diff options
| -rw-r--r-- | vorlesungen/10_mseliealgebra/slides.tex | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/7/Makefile.inc | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/7/chapter.tex | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/7/logarithmus.tex | 82 | ||||
| -rw-r--r-- | vorlesungen/slides/test.tex | 2 |
5 files changed, 86 insertions, 1 deletions
diff --git a/vorlesungen/10_mseliealgebra/slides.tex b/vorlesungen/10_mseliealgebra/slides.tex index ff2e629..936139a 100644 --- a/vorlesungen/10_mseliealgebra/slides.tex +++ b/vorlesungen/10_mseliealgebra/slides.tex @@ -19,6 +19,7 @@ \section{Exponentialabbildung} \folie{7/dg.tex} +\folie{7/logarithmus.tex} % Interpolation \folie{7/interpolation.tex} diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc index 4d291ed..ffd5091 100644 --- a/vorlesungen/slides/7/Makefile.inc +++ b/vorlesungen/slides/7/Makefile.inc @@ -23,6 +23,7 @@ chapter5 = \ ../slides/7/dg.tex \ ../slides/7/interpolation.tex \ ../slides/7/exponentialreihe.tex \ + ../slides/7/logarithmus.tex \ ../slides/7/zusammenhang.tex \ ../slides/7/quaternionen.tex \ ../slides/7/qdreh.tex \ diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex index 36e0bb1..3736e0f 100644 --- a/vorlesungen/slides/7/chapter.tex +++ b/vorlesungen/slides/7/chapter.tex @@ -22,6 +22,7 @@ \folie{7/dg.tex} \folie{7/interpolation.tex} \folie{7/exponentialreihe.tex} +\folie{7/logarithmus.tex} \folie{7/zusammenhang.tex} \folie{7/quaternionen.tex} \folie{7/qdreh.tex} diff --git a/vorlesungen/slides/7/logarithmus.tex b/vorlesungen/slides/7/logarithmus.tex new file mode 100644 index 0000000..58065d7 --- /dev/null +++ b/vorlesungen/slides/7/logarithmus.tex @@ -0,0 +1,82 @@ +% +% logarithmus.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Logarithmus} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Taylor-Reihe} +\begin{align*} +\frac{d}{dx}\log(1+x) +&= \frac{1}{1+x} +\\ +\uncover<2->{ +\Rightarrow\quad +\log (1+x) +&= +\int_0^x \frac{1}{1+t}\,dt} +\end{align*} +\begin{align*} +\uncover<3->{\frac{1}{1+t} +&= +1-t+t^2-t^3+\dots} +\\ +\uncover<4->{\log(1+x) +&=\int_0^x +1-t+t^2-t^3+\dots +\,dt} +\\ +&\only<5>{= +x-\frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}4 + \dots} +\uncover<6->{= +\sum_{k=1}^\infty (-1)^{k-1}\frac{x^k}{k}} +\\ +\uncover<7->{\log (I+A) +&= +\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}A^k} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Konvergenzradius} +Polstelle bei $x=-1$ +\( +\varrho =1 +\) +\end{block}} +\vspace{-5pt} +\begin{block}{\uncover<9->{Alternative: Spektraltheorie}} +\uncover<9->{ +Logarithmus $\log z$ in $\{z\in\mathbb{C}\;|\; \neg(\Re z\le 0\wedge\Im z=0)\}$ +definiert:} +\vspace{-15pt} +\uncover<8->{ +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\uncover<9->{ + \fill[color=red!20] (-2.1,-2.1) rectangle (2.5,2.1); +} +\draw[->] (-2.2,0) -- (2.9,0) coordinate[label={$\Re z$}]; +\draw[->] (0,-2.2) -- (0,2.4) coordinate[label={right:$\Im z$}]; +\fill[color=blue!40,opacity=0.5] (1,0) circle[radius=1]; +\draw[color=blue] (1,0) circle[radius=1]; +\uncover<9->{ + \draw[color=white,line width=5pt] (-2.2,0) -- (0.1,0); +} +\fill (1,0) circle[radius=0.08]; +\node at (2.3,1.9) {$\mathbb{C}$}; +\node at (1,0) [below] {$1$}; +\end{tikzpicture} +\end{center}} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 56cd611..7434b0b 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -3,5 +3,5 @@ % % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil % -\folie{7/interpolation.tex} +\folie{7/logarithmus.tex} |