade $\Leftrightarrow \hat{f}(\omega)=-2i\int_0^\infty f(t)\sin(\omega t)dt$ + \end{itemize} % Special case of the following function % $r(t) = \begin{cases} % 1/2 & \text{falls} \abs{t}<1 \\ @@ -337,7 +342,8 @@ \section{Laplacetransformation \quad $\mathcal L\bigl(f(t)\bigr) = F(s)$} $\sin(a t)$ & \kern-2em $\LT \frac{a}{s^2 + a^2}$ & $\cos(a t)$ & \kern-2em $\LT \frac{s}{s^2 + a^2}$\\[0.5em] $\sinh(a t)$ & \kern-2em $\LT \frac{a}{s^2 - a^2}$ & $\cosh(a t)$ & \kern-2em $\LT \frac{s}{s^2 - a^2}$\\[0.5em] $\frac{\sin(at)}{t}$ & \kern-2em $\LT\arctan\left(\frac{a}{s}\right)$ & $\frac{t^{n-1}}{(n-1)!}$ & \kern-2em $\LT \frac{1}{s^n}$ \\[0.5em] - $e^{-at} \sin(b t)$ & \kern-2em $\LT \frac{b}{(s+a)^2+b^2}$ \\ + $e^{-at} \sin(b t)$ & \kern-2em $\LT \frac{b}{(s+a)^2+b^2}$ + &$t^ne^{at}$ & \kern-2em $\LT \frac{n!(s-a)}{s^{n+1}}$\\ $e^{-at} \cos(b t)$ & \kern-2em $\LT \frac{s+a}{(s+a)^2+b^2}$\\ $\frac{ae^{-at}-be^{-bt}}{a-b}$ & $\kern-2em \LT \frac{s}{(s+a)(s+b)}$ \end{tabular}\\ @@ -487,7 +493,7 @@ \section{Funktionentheorie (Komplexe Funktionen)} \subsection{Existenz einer Stammfunktion und Wegunabhängigkeit} Ist $\cx f: G \ra \C$ holomorph auf dem einfach zsh. Gebiet $G$, so existiert zu $\cx f$ eine Stammfunktion $\cx F$, und es gilt für jede in $G$ verlaufende Kurve $\cx \gamma$ mit Anfangspunkt $\cx \gamma(a)$ und Endpunkt $\cx \gamma(b)$: \\ \begin{equation*} - \int \limits_{\cx \gamma} \cx f(\cx z) \diff \cx z = \cx F(\cx \gamma (b)) - \cx F(\cx \gamma (a)) + \int \limits_{\cx \gamma} \cx f(\cx z) \diff \cx z = \cx F(\cx \gamma (b)) - \cx F(\cx \gamma (a)) = \int\limits_a^bf(\gamma(t))\gamma'(t)dt \end{equation*} \end{sectionbox} From 16a31db07a0fa20abc2ac735769790a77118368c Mon Sep 17 00:00:00 2001 From: Darius Peters <38152878+dariusptrs@users.noreply.github.com> Date: Fri, 21 Feb 2025 13:59:40 +0100 Subject: [PATCH 2/6] continue after latexdiff-error --- .github/workflows/ci.yml | 1 + 1 file changed, 1 insertion(+) diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml index d3ac834..62c9df6 100644 --- a/.github/workflows/ci.yml +++ b/.github/workflows/ci.yml @@ -51,6 +51,7 @@ jobs: # Run latexdiff on each document by retrieving the original .tex file from the default branch. - name: Run latexdiff on documents + continue-on-error: true run: | # Install latexdiff if not present apt-get install -y latexdiff From 2e5bd5349a704bd1e745ccc55c3deeee48c869e9 Mon Sep 17 00:00:00 2001 From: Darius Peters <38152878+dariusptrs@users.noreply.github.com> Date: Fri, 21 Feb 2025 14:55:40 +0100 Subject: [PATCH 3/6] Update Analysis-3.tex --- Analysis-3.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Analysis-3.tex b/Analysis-3.tex index 3108ef8..36db3d1 100644 --- a/Analysis-3.tex +++ b/Analysis-3.tex @@ -347,7 +347,7 @@ \section{Laplacetransformation \quad $\mathcal L\bigl(f(t)\bigr) = F(s)$} $\sinh(a t)$ & \kern-2em $\LT \frac{a}{s^2 - a^2}$ & $\cosh(a t)$ & \kern-2em $\LT \frac{s}{s^2 - a^2}$\\[0.5em] $\frac{\sin(at)}{t}$ & \kern-2em $\LT\arctan\left(\frac{a}{s}\right)$ & $\frac{t^{n-1}}{(n-1)!}$ & \kern-2em $\LT \frac{1}{s^n}$ \\[0.5em] $e^{-at} \sin(b t)$ & \kern-2em $\LT \frac{b}{(s+a)^2+b^2}$ - &$t^ne^{at}$ & \kern-2em $\LT \frac{n!(s-a)}{s^{n+1}}$\\ + &$t^ne^{at}$ & \kern-2em $\LT \frac{n!}{(s-a)^{n+1}}$\\ $e^{-at} \cos(b t)$ & \kern-2em $\LT \frac{s+a}{(s+a)^2+b^2}$\\ $\frac{ae^{-at}-be^{-bt}}{a-b}$ & $\kern-2em \LT \frac{s}{(s+a)(s+b)}$ \end{tabular}\\ From 8b82c8a81338020362596092b9f42d3331648fea Mon Sep 17 00:00:00 2001 From: Darius Peters <38152878+dariusptrs@users.noreply.github.com> Date: Fri, 21 Feb 2025 14:55:48 +0100 Subject: [PATCH 4/6] Update Analysis-3.tex --- Analysis-3.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Analysis-3.tex b/Analysis-3.tex index 36db3d1..f2b597e 100644 --- a/Analysis-3.tex +++ b/Analysis-3.tex @@ -497,7 +497,7 @@ \section{Funktionentheorie (Komplexe Funktionen)} \subsection{Existenz einer Stammfunktion und Wegunabhängigkeit} Ist $\cx f: G \ra \C$ holomorph auf dem einfach zsh. Gebiet $G$, so existiert zu $\cx f$ eine Stammfunktion $\cx F$, und es gilt für jede in $G$ verlaufende Kurve $\cx \gamma$ mit Anfangspunkt $\cx \gamma(a)$ und Endpunkt $\cx \gamma(b)$: \\ \begin{equation*} - \int \limits_{\cx \gamma} \cx f(\cx z) \diff \cx z = \cx F(\cx \gamma (b)) - \cx F(\cx \gamma (a)) = \int\limits_a^bf(\gamma(t))\gamma'(t)dt + \int \limits_{\cx \gamma} \cx f(\cx z) \diff \cx z = \cx F(\cx \gamma (b)) - \cx F(\cx \gamma (a)) = \int\limits_a^bf(\gamma(t)) \cdot \dot{\gamma}(t) \diff t \end{equation*} \end{sectionbox} From c22d74282e3fba00f362681e8d68e3434c24f6bb Mon Sep 17 00:00:00 2001 From: Darius Peters <38152878+dariusptrs@users.noreply.github.com> Date: Fri, 21 Feb 2025 14:55:58 +0100 Subject: [PATCH 5/6] Update Analysis-3.tex --- Analysis-3.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Analysis-3.tex b/Analysis-3.tex index f2b597e..4ea8f23 100644 --- a/Analysis-3.tex +++ b/Analysis-3.tex @@ -263,7 +263,7 @@ \section{Fouriertransformation \quad $f(t) \ra F(\omega)$} \end{tabular} \textbf{Spezialfälle:} \begin{itemize} - \item $f$ gerade $\Leftrightarrow \hat{f}(\omega)=2\int_0^\infty f(t)\cos(\omega t)dt$ + \item $f$ gerade $\Leftrightarrow \hat{f}(\omega)=2\int_0^\infty f(t)\cos(\omega t)\diff t$ \item $f$ ungerade $\Leftrightarrow \hat{f}(\omega)=-2i\int_0^\infty f(t)\sin(\omega t)dt$ \end{itemize} % Special case of the following function From 73e173b06696ec06b575619b2234b56fef25a820 Mon Sep 17 00:00:00 2001 From: Darius Peters <38152878+dariusptrs@users.noreply.github.com> Date: Fri, 21 Feb 2025 14:56:04 +0100 Subject: [PATCH 6/6] Update Analysis-3.tex --- Analysis-3.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Analysis-3.tex b/Analysis-3.tex index 4ea8f23..b5b031f 100644 --- a/Analysis-3.tex +++ b/Analysis-3.tex @@ -264,7 +264,7 @@ \section{Fouriertransformation \quad $f(t) \ra F(\omega)$} \textbf{Spezialfälle:} \begin{itemize} \item $f$ gerade $\Leftrightarrow \hat{f}(\omega)=2\int_0^\infty f(t)\cos(\omega t)\diff t$ - \item $f$ ungerade $\Leftrightarrow \hat{f}(\omega)=-2i\int_0^\infty f(t)\sin(\omega t)dt$ + \item $f$ ungerade $\Leftrightarrow \hat{f}(\omega)=-2i\int_0^\infty f(t)\sin(\omega t)\diff t$ \end{itemize} % Special case of the following function % $r(t) = \begin{cases}