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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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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} + |