diff options
Diffstat (limited to 'vorlesungen/slides/fresnel/numerik.tex')
| -rw-r--r-- | vorlesungen/slides/fresnel/numerik.tex | 75 |
1 files changed, 58 insertions, 17 deletions
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 |