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author | tim30b <tim.toenz@ost.ch> | 2022-05-18 13:53:24 +0200 |
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committer | tim30b <tim.toenz@ost.ch> | 2022-05-18 13:53:24 +0200 |
commit | 8cabb06907dbe5d18df7a83d30edea9477d4e643 (patch) | |
tree | aaf669409ba40b168e76d435300f74a457f8e5ee /vorlesungen/slides/fresnel/numerik.tex | |
parent | Intro chapters (diff) | |
parent | typos (diff) | |
download | SeminarSpezielleFunktionen-8cabb06907dbe5d18df7a83d30edea9477d4e643.tar.gz SeminarSpezielleFunktionen-8cabb06907dbe5d18df7a83d30edea9477d4e643.zip |
Merge remote-tracking branch 'mueller/master'
Diffstat (limited to 'vorlesungen/slides/fresnel/numerik.tex')
-rw-r--r-- | vorlesungen/slides/fresnel/numerik.tex | 124 |
1 files changed, 124 insertions, 0 deletions
diff --git a/vorlesungen/slides/fresnel/numerik.tex b/vorlesungen/slides/fresnel/numerik.tex new file mode 100644 index 0000000..0bd4d5a --- /dev/null +++ b/vorlesungen/slides/fresnel/numerik.tex @@ -0,0 +1,124 @@ +% +% numerik.tex -- numerische Berechnung der Fresnel Integrale +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Numerik} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Taylor-Reihe} +\begin{align*} +\sin t^{\uncover<2->{\color<2>{red}2}} +&= +\sum_{k=0}^\infty +(-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<4->{ +S_1(t) +&= +\sum_{k=0}^\infty +(-1)^k \frac{t^{4k+3}}{(2k+1)!(4n+3)} +} +\\ +\cos t^{\uncover<3->{\color<3>{red}2}} +&= +\sum_{k=0}^\infty +(-1)^k \frac{t^{ +\ifthenelse{\boolean{presentation}}{\only<-2>{2k}}{} +\only<3->{\color<3>{red}4k}} +}{ +(2k)! +} +\\ +%\int \sin t^2\,dt +\uncover<5->{ +C_1(t) +&= +\sum_{k=0}^\infty +(-1)^k \frac{t^{4k+1}}{(2k)!(4k+1)} +} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{ +\begin{block}{Differentialgleichung} +\[ +\dot{\gamma}_1(t) += +\begin{pmatrix} +\cos t^2\\ \sin t^2 +\end{pmatrix} +\uncover<7->{ +\; +\to +\; +\gamma_1(t) += +\begin{pmatrix} +C_1(t)\\S_1(t) +\end{pmatrix} +} +\] +\end{block}} +\uncover<8->{% +\begin{block}{Hypergeometrische Reihen} +\begin{align*} +\uncover<9->{% +S(t) +&= +\frac{\pi z^3}{6} +\cdot +\mathstrut_1F_2\biggl( +\begin{matrix}\frac34\\\frac32,\frac74\end{matrix} +; +-\frac{\pi^2z^4}{16} +\biggr) +} +\\ +\uncover<10->{ +C(t) +&= +z +\cdot +\mathstrut_1F_2\biggl( +\begin{matrix}\frac14\\\frac12,\frac54\end{matrix} +; +-\frac{\pi^2z^4}{16} +\biggr)} +\end{align*} +\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 |