From db0fcc225416e260284ffa3a1da5919a2b1ac5a5 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Mon, 9 May 2022 11:03:14 +0200
Subject: =?UTF-8?q?Fresnel-Pr=C3=A4sentation?=
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---
 vorlesungen/slides/fresnel/numerik.tex | 75 ++++++++++++++++++++++++++--------
 1 file changed, 58 insertions(+), 17 deletions(-)

(limited to 'vorlesungen/slides/fresnel/numerik.tex')

diff --git a/vorlesungen/slides/fresnel/numerik.tex b/vorlesungen/slides/fresnel/numerik.tex
index 5c6f96d..0bd4d5a 100644
--- a/vorlesungen/slides/fresnel/numerik.tex
+++ b/vorlesungen/slides/fresnel/numerik.tex
@@ -1,5 +1,5 @@
 %
-% numerik.tex -- slide template
+% numerik.tex -- numerische Berechnung der Fresnel Integrale
 %
 % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 %
@@ -13,27 +13,38 @@
 \begin{column}{0.48\textwidth}
 \begin{block}{Taylor-Reihe}
 \begin{align*}
-\sin t^2
+\sin t^{\uncover<2->{\color<2>{red}2}}
 &=
 \sum_{k=0}^\infty
-(-1)^k \frac{t^{2k+1}}{(2k+1)!}
+(-1)^k \frac{t^{
+\ifthenelse{\boolean{presentation}}{\only<1>{2k+1}}{}
+\only<2->{\color<2>{red}4k+2}
+}
+}{
+(2k+1)!
+}
 \\
 %\int \sin t^2\,dt
-\uncover<2->{
-S(t)
+\uncover<4->{
+S_1(t)
 &=
 \sum_{k=0}^\infty
 (-1)^k \frac{t^{4k+3}}{(2k+1)!(4n+3)}
 }
 \\
-\cos t^2
+\cos t^{\uncover<3->{\color<3>{red}2}}
 &=
 \sum_{k=0}^\infty
-(-1)^k \frac{t^{2k}}{(2k)!}
+(-1)^k \frac{t^{
+\ifthenelse{\boolean{presentation}}{\only<-2>{2k}}{}
+\only<3->{\color<3>{red}4k}}
+}{
+(2k)!
+}
 \\
 %\int \sin t^2\,dt
-\uncover<3->{
-C(t)
+\uncover<5->{
+C_1(t)
 &=
 \sum_{k=0}^\infty
 (-1)^k \frac{t^{4k+1}}{(2k)!(4k+1)}
@@ -42,23 +53,34 @@ C(t)
 \end{block}
 \end{column}
 \begin{column}{0.48\textwidth}
-\uncover<4->{
+\uncover<6->{
 \begin{block}{Differentialgleichung}
 \[
-\dot{\gamma}(t)
+\dot{\gamma}_1(t)
 =
 \begin{pmatrix}
-\sin t^2\\ \cos t^2
+\cos t^2\\ \sin t^2
 \end{pmatrix}
+\uncover<7->{
+\;
+\to
+\;
+\gamma_1(t)
+=
+\begin{pmatrix}
+C_1(t)\\S_1(t)
+\end{pmatrix}
+}
 \]
 \end{block}}
-\uncover<5->{%
+\uncover<8->{%
 \begin{block}{Hypergeometrische Reihen}
 \begin{align*}
-\uncover<6->{%
+\uncover<9->{%
 S(t)
 &=
-\frac{\pi z^3}{6}\,
+\frac{\pi z^3}{6}
+\cdot
 \mathstrut_1F_2\biggl(
 \begin{matrix}\frac34\\\frac32,\frac74\end{matrix}
 ;
@@ -66,10 +88,11 @@ S(t)
 \biggr)
 }
 \\
-\uncover<7->{
+\uncover<10->{
 C(t)
 &=
-z\,
+z
+\cdot
 \mathstrut_1F_2\biggl(
 \begin{matrix}\frac14\\\frac12,\frac54\end{matrix}
 ;
@@ -79,5 +102,23 @@ z\,
 \end{block}}
 \end{column}
 \end{columns}
+\uncover<11->{%
+\begin{block}{Komplexe Fehlerfunktion}
+\[
+\left.
+\begin{matrix}
+S(z)\\
+C(z)
+\end{matrix}
+\right\}
+=
+\frac{1\pm i}{4}
+\left(
+\operatorname{erf}\biggl({\frac{1+i}2}\sqrt{\pi}z\biggr)
+\mp i
+\operatorname{erf}\biggl({\frac{1-i}2}\sqrt{\pi}z\biggr)
+\right)
+\]
+\end{block}}
 \end{frame}
 \egroup
-- 
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