erster Entwurf Kapitel Funktionentheorie

13 files changed, 390 insertions, 0 deletions

diff --git a/buch/chapters/080-funktionentheorie/images/Makefile b/buch/chapters/080-funktionentheorie/images/Makefile
new file mode 100644
index 0000000..66e6d0f
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/Makefile
@@ -0,0 +1,26 @@
+#
+# Makefile -- build images
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	nonanalytic.pdf integralanalytisch.pdf laurent.pdf \
+	fortsetzreziprok.pdf forts.pdf logforts.pdf
+
+nonanalytic.pdf:	nonanalytic.tex
+	pdflatex nonanalytic.tex
+
+integralanalytisch.pdf:	integralanalytisch.tex
+	pdflatex integralanalytisch.tex
+
+laurent.pdf:	laurent.tex
+	pdflatex laurent.tex
+
+fortsetzreziprok.pdf:	fortsetzreziprok.tex
+	pdflatex fortsetzreziprok.tex
+
+forts.pdf:	forts.tex
+	pdflatex forts.tex
+
+logforts.pdf:	logforts.tex
+	pdflatex logforts.tex
+
diff --git a/buch/chapters/080-funktionentheorie/images/forts.pdf b/buch/chapters/080-funktionentheorie/images/forts.pdf
new file mode 100644
index 0000000..db8199a
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/forts.pdf
diff --git a/buch/chapters/080-funktionentheorie/images/forts.tex b/buch/chapters/080-funktionentheorie/images/forts.tex
new file mode 100644
index 0000000..11c45e1
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/forts.tex
@@ -0,0 +1,86 @@
+%
+% forts.tex -- analytische Fortsetzung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\pgfmathparse{-(3+3*sin(180*(1.4))-(1.4))}
+\xdef\X{\pgfmathresult}
+\pgfmathparse{-(2+2*cos(180*(1.4))-(1.4))}
+\xdef\Y{\pgfmathresult}
+
+\def\kurve#1{
+	({3+3*sin(180*(#1))-(#1)+\X},{2+2*cos(180*(#1))-(#1)+\Y})
+}
+
+\def\punkt#1{
+	\fill[color=white]
+		({3+3*sin(180*(#1))-(#1)+\X},{2+2*cos(180*(#1))-(#1)+\Y})
+		circle[radius=0.08];
+	\draw[color=red]
+		({3+3*sin(180*(#1))-(#1)+\X},{2+2*cos(180*(#1))-(#1)+\Y})
+		circle[radius=0.08];
+}
+
+\def\kreis#1#2{
+	\fill[color=gray!50,opacity=0.5] #1 circle[radius=#2];
+}
+\def\rand#1#2{
+	\draw #1 circle[radius=#2];
+}
+
+\kreis{\kurve{-0.2}}{1.2}
+\kreis{\kurve{0.0}}{1.2}
+\kreis{\kurve{0.2}}{1.2}
+\kreis{\kurve{0.4}}{1.2}
+\kreis{\kurve{0.6}}{1.2}
+\kreis{\kurve{0.8}}{1.3}
+\kreis{\kurve{1.0}}{1.5}
+\kreis{\kurve{1.2}}{1.3}
+\kreis{\kurve{1.4}}{1.2}
+\rand{\kurve{-0.2}}{1.2}
+\rand{\kurve{0.0}}{1.2}
+\rand{\kurve{0.2}}{1.2}
+\rand{\kurve{0.4}}{1.2}
+\rand{\kurve{0.6}}{1.2}
+\rand{\kurve{0.8}}{1.3}
+\rand{\kurve{1.0}}{1.5}
+\rand{\kurve{1.2}}{1.3}
+\rand{\kurve{1.4}}{1.2}
+
+\draw[->] (-1.5,0) -- (8.5,0) coordinate[label={$\operatorname{Re}z$}];
+\draw[->] (0,-2.6) -- (0,5.6) coordinate[label={left:$\operatorname{Im}z$}];
+
+\draw[color=red,line width=1.4pt]
+	plot[domain=-0.2:1.4,samples=100] 
+	({3+3*sin(180*\x)-\x+\X},{2+2*cos(180*\x)-\x+\Y});
+
+\foreach \t in {-0.2,0,0.2,0.4,0.6,0.8,1.0,1.2,1.4}{
+	\punkt{\t}
+}
+
+\node[color=red] at \kurve{1.4} [above left] {$z_0$};
+\node[color=red] at \kurve{1.2} [below] {$z_1$};
+\node[color=red] at \kurve{1.0} [below] {$z_2$};
+\node[color=red] at \kurve{0.8} [below right] {$z_3$};
+\node[color=red] at \kurve{0.6} [right] {$z_4$};
+\node[color=red] at \kurve{0.4} [right] {$z_5$};
+\node[color=red] at \kurve{0.2} [above right] {$z_6$};
+\node[color=red] at \kurve{0.0} [above] {$z_7$};
+\node[color=red] at \kurve{-0.2} [above left] {$z_8$};
+
+\node[color=red] at \kurve{0.96} [above] {$\gamma$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/080-funktionentheorie/images/fortsetzreziprok.pdf b/buch/chapters/080-funktionentheorie/images/fortsetzreziprok.pdf
new file mode 100644
index 0000000..9450e80
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/fortsetzreziprok.pdf
diff --git a/buch/chapters/080-funktionentheorie/images/fortsetzreziprok.tex b/buch/chapters/080-funktionentheorie/images/fortsetzreziprok.tex
new file mode 100644
index 0000000..4fa4a68
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/fortsetzreziprok.tex
@@ -0,0 +1,65 @@
+%
+% fortsetzreziprok.tex -- analytische Fortsetzung der 1/z-Funktion
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{document}
+\def\skala{1}
+\def\u{3}
+\def\punkt#1#2{
+	\fill[color=white] #1 circle[radius=0.08];
+	\draw[color=#2] #1 circle[radius=0.08];
+}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\fill[color=gray!70,opacity=0.5] (\u,0) circle[radius=\u];
+\fill[color=gray!70,opacity=0.5] (0,\u) circle[radius=\u];
+\fill[color=gray!70,opacity=0.5] (-\u,0) circle[radius=\u];
+\fill[color=gray!70,opacity=0.5] (0,-\u) circle[radius=\u];
+
+%\draw[line width=0.5pt] (\u,0) circle[radius=\u];
+%\draw[line width=0.5pt] (0,\u) circle[radius=\u];
+%\draw[line width=0.5pt] (-\u,0) circle[radius=\u];
+%\draw[line width=0.5pt] (0,-\u) circle[radius=\u];
+
+\draw[->] ({-0.5-2*\u},0) -- ({2*\u+0.8},0)
+	coordinate[label={$\operatorname{Re}z$}];
+\draw[->] (0,{-0.5-2*\u}) -- (0,{2*\u+0.8})
+	coordinate[label={$\operatorname{Im}z$}];
+
+\draw[color=darkgreen,line width=1.4pt] (0,0) circle[radius=\u];
+\node[color=darkgreen] at (70:\u) [above] {$\gamma_+$};
+\node[color=darkgreen] at (-70:\u) [below] {$\gamma_-$};
+\node at (\u,0) [above right] {$1$};
+\node at (0,\u) [above left] {$i$};
+\node at (-\u,0) [above left] {$-1$};
+\node at (0,-\u) [below left] {$-i$};
+
+\punkt{(\u,0)}{black}
+\punkt{(-\u,0)}{black}
+\punkt{(0,\u)}{black}
+\punkt{(0,-\u)}{black}
+
+\punkt{({0.5*\u},{0.5*\u})}{red}
+\punkt{({-0.5*\u},{0.5*\u})}{red}
+\punkt{({-0.5*\u},{-0.5*\u})}{red}
+\punkt{({0.5*\u},{-0.5*\u})}{red}
+
+\node[color=red] at ({0.5*\u},{0.5*\u}) [above] {$(1+i)\frac{\sqrt{2}}{2}$};
+\node[color=red] at ({0.5*\u},{-0.5*\u}) [below] {$(1-i)\frac{\sqrt{2}}{2}$};
+\node[color=red] at ({-0.5*\u},{-0.5*\u}) [below] {$(-1-i)\frac{\sqrt{2}}{2}$};
+\node[color=red] at ({-0.5*\u},{0.5*\u}) [above] {$(-1+i)\frac{\sqrt{2}}{2}$};
+
+\fill[color=white] (0,0) circle[radius=0.2];
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/080-funktionentheorie/images/integralanalytisch.pdf b/buch/chapters/080-funktionentheorie/images/integralanalytisch.pdf
new file mode 100644
index 0000000..051562f
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/integralanalytisch.pdf
diff --git a/buch/chapters/080-funktionentheorie/images/integralanalytisch.tex b/buch/chapters/080-funktionentheorie/images/integralanalytisch.tex
new file mode 100644
index 0000000..a407eed
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/integralanalytisch.tex
@@ -0,0 +1,51 @@
+%
+% integralanalytisch.tex -- Illustration zum Beweis, das das Cauchy-Integral
+%                           auf eine analytische Funktion führt
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\fill[color=blue!20] (0,0) circle[radius=1.5];
+\draw[color=blue,line width=0.7pt] (0,0) circle[radius=1.5];
+
+\draw[->] (0,0) -- (-150:1.5);
+\node at (-150:1.0) [below] {$\varrho$};
+
+\begin{scope}
+\clip (-4,-3) rectangle (4,3);
+\draw[color=red, line width=1.4pt]
+	 (-3,0.5) 
+	.. controls (-4,-0.5) and (-3,-2) ..
+	(-2,-2)
+	.. controls (-1,-2) and (-1,-1.5) ..
+	(0,-1.5)
+	.. controls (1.0,-1.5) and (1.0,-3) ..
+	(2,-3)
+	.. controls (5,-3) and (3,5) ..
+	(-1,2);
+\end{scope}
+
+\node[color=red] at (3.2,-1.5) {$\gamma$};
+
+\coordinate (Z) at (1,0.5);
+
+\fill[color=white] (Z) circle[radius=0.05];
+\draw (Z) circle[radius=0.05];
+\node at (Z) [above] {$z$};
+
+\draw[->] (0,-3.1) -- (0,3.3) coordinate[label={left:$\operatorname{Im}z$}];
+\draw[->] (-4.1,0) -- (4.3,0) coordinate[label={$\operatorname{Re}z$}];
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/080-funktionentheorie/images/laurent.pdf b/buch/chapters/080-funktionentheorie/images/laurent.pdf
new file mode 100644
index 0000000..7cc7652
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/laurent.pdf
diff --git a/buch/chapters/080-funktionentheorie/images/laurent.tex b/buch/chapters/080-funktionentheorie/images/laurent.tex
new file mode 100644
index 0000000..b403aea
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/laurent.tex
@@ -0,0 +1,57 @@
+%
+% laurent.tex -- Laurent-Reihen und Cauchy-Integral
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc,decorations.markings}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\coordinate (Z0) at (1.5,1);
+
+\def\d{0.03}
+
+\draw[->] (-2.1,0) -- (4.8,0) coordinate[label={$\operatorname{Re}z$}];
+\draw[->] (0,-2.1) -- (0,4.3) coordinate[label={$\operatorname{Im}z$}];
+
+\fill[color=red!20,opacity=0.7] (Z0) circle[radius=2.5];
+
+\fill[color=white] (Z0) circle[radius=0.2];
+
+
+\draw[color=red] (Z0) circle[radius=2.5];
+\draw[color=red] (Z0) circle[radius=0.2];
+\fill[color=white] ($(Z0)+(0,-\d)$) rectangle ($(Z0)+(3,\d)$);
+\begin{scope}[decoration={
+    markings,
+    mark=at position 0.5 with {\arrow{>}}}
+    ]
+\draw[color=red,postaction={decorate}]
+	($(Z0)+({asin(-\d/2.5)}:2.5)$)
+	--
+	($(Z0)+({asin(-\d/0.2)}:0.2)$);
+\draw[color=red,postaction={decorate}]
+	($(Z0)+({asin(\d/0.2)}:0.2)$)
+	--
+	($(Z0)+({asin(\d/2.5)}:2.5)$);
+\end{scope}
+
+\draw (Z0) circle[radius=0.05];
+\node at ($(Z0)+(-0.1,-0.1)$) [below left] {$z_0$};
+
+\node[color=red] at (2.75,1) [above] {$l_1$};
+\node[color=red] at (2.75,1) [below] {$l_2$};
+
+\node[color=red] at ($(Z0)+(45:2.5)$) [above right] {$\gamma_1$};
+\node[color=red] at ($(Z0)+(0,0.2)$) [above] {$\gamma_2$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/080-funktionentheorie/images/logforts.pdf b/buch/chapters/080-funktionentheorie/images/logforts.pdf
new file mode 100644
index 0000000..82428cc
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/logforts.pdf
diff --git a/buch/chapters/080-funktionentheorie/images/logforts.tex b/buch/chapters/080-funktionentheorie/images/logforts.tex
new file mode 100644
index 0000000..241dae6
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/logforts.tex
@@ -0,0 +1,65 @@
+%
+% logforts.tex -- analytische Fortsetzung der Logarithmus-Funktion
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\begin{document}
+\def\skala{2}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+\def\r{1.2}
+\def\a{65}
+
+\fill[color=gray!40] (0,0) -- (0:0.4) arc (0:\a:0.4) -- cycle;
+\node at ({\a/2}:0.3) {$t$};
+
+\draw[->] (-2.5,0) -- (2.5,0) coordinate[label={$\operatorname{Re}z$}];
+\draw[->] (0,-1.6) -- (0,1.6) coordinate[label={right:$\operatorname{Im}z$}];
+
+\draw (1,{-0.1/\skala}) -- (1,{0.1/\skala});
+\draw (2,{-0.1/\skala}) -- (2,{0.1/\skala});
+\draw (-1,{-0.1/\skala}) -- (-1,{0.1/\skala});
+\draw (-2,{-0.1/\skala}) -- (-2,{0.1/\skala});
+\node at (1,0) [below] {$1$};
+\node at (2,0) [below] {$2$};
+\node at (-1,0) [below] {$-1$};
+\node at (-2,0) [below] {$-2$};
+\draw ({-0.1/\skala},1) -- ({0.1/\skala},1);
+\node at (0,1) [left] {$1$};
+\draw ({-0.1/\skala},-1) -- ({0.1/\skala},-1);
+\node at (0,-1) [left] {$-1$};
+
+\draw[->,color=red,line width=1.4pt] (0:\r) arc (0:357:\r);
+
+\fill[color=white] (0:\r) circle[radius=0.03];
+\draw (0:\r) circle[radius=0.03];
+\node at (0:\r) [above right] {$y(r)=\log r$};
+
+\def\punkt#1{
+	\fill[color=white] #1 circle[radius=0.03];
+	\draw[color=red] #1 circle[radius=0.03];
+}
+\draw[->] (0,0) -- (\a:\r);
+\punkt{(\a:\r)}
+\node at ($(\a:\r)+(0,-0.2)$) [above right] {$\displaystyle y(\gamma(t)) = \int_{\gamma_{|[0,t]}}\frac{1}{z}\,dz$};
+
+\punkt{(135:\r)}
+\node at (135:\r) [above left] {$y=\gamma(\frac34\pi))=\log r +\frac34\pi i$};
+
+\punkt{(252:\r)}
+\node at (252:\r) [below left] {$y=\gamma(\frac75\pi))=\log r +\frac75\pi i$};
+
+\draw[color=red,line width=0.4pt] (1.4,-1.1) -- (1.4,-0.2) -- (357:\r);
+\punkt{(357:\r)}
+
+\node at (1.4,-1.1) [below] {$y=\gamma(\frac{119}{60}\pi))=\log r +\frac{119}{60}\pi i$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/080-funktionentheorie/images/nonanalytic.pdf b/buch/chapters/080-funktionentheorie/images/nonanalytic.pdf
new file mode 100644
index 0000000..9cac699
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/nonanalytic.pdf
diff --git a/buch/chapters/080-funktionentheorie/images/nonanalytic.tex b/buch/chapters/080-funktionentheorie/images/nonanalytic.tex
new file mode 100644
index 0000000..258e0f5
--- /dev/null
+++ b/buch/chapters/080-funktionentheorie/images/nonanalytic.tex
@@ -0,0 +1,40 @@
+%
+% nonanalytic.tex -- nicht analytische reelle C^\infty-Funktion
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\draw[color=red,line width=1.4pt] plot[domain=0.01:2.5,samples=100]
+	({2*\x},{4*exp(-1/(\x*\x))});
+\draw[color=red,line width=1.4pt] plot[domain=0.01:2.5,samples=100]
+	({-2*\x},{4*exp(-1/(\x*\x))});
+
+\draw (2,-0.1) -- (2,0.1);
+\draw (4,-0.1) -- (4,0.1);
+\draw (-2,-0.1) -- (-2,0.1);
+\draw (-4,-0.1) -- (-4,0.1);
+
+\node at (2,0) [below] {$1$};
+\node at (4,0) [below] {$2$};
+\node at (-2,0) [below] {$-1$};
+\node at (-4,0) [below] {$-2$};
+
+\draw (-0.1,4) -- (0.1,4);
+\node at (-0.1,4) [left] {$1$};
+
+\draw[->] (-5.1,0) -- (5.4,0) coordinate[label={$x$}];
+\draw[->] (0,-0.1) -- (0,4.4) coordinate[label={right:$y$}];
+
+\end{tikzpicture}
+\end{document}
+