Merge branch 'AndreasFMueller:master' into master

13 files changed, 879 insertions, 1 deletions

diff --git a/buch/chapters/040-rekursion/images/0f1.cpp b/buch/chapters/040-rekursion/images/0f1.cpp
new file mode 100644
index 0000000..24ca3f1
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/0f1.cpp
@@ -0,0 +1,94 @@
+/*
+ * 0f1.cpp
+ *
+ * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+#include <cstring>
+#include <cstdio>
+#include <cstdlib>
+#include <cmath>
+#include <string>
+#include <iostream>
+#include <fstream>
+
+static int	N = 100;
+static double	xmin = -50;
+static double	xmax =  30;
+static int	points = 200;
+
+double	f(double b, double x) {
+	double	s = 1;
+	double	p = 1;
+	for (int k = 1; k < N; k++) {
+		p = p * x / (k * (b + k - 1.));
+		s += p;
+	}
+	return s;
+}
+
+typedef	std::pair<double, double> point_t;
+
+point_t	F(double b, double x) {
+	return std::make_pair(x, f(b, x));
+}
+
+std::string	ff(double f) {
+	if (f > 1000) { f = 1000; }
+	if (f < -1000) { f = -1000; }
+	char	b[128];
+	snprintf(b, sizeof(b), "%.4f", f);
+	return std::string(b);
+}
+
+std::ostream&	operator<<(std::ostream& out, const point_t& p) {
+	char	b[128];
+	out << "({" << ff(p.first) << "*\\dx},{" << ff(p.second) << "*\\dy})";
+	return out;
+}
+
+void	curve(std::ostream& out, double b, const std::string& name) {
+	double	h = (xmax - xmin) / points;
+	out << "\\def\\kurve" << name << "{";
+	out << std::endl << "\t" << F(b, xmin);
+	for (int i = 1; i <= points; i++) {
+		double	x = xmin + h * i;
+		out << std::endl << "\t-- " << F(b, x);
+	}
+	out << std::endl;
+	out << "}" << std::endl;
+}
+
+int	main(int argc, char *argv[]) {
+	std::ofstream	out("0f1data.tex");
+
+	double	s = 13/(xmax-xmin);
+	out << "\\def\\dx{" << ff(s) << "}" << std::endl;
+	out << "\\def\\dy{" << ff(s) << "}" << std::endl;
+	out << "\\def\\xmin{" << ff(s * xmin) << "}" << std::endl;
+	out << "\\def\\xmax{" << ff(s * xmax) << "}" << std::endl;
+
+	curve(out, 0.5, "one");
+	curve(out, 1.5, "two");
+	curve(out, 2.5, "three");
+	curve(out, 3.5, "four");
+	curve(out, 4.5, "five");
+	curve(out, 5.5, "six");
+	curve(out, 6.5, "seven");
+	curve(out, 7.5, "eight");
+	curve(out, 8.5, "nine");
+	curve(out, 9.5, "ten");
+
+	curve(out,-0.5, "none");
+	curve(out,-1.5, "ntwo");
+	curve(out,-2.5, "nthree");
+	curve(out,-3.5, "nfour");
+	curve(out,-4.5, "nfive");
+	curve(out,-5.5, "nsix");
+	curve(out,-6.5, "nseven");
+	curve(out,-7.5, "neight");
+	curve(out,-8.5, "nnine");
+	curve(out,-9.5, "nten");
+	
+	out.close();
+	return EXIT_SUCCESS;
+}
diff --git a/buch/chapters/040-rekursion/images/0f1.pdf b/buch/chapters/040-rekursion/images/0f1.pdf
new file mode 100644
index 0000000..2c35813
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/0f1.pdf
diff --git a/buch/chapters/040-rekursion/images/0f1.tex b/buch/chapters/040-rekursion/images/0f1.tex
new file mode 100644
index 0000000..1bc8b87
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/0f1.tex
@@ -0,0 +1,86 @@
+%
+% 0f1.tex -- template for standalon tikz images
+%
+% (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}
+\input{0f1data.tex}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\begin{scope}
+\clip (\xmin,-1) rectangle (\xmax,5);
+\draw[color=blue!5!red,line width=1.4pt] \kurveone;
+\draw[color=blue!16!red,line width=1.4pt] \kurvetwo;
+\draw[color=blue!26!red,line width=1.4pt] \kurvethree;
+\draw[color=blue!37!red,line width=1.4pt] \kurvefour;
+\draw[color=blue!47!red,line width=1.4pt] \kurvefive;
+\draw[color=blue!57!red,line width=1.4pt] \kurvesix;
+\draw[color=blue!68!red,line width=1.4pt] \kurveseven;
+\draw[color=blue!78!red,line width=1.4pt] \kurveeight;
+\draw[color=blue!89!red,line width=1.4pt] \kurvenine;
+\draw[color=blue!100!red,line width=1.4pt] \kurveten;
+\def\ds{0.4}
+\begin{scope}[yshift=0.5cm]
+\node[color=blue!5!red] at (\xmin,{1*\ds}) [right] {$\alpha=0.5$};
+\node[color=blue!16!red] at (\xmin,{2*\ds}) [right] {$\alpha=1.5$};
+\node[color=blue!26!red] at (\xmin,{3*\ds}) [right] {$\alpha=2.5$};
+\node[color=blue!37!red] at (\xmin,{4*\ds}) [right] {$\alpha=2.5$};
+\node[color=blue!47!red] at (\xmin,{5*\ds}) [right] {$\alpha=3.5$};
+\node[color=blue!57!red] at (\xmin,{6*\ds}) [right] {$\alpha=5.5$};
+\node[color=blue!68!red] at (\xmin,{7*\ds}) [right] {$\alpha=6.5$};
+\node[color=blue!78!red] at (\xmin,{8*\ds}) [right] {$\alpha=7.5$};
+\node[color=blue!89!red] at (\xmin,{9*\ds}) [right] {$\alpha=8.5$};
+\node[color=blue!100!red]at (\xmin,{10*\ds}) [right] {$\alpha=9.5$};
+\end{scope}
+\node at (-1.7,4.5) {$\displaystyle
+y=\mathstrut_0F_1\biggl(\begin{matrix}\text{---}\\\alpha\end{matrix};x\biggr)$};
+\end{scope}
+
+\draw[->] (\xmin-0.2,0) -- (\xmax+0.3,0) coordinate[label=$x$];
+\draw[->] (0,-0.5) -- (0,5.3) coordinate[label={right:$y$}];
+
+\begin{scope}[yshift=-6.5cm]
+\begin{scope}
+\clip (\xmin,-5) rectangle (\xmax,5);
+
+\draw[color=darkgreen!5!red,line width=1.4pt] \kurvenone;
+\draw[color=darkgreen!16!red,line width=1.4pt] \kurventwo;
+\draw[color=darkgreen!26!red,line width=1.4pt] \kurventhree;
+\draw[color=darkgreen!37!red,line width=1.4pt] \kurvenfour;
+\draw[color=darkgreen!47!red,line width=1.4pt] \kurvenfive;
+\draw[color=darkgreen!57!red,line width=1.4pt] \kurvensix;
+\draw[color=darkgreen!68!red,line width=1.4pt] \kurvenseven;
+\draw[color=darkgreen!78!red,line width=1.4pt] \kurveneight;
+\draw[color=darkgreen!89!red,line width=1.4pt] \kurvennine;
+\draw[color=darkgreen!100!red,line width=1.4pt] \kurventen;
+\end{scope}
+
+\draw[->] (\xmin-0.2,0) -- (\xmax+0.3,0) coordinate[label=$x$];
+\draw[->] (0,-5.2) -- (0,5.3) coordinate[label={right:$y$}];
+\def\ds{-0.4}
+\begin{scope}[yshift=-0.5cm]
+\node[color=darkgreen!5!red] at (\xmax,{1*\ds}) [left] {$\alpha=-0.5$};
+\node[color=darkgreen!16!red] at (\xmax,{2*\ds}) [left] {$\alpha=-1.5$};
+\node[color=darkgreen!26!red] at (\xmax,{3*\ds}) [left] {$\alpha=-2.5$};
+\node[color=darkgreen!37!red] at (\xmax,{4*\ds}) [left] {$\alpha=-2.5$};
+\node[color=darkgreen!47!red] at (\xmax,{5*\ds}) [left] {$\alpha=-3.5$};
+\node[color=darkgreen!57!red] at (\xmax,{6*\ds}) [left] {$\alpha=-5.5$};
+\node[color=darkgreen!68!red] at (\xmax,{7*\ds}) [left] {$\alpha=-6.5$};
+\node[color=darkgreen!78!red] at (\xmax,{8*\ds}) [left] {$\alpha=-7.5$};
+\node[color=darkgreen!89!red] at (\xmax,{9*\ds}) [left] {$\alpha=-8.5$};
+\node[color=darkgreen!100!red]at (\xmax,{10*\ds}) [left] {$\alpha=-9.5$};
+\end{scope}
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/040-rekursion/images/Makefile b/buch/chapters/040-rekursion/images/Makefile
index 9608a94..54ed23b 100644
--- a/buch/chapters/040-rekursion/images/Makefile
+++ b/buch/chapters/040-rekursion/images/Makefile
@@ -3,7 +3,8 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-all:	gammaplot.pdf fibonacci.pdf
+all:	gammaplot.pdf fibonacci.pdf order.pdf beta.pdf loggammaplot.pdf \
+	0f1.pdf
 
 gammaplot.pdf:	gammaplot.tex gammapaths.tex
 	pdflatex gammaplot.tex
@@ -16,3 +17,30 @@ fibonaccigrid.tex:	fibonacci.m
 
 fibonacci.pdf:	fibonacci.tex fibonaccigrid.tex
 	pdflatex fibonacci.tex
+
+order.pdf:      order.tex orderpath.tex
+	pdflatex order.tex
+
+orderpath.tex:  order.m
+	octave order.m
+
+beta.pdf:       beta.tex betapaths.tex
+	pdflatex beta.tex
+
+betapaths.tex:  betadist.m
+	octave betadist.m
+
+loggammaplot.pdf:	loggammaplot.tex loggammadata.tex
+	pdflatex loggammaplot.tex
+
+loggammadata.tex:	loggammaplot.m
+	octave loggammaplot.m
+
+0f1:	0f1.cpp
+	g++ -O -Wall -g -o 0f1 0f1.cpp
+
+0f1data.tex:	0f1
+	./0f1
+
+0f1.pdf:	0f1.tex 0f1data.tex
+	pdflatex 0f1.tex
diff --git a/buch/chapters/040-rekursion/images/beta.pdf b/buch/chapters/040-rekursion/images/beta.pdf
new file mode 100644
index 0000000..0e6567b
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/beta.pdf
diff --git a/buch/chapters/040-rekursion/images/beta.tex b/buch/chapters/040-rekursion/images/beta.tex
new file mode 100644
index 0000000..1e1a1b3
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/beta.tex
@@ -0,0 +1,236 @@
+%
+% beta.tex -- display some symmetric beta distributions
+%
+% (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}
+\input{betapaths.tex}
+\begin{document}
+\def\skala{12}
+\definecolor{colorone}{rgb}{1.0,0.6,0.0}
+\definecolor{colortwo}{rgb}{1.0,0.0,0.0}
+\definecolor{colorthree}{rgb}{0.6,0.0,0.6}
+\definecolor{colorfour}{rgb}{0.6,0.0,1.0}
+\definecolor{colorfive}{rgb}{0.0,0.0,1.0}
+\definecolor{colorsix}{rgb}{0.4,0.6,1.0}
+\definecolor{colorseven}{rgb}{0.0,0.0,0.0}
+\definecolor{coloreight}{rgb}{0.0,0.8,0.8}
+\definecolor{colornine}{rgb}{0.0,0.8,0.2}
+\definecolor{colorten}{rgb}{0.2,0.4,0.0}
+\definecolor{coloreleven}{rgb}{0.6,1.0,0.0}
+\definecolor{colortwelve}{rgb}{1.0,0.8,0.4}
+
+\def\achsen{
+	\foreach \x in {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9}{
+		\draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala});
+		\node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$};
+	}
+	\foreach \y in {1,2,3,4}{
+		\draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy});
+		\node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$};
+	}
+	\def\x{1}
+	\draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala});
+	\node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$};
+	\def\x{0}
+	\node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$};
+
+	\draw[->] ({-0.1/\skala},0) -- ({1*\dx+0.4/\skala},0)
+		coordinate[label={$x$}];
+	\draw[->] (0,{-0.1/\skala}) -- (0,{\betamax*\dy+0.4/\skala},0)
+		coordinate[label={right:$\beta(a,b,x)$}];
+}
+
+\def\farbcoord#1#2{
+	({\dx*(0.63+((#1)/5)*0.27)},{\dx*(0.18+((#2)/5)*0.27)})
+}
+\def\farbviereck{
+	\foreach \x in {1,2,3,4}{
+		\draw[color=gray!30] \farbcoord{\x}{0} -- \farbcoord{\x}{4};
+		\draw[color=gray!30] \farbcoord{0}{\x} -- \farbcoord{4}{\x};
+	}
+	\draw[->] \farbcoord{0}{0} -- \farbcoord{4.4}{0}
+		coordinate[label={$a$}];
+	\draw[->] \farbcoord{0}{0} -- \farbcoord{0}{4.4}
+		coordinate[label={left: $b$}];
+	\foreach \x in {1,2,3,4}{
+		\node[color=gray] at \farbcoord{4}{\x} [right] {\tiny $b=\x$};
+		%\fill[color=white,opacity=0.7] 
+		%	\farbcoord{(\x-0.1)}{3.3}
+		%	rectangle
+		%	\farbcoord{(\x+0.1)}{4};
+		\node[color=gray] at \farbcoord{\x}{4} [right,rotate=90]
+			{\tiny $a=\x$};
+	}
+}
+\def\farbpunkt#1#2#3{
+	\fill[color=#3] \farbcoord{#1}{#2} circle[radius={0.1/\skala}];
+}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\dx{1.15}
+\def\dy{0.1}
+\def\opa{0.1}
+
+\def\betamax{4.9}
+
+\begin{scope}
+\clip (0,0) rectangle ({1*\dx},{\betamax*\dy});
+\fill[color=colorone,opacity=\opa]    (0,0) -- \betaaa -- (\dx,0) -- cycle;
+\fill[color=colortwo,opacity=\opa]    (0,0) -- \betabb -- (\dx,0) -- cycle;
+\fill[color=colorthree,opacity=\opa]  (0,0) -- \betacc -- (\dx,0) -- cycle;
+\fill[color=colorfour,opacity=\opa]   (0,0) -- \betadd -- (\dx,0) -- cycle;
+\fill[color=colorfive,opacity=\opa]   (0,0) -- \betaee -- (\dx,0) -- cycle;
+\fill[color=colorsix,opacity=\opa]    (0,0) -- \betaff -- (\dx,0) -- cycle;
+\fill[color=colorseven,opacity=\opa]  (0,0) -- \betagg -- (\dx,0) -- cycle;
+\fill[color=coloreight,opacity=\opa]  (0,0) -- \betahh -- (\dx,0) -- cycle;
+\fill[color=colornine,opacity=\opa]   (0,0) -- \betaii -- (\dx,0) -- cycle;
+\fill[color=colorten,opacity=\opa]    (0,0) -- \betajj -- (\dx,0) -- cycle;
+\fill[color=coloreleven,opacity=\opa] (0,0) -- \betakk -- (\dx,0) -- cycle;
+\fill[color=colortwelve,opacity=\opa] (0,0) -- \betall -- (\dx,0) -- cycle;
+
+\draw[color=colorone]    \betaaa;
+\draw[color=colortwo]    \betabb;
+\draw[color=colorthree]  \betacc;
+\draw[color=colorfour]   \betadd;
+\draw[color=colorfive]   \betaee;
+\draw[color=colorsix]    \betaff;
+\draw[color=colorseven]  \betagg;
+\draw[color=coloreight]  \betahh;
+\draw[color=colornine]   \betaii;
+\draw[color=colorten]    \betajj;
+\draw[color=coloreleven] \betakk;
+\draw[color=colortwelve] \betall;
+
+\end{scope}
+
+\achsen
+
+\farbviereck
+
+\farbpunkt{\alphatwelve}{\betatwelve}{colortwelve}
+\farbpunkt{\alphaeleven}{\betaeleven}{coloreleven}
+\farbpunkt{\alphaten}{\betaten}{colorten}
+\farbpunkt{\alphanine}{\betanine}{colornine}
+\farbpunkt{\alphaeight}{\betaeight}{coloreight}
+\farbpunkt{\alphaseven}{\betaseven}{colorseven}
+\farbpunkt{\alphasix}{\betasix}{colorsix}
+\farbpunkt{\alphafive}{\betafive}{colorfive}
+\farbpunkt{\alphafour}{\betafour}{colorfour}
+\farbpunkt{\alphathree}{\betathree}{colorthree}
+\farbpunkt{\alphatwo}{\betatwo}{colortwo}
+\farbpunkt{\alphaone}{\betaone}{colorone}
+
+
+\def\betamax{4.9}
+
+\begin{scope}[yshift=-0.6cm]
+
+\begin{scope}
+\clip (0,0) rectangle ({1*\dx},{\betamax*\dy});
+\fill[color=colorone,opacity=\opa]    (0,0) -- \betaea -- (\dx,0) -- cycle;
+\fill[color=colortwo,opacity=\opa]    (0,0) -- \betaeb -- (\dx,0) -- cycle;
+\fill[color=colorthree,opacity=\opa]  (0,0) -- \betaec -- (\dx,0) -- cycle;
+\fill[color=colorfour,opacity=\opa]   (0,0) -- \betaed -- (\dx,0) -- cycle;
+\fill[color=colorfive,opacity=\opa]   (0,0) -- \betaee -- (\dx,0) -- cycle;
+\fill[color=colorsix,opacity=\opa]    (0,0) -- \betaef -- (\dx,0) -- cycle;
+\fill[color=colorseven,opacity=\opa]  (0,0) -- \betaeg -- (\dx,0) -- cycle;
+\fill[color=coloreight,opacity=\opa]  (0,0) -- \betaeh -- (\dx,0) -- cycle;
+\fill[color=colornine,opacity=\opa]   (0,0) -- \betaei -- (\dx,0) -- cycle;
+\fill[color=colorten,opacity=\opa]    (0,0) -- \betaej -- (\dx,0) -- cycle;
+\fill[color=coloreleven,opacity=\opa] (0,0) -- \betaek -- (\dx,0) -- cycle;
+\fill[color=colortwelve,opacity=\opa] (0,0) -- \betael -- (\dx,0) -- cycle;
+
+\draw[color=colorone]    \betaea;
+\draw[color=colortwo]    \betaeb;
+\draw[color=colorthree]  \betaec;
+\draw[color=colorfour]   \betaed;
+\draw[color=colorfive]   \betaee;
+\draw[color=colorsix]    \betaef;
+\draw[color=colorseven]  \betaeg;
+\draw[color=coloreight]  \betaeh;
+\draw[color=colornine]   \betaei;
+\draw[color=colorten]    \betaej;
+\draw[color=coloreleven] \betaek;
+\draw[color=colortwelve] \betael;
+\end{scope}
+
+\achsen
+
+\farbviereck
+
+\farbpunkt{\alphafive}{\betatwelve}{colortwelve}
+\farbpunkt{\alphafive}{\betaeleven}{coloreleven}
+\farbpunkt{\alphafive}{\betaten}{colorten}
+\farbpunkt{\alphafive}{\betanine}{colornine}
+\farbpunkt{\alphafive}{\betaeight}{coloreight}
+\farbpunkt{\alphafive}{\betaseven}{colorseven}
+\farbpunkt{\alphafive}{\betasix}{colorsix}
+\farbpunkt{\alphafive}{\betafive}{colorfive}
+\farbpunkt{\alphafive}{\betafour}{colorfour}
+\farbpunkt{\alphafive}{\betathree}{colorthree}
+\farbpunkt{\alphafive}{\betatwo}{colortwo}
+\farbpunkt{\alphafive}{\betaone}{colorone}
+
+\end{scope}
+
+\begin{scope}[yshift=-1.2cm]
+
+\begin{scope}
+\clip (0,0) rectangle ({1*\dx},{\betamax*\dy});
+\fill[color=colorone,opacity=\opa]    (0,0) -- \betaal -- (\dx,0) -- cycle;
+\fill[color=colortwo,opacity=\opa]    (0,0) -- \betabl -- (\dx,0) -- cycle;
+\fill[color=colorthree,opacity=\opa]  (0,0) -- \betacl -- (\dx,0) -- cycle;
+\fill[color=colorfour,opacity=\opa]   (0,0) -- \betadl -- (\dx,0) -- cycle;
+\fill[color=colorfive,opacity=\opa]   (0,0) -- \betael -- (\dx,0) -- cycle;
+\fill[color=colorsix,opacity=\opa]    (0,0) -- \betafl -- (\dx,0) -- cycle;
+\fill[color=colorseven,opacity=\opa]  (0,0) -- \betagl -- (\dx,0) -- cycle;
+\fill[color=coloreight,opacity=\opa]  (0,0) -- \betahl -- (\dx,0) -- cycle;
+\fill[color=colornine,opacity=\opa]   (0,0) -- \betail -- (\dx,0) -- cycle;
+\fill[color=colorten,opacity=\opa]    (0,0) -- \betajl -- (\dx,0) -- cycle;
+\fill[color=coloreleven,opacity=\opa] (0,0) -- \betakl -- (\dx,0) -- cycle;
+\fill[color=colortwelve,opacity=\opa] (0,0) -- \betall -- (\dx,0) -- cycle;
+
+\draw[color=colorone]    \betaal;
+\draw[color=colortwo]    \betabl;
+\draw[color=colorthree]  \betacl;
+\draw[color=colorfour]   \betadl;
+\draw[color=colorfive]   \betael;
+\draw[color=colorsix]    \betafl;
+\draw[color=colorseven]  \betagl;
+\draw[color=coloreight]  \betahl;
+\draw[color=colornine]   \betail;
+\draw[color=colorten]    \betajl;
+\draw[color=coloreleven] \betakl;
+\draw[color=colortwelve] \betall;
+\end{scope}
+
+\achsen
+
+\farbviereck
+
+\farbpunkt{\alphatwelve}{\betatwelve}{colortwelve}
+\farbpunkt{\alphaeleven}{\betatwelve}{coloreleven}
+\farbpunkt{\alphaten}{\betatwelve}{colorten}
+\farbpunkt{\alphanine}{\betatwelve}{colornine}
+\farbpunkt{\alphaeight}{\betatwelve}{coloreight}
+\farbpunkt{\alphaseven}{\betatwelve}{colorseven}
+\farbpunkt{\alphasix}{\betatwelve}{colorsix}
+\farbpunkt{\alphafive}{\betatwelve}{colorfive}
+\farbpunkt{\alphafour}{\betatwelve}{colorfour}
+\farbpunkt{\alphathree}{\betatwelve}{colorthree}
+\farbpunkt{\alphatwo}{\betatwelve}{colortwo}
+\farbpunkt{\alphaone}{\betatwelve}{colorone}
+
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/040-rekursion/images/betadist.m b/buch/chapters/040-rekursion/images/betadist.m
new file mode 100644
index 0000000..5b466a6
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/betadist.m
@@ -0,0 +1,58 @@
+#
+# betadist.m
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+global N;
+N = 201;
+global nmin;
+global nmax;
+nmin = -4;
+nmax = 7;
+n = nmax - nmin + 1
+A = 3;
+
+t = (nmin:nmax) / nmax;
+alpha = 1 + A * t .* abs(t)
+#alpha(1) = 0.01;
+
+#alpha = [ 1, 1.03, 1.05, 1.1, 1.25, 1.5, 2, 2.5, 3, 4, 5 ];
+beta = alpha;
+names = [ "one"; "two"; "three"; "four"; "five"; "six"; "seven"; "eight";
+	  "nine"; "ten"; "eleven"; "twelve" ]
+
+function retval = Beta(a, b, x)
+	retval = x^(a-1) * (1-x)^(b-1) / beta(a, b);
+	if (retval > 100)
+		retval = 100
+	end
+end
+
+function plotbeta(fn, a, b, name)
+	global N;
+	fprintf(fn, "\\def\\beta%s{\n", strtrim(name));
+	fprintf(fn, "\t({%.4f*\\dx},{%.4f*\\dy})", 0, Beta(a, b, 0));
+	for x = (1:N-1)/(N-1)
+		X = (1-cos(pi * x))/2;
+		fprintf(fn, "\n\t--({%.4f*\\dx},{%.4f*\\dy})",
+			X, Beta(a, b, X));
+	end
+	fprintf(fn, "\n}\n");
+end
+
+fn = fopen("betapaths.tex", "w");
+
+for i = (1:n)
+	fprintf(fn, "\\def\\alpha%s{%f}\n", strtrim(names(i,:)), alpha(i));
+	fprintf(fn, "\\def\\beta%s{%f}\n", strtrim(names(i,:)), beta(i));
+end
+
+for i = (1:n)
+	for j = (1:n)
+		printf("working on %d,%d:\n", i, j);
+		plotbeta(fn, alpha(i), beta(j),
+			char(['a' + i - 1, 'a' + j - 1]));
+	end
+end
+
+fclose(fn);
diff --git a/buch/chapters/040-rekursion/images/loggammaplot.m b/buch/chapters/040-rekursion/images/loggammaplot.m
new file mode 100644
index 0000000..5456e4f
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/loggammaplot.m
@@ -0,0 +1,43 @@
+#
+# loggammaplot.m
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+xmax = 10;
+xmin = 0.1;
+N = 500;
+
+fn = fopen("loggammadata.tex", "w");
+
+fprintf(fn, "\\def\\loggammapath{\n ({%.4f*\\dx},{%.4f*\\dy})",
+	xmax, log(gamma(xmax)));
+xstep = (xmax - 1) / N;
+for x = (xmax:-xstep:1)
+	fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", x, log(gamma(x)));
+endfor
+for k = (0:0.2:10)
+	x = exp(-k);
+	fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", x, log(gamma(x)));
+endfor
+fprintf(fn, "\n}\n");
+
+function retval = lgp(fn, x0, name)
+	fprintf(fn, "\\def\\loggammaplot%s{\n", name);
+	fprintf(fn, "\\draw[color=red,line width=1pt] ");
+	for k = (-7:0.1:7)
+		x = x0 + 0.5 * tanh(k);
+		if (k > -5)
+			fprintf(fn, "\n\t-- ");
+		end
+		fprintf(fn, "({%.4f*\\dx},{%.4f*\\dy})", x, log(gamma(x)));
+	endfor
+	fprintf(fn, ";\n}\n");
+endfunction
+
+lgp(fn, -0.5, "zero");
+lgp(fn, -1.5, "one");
+lgp(fn, -2.5, "two");
+lgp(fn, -3.5, "three");
+lgp(fn, -4.5, "four");
+
+fclose(fn);
diff --git a/buch/chapters/040-rekursion/images/loggammaplot.pdf b/buch/chapters/040-rekursion/images/loggammaplot.pdf
new file mode 100644
index 0000000..a2963f2
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/loggammaplot.pdf
diff --git a/buch/chapters/040-rekursion/images/loggammaplot.tex b/buch/chapters/040-rekursion/images/loggammaplot.tex
new file mode 100644
index 0000000..8ca4e1c
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/loggammaplot.tex
@@ -0,0 +1,89 @@
+%
+% tikztemplate.tex -- template for standalon tikz images
+%
+% (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}
+\input{loggammadata.tex}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+% add image content here
+
+\def\dx{1}
+\def\dy{0.6}
+\def\xmax{8}
+\def\xmin{-4.9}
+\def\ymax{8}
+\def\ymin{-3.1}
+
+\fill[color=blue!20] ({\xmin*\dx},{\ymin*\dy}) rectangle ({-4*\dx},{\ymax*\dy});
+\fill[color=blue!20] ({-3*\dx},{\ymin*\dy}) rectangle ({-2*\dx},{\ymax*\dy});
+\fill[color=blue!20] ({-1*\dx},{\ymin*\dy}) rectangle ({-0*\dx},{\ymax*\dy});
+
+\draw[->] ({\xmin*\dx-0.1},0) -- ({\xmax*\dx+0.3},0)
+	coordinate[label={$x$}];
+\draw[->] (0,{\ymin*\dy-0.1}) -- (0,{\ymax*\dy+0.3})
+	coordinate[label={right:$y$}];
+
+\begin{scope}
+\clip ({\xmin*\dx},{\ymin*\dy}) rectangle ({\xmax*\dx},{\ymax*\dy});
+
+\foreach \x in {-1,-2,-3,-4}{
+	\draw[color=blue,line width=0.3pt]
+		({\x*\dx},{\ymin*\dy}) -- ({\x*\dx},{\ymax*\dy});
+}
+
+\draw[color=red,line width=1pt] \loggammapath;
+
+\loggammaplotzero
+\loggammaplotone
+\loggammaplottwo
+\loggammaplotthree
+\loggammaplotfour
+
+\end{scope}
+
+\foreach \y in {0.1,10,100,1000,1000}{
+	\draw[line width=0.3pt]
+		({\xmin*\dx},{ln(\y)*\dy}) 
+		--
+		({\xmax*\dx},{ln(\y)*\dy}) ;
+}
+
+\foreach \x in {1,...,8}{
+	\draw ({\x*\dx},{-0.05}) -- ({\x*\dx},{0.05});
+	\node at ({\x*\dx},0) [below] {$\x$};
+}
+
+\foreach \x in {-1,...,-4}{
+	\draw ({\x*\dx},{-0.05}) -- ({\x*\dx},{0.05});
+}
+\foreach \x in {-1,...,-3}{
+	\node at ({\x*\dx},0) [below right] {$\x$};
+}
+\node at ({-4*\dx},0) [below left] {$-4$};
+
+\def\htick#1#2{
+	\draw (-0.05,{ln(#1)*\dy}) -- (0.05,{ln(#1)*\dy});
+	\node at (0,{ln(#1)*\dy}) [above right] {#2};
+}
+
+\htick{10}{$10^1$}
+\htick{100}{$10^2$}
+\htick{1000}{$10^3$}
+\htick{0.1}{$10^{-1}$}
+
+\node[color=red] at ({3*\dx},{ln(30)*\dy}) {$y=\log|\Gamma(x)|$};
+
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/040-rekursion/images/order.m b/buch/chapters/040-rekursion/images/order.m
new file mode 100644
index 0000000..762f458
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/order.m
@@ -0,0 +1,119 @@
+#
+# order.m
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+global N;
+N = 10;
+global subdivisions;
+subdivisions = 100;
+global P;
+P = 0.5
+
+function retval = orderF(p, n, k)
+	retval = 0;
+	for i = (k:n)
+		retval = retval + nchoosek(n,i) * p^i * (1-p)^(n-i);
+	end
+end
+
+function retval = orderd(p, n, k)
+	retval = 0;
+	for i = (k:n)
+		s = i * p^(i-1) * (1-p)^(n-i);
+		s = s - p^i * (n-i) * (1-p)^(n-i-1);
+		retval = retval + nchoosek(n,i) * s;
+	end
+end
+
+function retval = orders(p, n, k)
+	retval = k * nchoosek(n, k) * p^(k-1) * (1-p)^(n-k);
+end
+
+function orderpath(fn, k, name)
+	fprintf(fn, "\\def\\order%s{\n\t(0,0)", name);
+	global N;
+	global subdivisions;
+	for i = (0:subdivisions)
+		p = i/subdivisions;
+		fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})",
+			p, orderF(p, N, k));
+	end
+	fprintf(fn, "\n}\n");
+end
+
+function orderdpath(fn, k, name)
+	fprintf(fn, "\\def\\orderd%s{\n\t(0,0)", name);
+	global N;
+	global subdivisions;
+	for i = (1:subdivisions-1)
+		p = i/subdivisions;
+		fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})",
+			p, orderd(p, N, k));
+	end
+	fprintf(fn, "\n\t-- ({1*\\dx},0)");
+	fprintf(fn, "\n}\n");
+end
+
+function orderspath(fn, k, name)
+	fprintf(fn, "\\def\\orders%s{\n\t(0,0)", name);
+	global N;
+	global subdivisions;
+	for i = (1:subdivisions-1)
+		p = i/subdivisions;
+		fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})",
+			p, orders(p, N, k));
+	end
+	fprintf(fn, "\n\t-- ({1*\\dx},0)");
+	fprintf(fn, "\n}\n");
+end
+
+fn = fopen("orderpath.tex", "w");
+
+orderpath(fn,   0, "zero");
+orderdpath(fn,  0, "zero");
+orderspath(fn,  0, "zero");
+
+orderpath(fn,   1, "one");
+orderdpath(fn,  1, "one");
+orderspath(fn,  1, "one");
+
+orderpath(fn,   2, "two");
+orderdpath(fn,  2, "two");
+orderspath(fn,  2, "two");
+
+orderpath(fn,   3, "three");
+orderdpath(fn,  3, "three");
+orderspath(fn,  3, "three");
+
+orderpath(fn,   4, "four");
+orderdpath(fn,  4, "four");
+orderspath(fn,  4, "four");
+
+orderpath(fn,   5, "five");
+orderdpath(fn,  5, "five");
+orderspath(fn,  5, "five");
+
+orderpath(fn,   6, "six");
+orderdpath(fn,  6, "six");
+orderspath(fn,  6, "six");
+
+orderpath(fn,   7, "seven");
+orderdpath(fn,  7, "seven");
+orderspath(fn,  7, "seven");
+
+orderpath(fn,   8, "eight");
+orderdpath(fn,  8, "eight");
+orderspath(fn,  8, "eight");
+
+orderpath(fn,   9, "nine");
+orderdpath(fn,  9, "nine");
+orderspath(fn,  9, "nine");
+
+orderpath(fn,  10, "ten");
+orderdpath(fn, 10, "ten");
+orderspath(fn, 10, "ten");
+
+fclose(fn);
+
+
diff --git a/buch/chapters/040-rekursion/images/order.pdf b/buch/chapters/040-rekursion/images/order.pdf
new file mode 100644
index 0000000..88b2b08
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/order.pdf
diff --git a/buch/chapters/040-rekursion/images/order.tex b/buch/chapters/040-rekursion/images/order.tex
new file mode 100644
index 0000000..0284735
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/order.tex
@@ -0,0 +1,125 @@
+%
+% order.tex -- Verteilungsfunktion für Ordnungsstatistik
+%
+% (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{8}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\n{10}
+\def\E#1#2{
+	\draw[color=#2]
+		({\dx*#1/(\n+1)},{-0.1/\skala}) -- ({\dx*#1/(\n+1)},{4.4*\dy});
+	\node[color=#2] at ({\dx*#1/(\n+1)},{3.2*\dy})
+		[rotate=90,above right] {$k=#1$};
+}
+\def\var#1#2{
+	\pgfmathparse{\dx*sqrt(#1*(\n-#1+1)/((\n+1)*(\n+1)*(\n+2)))}
+	\xdef\var{\pgfmathresult}
+	\fill[color=#2,opacity=0.5]
+		({\dx*#1/(\n+1)-\var},0) rectangle ({\dx*#1/(\n+1)+\var},{4.4*\dy});
+}
+
+\input{orderpath.tex}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\dx{1.6}
+\def\dy{0.5}
+
+\def\pfad#1#2{
+\draw[color=#2,line width=1.4pt] ({-0.1/\skala},0) 
+	--
+	#1
+	--
+	({1*\dx+0.1/\skala},0.5);
+}
+
+\pfad{\orderzero}{darkgreen!20}
+\pfad{\orderone}{darkgreen!20}
+\pfad{\ordertwo}{darkgreen!20}
+\pfad{\orderthree}{darkgreen!20}
+\pfad{\orderfour}{darkgreen!20}
+\pfad{\orderfive}{darkgreen!20}
+\pfad{\ordersix}{darkgreen!20}
+\pfad{\ordereight}{darkgreen!20}
+\pfad{\ordernine}{darkgreen!20}
+\pfad{\orderten}{darkgreen!20}
+\pfad{\orderseven}{darkgreen}
+
+\draw[->] ({-0.1/\skala},0) -- ({1.03*\dx},0) coordinate[label={$x$}];
+\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={right:$F(X)$}];
+\foreach \x in {0,0.2,0.4,0.6,0.8,1}{
+	\draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala});
+	\node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$};
+}
+\foreach \y in {0.5,1}{
+	\draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy});
+	\node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$};
+}
+
+\node[color=darkgreen] at ({0.64*\dx},{0.56*\dy}) [rotate=42] {$k=7$};
+
+\begin{scope}[yshift=-0.7cm]
+\def\dy{0.125}
+
+\foreach \k in {1,2,3,4,5,6,8,9,10}{
+	\E{\k}{blue!30}
+}
+\def\k{7}
+\var{\k}{orange!40}
+\node[color=blue] at ({\dx*\k/(\n+1)},{4.3*\dy}) [above] {$E(X_{7:n})$};
+
+\def\pfad#1#2{
+	\draw[color=#2,line width=1.4pt] ({-0.1/\skala},0) 
+		--
+		#1
+		--
+		({1*\dx+0.1/\skala},0.0);
+}
+
+\begin{scope}
+\clip ({-0.1/\skala},{-0.1/\skala})
+	rectangle ({1*\dx+0.1/\skala},{0.56+0.1/\skala});
+
+\pfad{\orderdzero}{red!20}
+\pfad{\orderdone}{red!20}
+\pfad{\orderdtwo}{red!20}
+\pfad{\orderdthree}{red!20}
+\pfad{\orderdfour}{red!20}
+\pfad{\orderdfive}{red!20}
+\pfad{\orderdsix}{red!20}
+\pfad{\orderdeight}{red!20}
+\pfad{\orderdnine}{red!20}
+\pfad{\orderdten}{red!20}
+\E{\k}{blue}
+\pfad{\orderdseven}{red}
+
+\end{scope}
+
+\draw[->] ({-0.1/\skala},0) -- ({1.03*\dx},0) coordinate[label={$x$}];
+\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={right:$\varphi(X)$}];
+\foreach \x in {0,0.2,0.4,0.6,0.8,1}{
+	\draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala});
+	\node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$};
+}
+\foreach \y in {1,2,3,4}{
+	\draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy});
+	\node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$};
+}
+
+\node[color=red] at ({0.67*\dx},{2.7*\dy}) [above] {$k=7$};
+
+
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+