3 files changed, 164 insertions, 0 deletions
diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc
index 1707e67..76b9032 100644
--- a/vorlesungen/slides/5/Makefile.inc
+++ b/vorlesungen/slides/5/Makefile.inc
@@ -22,6 +22,8 @@ chapter5 =								\
 	../slides/5/reellenormalform.tex				\
 	../slides/5/cayleyhamilton.tex					\
 									\
+	../slides/5/spektrum.tex					\
+									\
 	../slides/5/konvergenzradius.tex				\
 	../slides/5/krbeispiele.tex					\
 	../slides/5/spektralgelfand.tex					\
diff --git a/vorlesungen/slides/5/krbeispiele.tex b/vorlesungen/slides/5/krbeispiele.tex
new file mode 100644
index 0000000..b51df78
--- /dev/null
+++ b/vorlesungen/slides/5/krbeispiele.tex
@@ -0,0 +1,99 @@
+%
+% krbeispiele.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Konvergenzradius --- Beispiele}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Exponentialreihe}
+\vspace{-20pt}
+\begin{align*}
+e^z &= \sum_{k=0}^\infty \frac{z^k}{k!}
+\\
+\uncover<2->{
+\frac1k\log k!
+}
+&\uncover<3->{=\frac1k\sum_{x=1}^k {\color{blue}\log x}}
+\uncover<6->{>\frac1k\int_1^k{\color{red}\log x}\,dx}
+\\
+&
+\ifthenelse{\boolean{presentation}}{
+\only<7>{=\frac1k[x\log x -x]_1^k}
+}{}
+\only<8->{=
+\log k -1 +\frac1k}
+\uncover<9->{\to \infty\phantom{\frac1k}}
+\\
+\uncover<10->{(k!)^{\frac1k}
+&\to\infty}\uncover<11->{ \quad\Rightarrow\quad R = \infty}
+\end{align*}
+\vspace{-40pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.7]
+\uncover<4->{
+\foreach \x in {2,...,9}{
+	\fill[color=blue!20] ({\x-1},0) rectangle ({\x},{ln(\x)});
+	\draw[color=blue] ({\x-1},0) rectangle ({\x},{ln(\x)});
+	\node at ({\x-0.5},{ln(\x)}) [above] {\tiny $\log\x$};
+	\draw (\x,-0.1) -- (\x,0.1);
+	\node at (\x,0) [below] {\tiny$\x$};
+}
+\draw (1,-0.1) -- (1,0.1);
+\uncover<5->{
+\begin{scope}
+	\clip (0,-1) rectangle (9.5,2.5);
+	\fill[color=red!40,opacity=0.5] (0,0) -- (0,-1)
+		-- plot[domain=0.1:9.1,samples=100] ({\x},{ln(\x)})
+		-- (9.1,0) -- cycle;
+	\draw[color=red] plot[domain=0.1:9.1,samples=100] ({\x},{ln(\x)});
+\end{scope}
+}
+\draw[->] (-0.2,0) -- (9.4,0) coordinate[label={$x$}];
+\draw[->] (0,-1) -- (0,2.5) coordinate[label={right:$y$}];
+}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<12->{%
+\begin{block}{Geometrische Reihe}
+\vspace{-15pt}
+\begin{align*}
+\uncover<13->{
+\frac{1}{{\color{blue}1}-z}
+&=
+\sum_{k=0}^\infty
+z^k}
+\\
+\uncover<14->{
+a_k&=1}
+\uncover<15->{\quad\Rightarrow\quad
+|a_k|^{\frac1k}=1}
+\\
+\uncover<16->{
+\limsup_{k\to\infty} &= |a_k|^{\frac1k}=1}\uncover<17->{ = \frac1R}
+\uncover<18->{\quad\Rightarrow\quad R=1}
+\end{align*}
+%\uncover<19->{Polstelle bei $z=1$ limitiert Konvergenzradius}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}
+\clip (-2.2,-1.5) rectangle (2.2,1.5);
+\fill[color=red!20] (0,0) circle[radius=2];
+\draw[color=red] (0,0) circle[radius=2];
+\end{scope}
+\draw[->] (-2.2,0) -- (2.5,0) coordinate[label={$\operatorname{Re}z$}];
+\draw[->] (0,-1.6) -- (0,1.8) coordinate[label={right:$\operatorname{Im}z$}];
+\fill[color=blue!20] (2,0) circle[radius=0.08];
+\draw[color=blue] (2,0) circle[radius=0.08];
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/spektrum.tex b/vorlesungen/slides/5/spektrum.tex
new file mode 100644
index 0000000..f427c9a
--- /dev/null
+++ b/vorlesungen/slides/5/spektrum.tex
@@ -0,0 +1,63 @@
+%
+% spektrum.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Spektrum}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+$A\colon V\to V$ beschränkter Operator zwischen Banach-Räumen
+\[
+\operatorname{Sp}A
+=
+\left\{
+\lambda\in\mathbb{C}
+\;\left|\;
+\begin{minipage}{2cm}\raggedright
+$A-\lambda I$ nicht invertierbar
+\end{minipage}
+\right.
+\right\}
+\]
+\end{block}
+\begin{block}{Endlichdimensionale Räume}
+\vspace{-15pt}
+\begin{align*}
+&\lambda\in\operatorname{Sp}A
+\\
+\Leftrightarrow\quad&\text{$(A-\lambda I)$ nicht invertierbar}
+\\
+\Leftrightarrow\quad&\text{$(A-\lambda I)$ singulär}
+\\
+\Leftrightarrow\quad&\ker(A-\lambda I)\ne 0
+\\
+\Leftrightarrow\quad&\exists v\in V, v\ne 0, Av=\lambda v
+\end{align*}
+$\Rightarrow$ $\operatorname{Sp}A$ ist die Menge der Eigenwerte
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Unendlichdimensional}
+Es gibt eine Folge $x_n\in V$ von Einheitsvektoren
+$\|x_n\|=1$
+mit
+\begin{align*}
+\lim_{n\to\infty} (A - \lambda)x_n &= 0
+\end{align*}
+\end{block}
+\begin{block}{Spektrum und Norm}
+\[
+\operatorname{Sp}(A)
+\subset
+\{\lambda\in\mathbb{C}\;|\;
+|\lambda|\le \|A\|\}
+\]
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}