Merge branch 'master' of github.com:AndreasFMueller/SeminarSpezielleFunktionen

10 files changed, 804 insertions, 1 deletions

diff --git a/buch/chapters/110-elliptisch/images/Makefile b/buch/chapters/110-elliptisch/images/Makefile
index afa70ba..931564b 100644
--- a/buch/chapters/110-elliptisch/images/Makefile
+++ b/buch/chapters/110-elliptisch/images/Makefile
@@ -3,11 +3,34 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-all:	lemniskate.pdf ellipse.pdf
+all:	lemniskate.pdf ellipsenumfang.pdf unvollstaendig.pdf rechteck.pdf \
+	ellipse.pdf
 
 lemniskate.pdf:	lemniskate.tex
 	pdflatex lemniskate.tex
 
+ellipsenumfang.pdf:	ellipsenumfang.tex ekplot.tex
+	pdflatex ellipsenumfang.tex
+
+ekplot.tex:	ellipsenumfang.m
+	octave ellipsenumfang.m
+
+rechteck:	rechteck.cpp
+	g++ `pkg-config --cflags gsl` `pkg-config --libs gsl` -O -Wall -g -std=c++11 rechteck.cpp -o rechteck
+
+rechteckpfade.tex:	rechteck
+	./rechteck --outfile rechteckpfade.tex
+
+rechteck.pdf:	rechteck.tex rechteckpfade.tex
+	pdflatex rechteck.tex
+
+unvollstaendig.pdf:	unvollstaendig.tex unvollpath.tex
+	pdflatex unvollstaendig.tex
+
+unvollpath.tex:	unvollstaendig.m
+	octave unvollstaendig.m
+
 ellipse.pdf:	ellipse.tex
 	pdflatex ellipse.tex
 
+
diff --git a/buch/chapters/110-elliptisch/images/ellipsenumfang.m b/buch/chapters/110-elliptisch/images/ellipsenumfang.m
new file mode 100644
index 0000000..9ac8fe2
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/ellipsenumfang.m
@@ -0,0 +1,21 @@
+#
+# ellipsenumfang
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+f = fopen("ekplot.tex", "w");
+fprintf(f, "\\def\\ekpath{\n");
+fprintf(f, "(0,{\\dy*%.4f})\n", pi / 2);
+for epsilon = (1:100) / 100
+	[k, e] = ellipke(epsilon^2);
+	fprintf(f, "--({\\dx*%.4f},{\\dy*%.4f})\n", epsilon, e);
+endfor
+fprintf(f, "\n}\n");
+fprintf(f, "\\def\\punkte{\n");
+for epsilon = (0.05:0.1:0.95)
+	[k, e] = ellipke(epsilon^2);
+	fprintf(f, "\\fill[color=blue] ({\\dx*%.4f},{\\dy*%.4f}) circle[radius=0.08];\n", epsilon, e);
+	fprintf(f, "\\draw[color=blue,line width=0.2pt] ({\\dx*%.4f},{\\dy*%.4f}) -- ({\\dx*%.4f},{\\dy*1.85});\n", epsilon, e, epsilon);
+endfor
+fprintf(f,"}\n");
+fclose(f);
diff --git a/buch/chapters/110-elliptisch/images/ellipsenumfang.pdf b/buch/chapters/110-elliptisch/images/ellipsenumfang.pdf
new file mode 100644
index 0000000..fc27f21
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/ellipsenumfang.pdf
diff --git a/buch/chapters/110-elliptisch/images/ellipsenumfang.tex b/buch/chapters/110-elliptisch/images/ellipsenumfang.tex
new file mode 100644
index 0000000..0d1b807
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/ellipsenumfang.tex
@@ -0,0 +1,83 @@
+%
+% ellipsenumfang.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}
+\input{ekplot.tex}
+\def\skala{1.19}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\pgfkeys{/pgf/number format/.cd, fixed, fixed zerofill, precision=1}
+
+\def\dx{10}
+\def\dy{4}
+
+%\draw[line width=0.4pt] (0,\dy) -- (10,\dy);
+%\draw[line width=0.4pt] (\dx,0) -- (10,\dy);
+
+\draw[->] (0,{\dy-0.1}) -- (0,7.0) coordinate[label={left:$E(k=\varepsilon)$}];
+\draw[->] (-0.1,\dy) -- (10.5,\dy) coordinate[label={$\varepsilon$}];
+
+\foreach \i in {0,...,10}{
+	\pgfmathparse{\i/10}
+	\xdef\wert{\pgfmathresult}
+	\draw (\i,{\dy-0.1}) -- (\i,{\dy+0.1});
+	\node at (\i,{\dy-0.1}) [below] {$\pgfmathprintnumber{\wert}$};
+}
+
+\draw[color=red,line width=1.4pt] \ekpath;
+\fill[color=red] (\dx,\dy) circle[radius=0.07];
+
+\foreach \y in {1.0,1.2,1.4}{
+	\draw (-0.1,{\dy*\y})  -- (0.1,{\dy*\y});
+	\node at (-0.1,{\dy*\y}) [left] {$\pgfmathprintnumber{\y}$};
+}
+
+\draw (-0.1,{\dy*3.14159/2}) -- (0.1,{\dy*3.14159/2});
+\node at (0,{\dy*3.14159/2}) [left] {$\displaystyle\frac{\pi}2$};
+
+
+\punkte
+
+\foreach \exzentrizitaet in {0.05,0.15,...,0.95}{
+	\pgfmathparse{sqrt(1-\exzentrizitaet*\exzentrizitaet)}
+	\xdef\halbachse{\pgfmathresult}
+
+	\pgfmathparse{\exzentrizitaet*\dx}
+	\xdef\mitte{\pgfmathresult}
+	%\node at (\mitte,1) {$\pgfmathprintnumber{\mitte}$};
+
+	\pgfmathparse{(1-\exzentrizitaet)}
+	\xdef\breite{\halbachse}
+	%\node at (\mitte,0.5) {$\pgfmathprintnumber{\breite}$};
+
+	\begin{scope}
+
+		\clip ({\mitte-(\breite/2)},{1.8*\dy})
+			rectangle ({\mitte+(\breite/2)+0.1},{1.8*\dy+1.1});
+		\fill[color=blue!20] ({\mitte-\breite/2},{1.8*\dy})
+			ellipse({\breite} and 1);
+		\draw[color=blue,line width=1.4pt]
+			({\mitte-\breite/2},{1.8*\dy})
+				ellipse({\breite} and 1);
+		\draw[color=blue,line width=0.2pt]
+			({\mitte-\breite/2},{1.8*\dy+1})
+			--
+			({\mitte-\breite/2},{1.8*\dy})
+			--
+			({\mitte+\breite/2},{1.8*\dy});
+	\end{scope}
+}
+
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/110-elliptisch/images/rechteck.cpp b/buch/chapters/110-elliptisch/images/rechteck.cpp
new file mode 100644
index 0000000..c65ae0f
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/rechteck.cpp
@@ -0,0 +1,415 @@
+/*
+ * rechteck.cpp
+ *
+ * (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+#include <cmath>
+#include <cstdlib>
+#include <cstdio>
+#include <complex>
+#include <iostream>
+#include <fstream>
+#include <list>
+#include <getopt.h>
+#include <gsl/gsl_sf_ellint.h>
+
+double	ast = 1;
+
+/**
+ * \brief Base class for integrands
+ */
+class integrand {
+protected:
+	double	_k;
+public:
+	double	k() const { return _k; }
+	integrand(double k) : _k(k) { }
+	virtual std::complex<double>	operator()(const std::complex<double>& z) const = 0;
+	static double	kprime(double k);
+};
+
+double	integrand::kprime(double k) {
+	return sqrt(1 - k*k);
+}
+
+/**
+ * \brief Elliptic integral of the first kind
+ */
+class integrand1 : public integrand {
+public:
+	integrand1(double k) : integrand(k) { }
+	virtual std::complex<double>	operator()(
+		const std::complex<double>& z) const {
+		std::complex<double>	square = z * z;
+		std::complex<double>	eins(1.0);
+		std::complex<double>	result = eins
+			/ sqrt((eins - square) * (eins - k() * k() * square));
+		return ast * ((result.imag() < 0) ? -result : result);
+	}
+};
+
+/**
+ * \brief A class to trace curves
+ */
+class curvetracer {
+	const integrand&	_f;
+public:
+	typedef std::list<std::complex<double> >	curve_t;
+	curvetracer(const integrand& f) : _f(f) { }
+
+	std::complex<double>	startpoint(const std::complex<double>& z,
+		int n = 100) const;
+
+	std::complex<double>	segment(const std::complex<double>& z1,
+		const std::complex<double>& z2, int n) const;
+
+	std::pair<std::complex<double>,std::complex<double> >	segmentl(
+		const std::complex<double>& start,
+		const std::complex<double>& dir,
+		double stepsize, int n = 10) const;
+
+	curve_t	trace(const std::complex<double>& startz,
+		const std::complex<double>& dir,
+		const std::complex<double>& startw,
+		int maxsteps) const;
+
+	static curve_t	mirrorx(const curve_t& c);
+	static curve_t	mirrory(const curve_t& c);
+	static curve_t	mirror(const curve_t& c);
+};
+
+/**
+ * \brief Perform integration for a 
+ *
+ * \param z1	the start point of the segment in the domain
+ * \param z2	the end point of the segment in the domain
+ * \param n	the number of integration steps to use
+ * \return	the increment along that segment
+ */
+std::complex<double>	curvetracer::segment(const std::complex<double>& z1,
+		const std::complex<double>& z2, int n = 100) const {
+	std::complex<double>	dz = z2 - z1;
+	std::complex<double>	summe(0);
+	double	h = 1. / (2 * n);
+	for (int i = 1; i < 2 * n; i += 2) {
+		double	t = i * h;
+		std::complex<double>	z = (1 - t) * z1 + t * z2;
+		summe += _f(z);
+	}
+	return dz * h * summe * 2.;
+}
+
+/**
+ * \brief Exception thrown when the number of iterations is exceeded
+ */
+class toomanyiterations : public std::runtime_error {
+public:
+	toomanyiterations(const std::string& cause)
+		: std::runtime_error(cause) {
+	}
+};
+
+/**
+ * \brief Perform integration with a given target length
+ *
+ * \param start		the start point
+ * \param dir		the direction in which to integrate
+ * \param stepsize	the required length of the step
+ * \param n		the number of integration steps
+ * \return		the increment in the range by this segment
+ */
+std::pair<std::complex<double>, std::complex<double> >	curvetracer::segmentl(
+	const std::complex<double>& start,
+	const std::complex<double>& dir,
+	double stepsize,
+	int n) const {
+	std::complex<double>	s(0.);
+	std::complex<double>	z = start;
+	int	counter = 100;
+	while (abs(s) < stepsize) {
+		s = s + segment(z, z + dir, n);
+		z = z + dir;
+		if (counter-- == 0) {
+			throw toomanyiterations("too many iterations");
+		}
+	}
+	return std::make_pair(z, s);
+}
+
+/**
+ * \brief Trace a curve from a starting point in the domain
+ *
+ * \param startz	the start point of the curve in the domain
+ * \param dir		the direction of the curve in the domain
+ * \param startw	the initial function value where the curve starts in
+ *			the range
+ * \param maxsteps	the maximum number of dir-steps before giving up
+ * \return		the curve as a list of points
+ */
+curvetracer::curve_t	curvetracer::trace(const std::complex<double>& startz,
+			const std::complex<double>& dir,
+			const std::complex<double>& startw,
+			int maxsteps) const {
+	curve_t	result;
+	std::complex<double>	z = startz;
+	std::complex<double>	w = startw;
+	result.push_back(w);
+	while (maxsteps-- > 0) {
+		try {
+			auto	seg = segmentl(z, dir, abs(dir), 40);
+			z = seg.first;
+			w = w + seg.second;
+			result.push_back(w);
+		} catch (const toomanyiterations& x) {
+			std::cerr << "iterations exceeded after ";
+			std::cerr << result.size();
+			std::cerr << " points";
+			maxsteps = 0;
+		}
+	}
+	return result;
+}
+
+/**
+ * \brief Find the initial point for a coordinate curve
+ *
+ * \param k	the elliptic integral parameter
+ * \param z	the desired starting point argument
+ */
+std::complex<double>	curvetracer::startpoint(const std::complex<double>& z,
+		int n) const {
+	std::cerr << "start at " << z.real() << "," << z.imag() << std::endl;
+	return segment(std::complex<double>(0.), z, n);
+}
+
+
+curvetracer::curve_t	curvetracer::mirrorx(const curve_t& c) {
+	curve_t	result;
+	for (auto z : c) {
+		result.push_back(-std::conj(z));
+	}
+	return result;
+}
+
+curvetracer::curve_t	curvetracer::mirrory(const curve_t& c) {
+	curve_t	result;
+	for (auto z : c) {
+		result.push_back(std::conj(z));
+	}
+	return result;
+}
+
+curvetracer::curve_t	curvetracer::mirror(const curve_t& c) {
+	curve_t	result;
+	for (auto z : c) {
+		result.push_back(-z);
+	}
+	return result;
+}
+
+/**
+ * \brief Class to draw the curves to a file
+ */
+class curvedrawer {
+	std::ostream	*_out;
+	std::string	_color;
+public:
+	curvedrawer(std::ostream *out) : _out(out), _color("red") { }
+	const std::string&	color() const { return _color; }
+	void	color(const std::string& c) { _color = c; }
+	void	operator()(const curvetracer::curve_t& curve);
+	std::ostream	*out() { return _out; }
+};
+
+/**
+ * \brief Operator to draw a curve
+ *
+ * \param curve		the curve to draw
+ */
+void	curvedrawer::operator()(const curvetracer::curve_t& curve) {
+	double	first = true;
+	for (auto z : curve) {
+		if (first) {
+			*_out << "\\draw[color=" << _color << "] ";
+			first = false;
+		} else {
+			*_out << std::endl << "    -- ";
+		}
+		*_out << "({" << z.real() << "*\\dx},{" << z.imag() << "*\\dy})";
+	}
+	*_out << ";" << std::endl;
+}
+
+static struct option	longopts[] = {
+{ "outfile",	required_argument,	NULL,	'o' },
+{ "k",		required_argument,	NULL,	'k' },
+{ "deltax",	required_argument,	NULL,	'd' },
+{ NULL,		0,			NULL,	 0  }
+};
+
+/**
+ * \brief Main function
+ */
+int	main(int argc, char *argv[]) {
+	double	k = 0.625;
+	double	deltax = 0.2;
+
+	int	c;
+	int	longindex;
+	std::string	outfilename;
+	while (EOF != (c = getopt_long(argc, argv, "o:k:d:", longopts,
+		&longindex)))
+		switch (c) {
+		case 'd':
+			deltax = std::stod(optarg);
+			break;
+		case 'o':
+			outfilename = std::string(optarg);
+			break;
+		case 'k':
+			k = std::stod(optarg);
+			break;
+		}
+
+	double	kprime = integrand::kprime(k);
+
+	double	xmax = gsl_sf_ellint_Kcomp(k, GSL_PREC_DOUBLE);
+	double	ymax = gsl_sf_ellint_Kcomp(kprime, GSL_PREC_DOUBLE);
+	std::cout << "xmax = " << xmax << std::endl;
+	std::cout << "ymax = " << ymax << std::endl;
+
+	curvedrawer	*cdp;
+	std::ofstream	*outfile = NULL;
+	if (outfilename.c_str()) {
+		outfile = new std::ofstream(outfilename.c_str());
+	}
+	if (outfile) {
+		cdp = new curvedrawer(outfile);
+	} else {
+		cdp = new curvedrawer(&std::cout);
+	}
+
+	integrand1	f(k);
+	curvetracer	ct(f);
+
+	// fill
+	(*cdp->out()) << "\\fill[color=red!10] ({" << (-xmax) << "*\\dx},0) "
+		<< "rectangle ({" << xmax << "*\\dx},{" << ymax << "*\\dy});"
+		<< std::endl;
+	(*cdp->out()) << "\\fill[color=blue!10] ({" << (-xmax) << "*\\dx},{"
+		<< (-ymax) << "*\\dy}) rectangle ({" << xmax << "*\\dx},0);"
+		<< std::endl;
+
+	// "circles"
+	std::complex<double>	dir(0.01, 0);
+	for (double im = deltax; im < 3; im += deltax) {
+		std::complex<double>	startz(0, im);
+		std::complex<double>	startw = ct.startpoint(startz);
+		curvetracer::curve_t	curve = ct.trace(startz, dir,
+			startw, 1000);
+		cdp->color("red");
+		(*cdp)(curve);
+		(*cdp)(curvetracer::mirrorx(curve));
+		cdp->color("blue");
+		(*cdp)(curvetracer::mirrory(curve));
+		(*cdp)(curvetracer::mirror(curve));
+	}
+
+	// imaginary axis
+	(*cdp->out()) << "\\draw[color=red] (0,0) -- (0,{" << ymax
+		<< "*\\dy});" << std::endl;
+	(*cdp->out()) << "\\draw[color=blue] (0,0) -- (0,{" << (-ymax)
+		<< "*\\dy});" << std::endl;
+
+	// arguments between 0 and 1
+	dir = std::complex<double>(0, 0.01);
+	for (double re = 0.2; re < 1; re += 0.2) {
+		std::complex<double>	startz(re, 0.001);
+		std::complex<double>	startw = ct.startpoint(startz);
+		startw = std::complex<double>(startw.real(), 0);
+		curvetracer::curve_t	curve = ct.trace(startz, dir,
+			startw, 1000);
+		cdp->color("red");
+		(*cdp)(curve);
+		(*cdp)(curvetracer::mirrorx(curve));
+		cdp->color("blue");
+		(*cdp)(curvetracer::mirrory(curve));
+		(*cdp)(curvetracer::mirror(curve));
+	}
+
+	// argument 1 (singularity)
+	{
+		std::complex<double>	startz(1.);
+		std::complex<double>	startw(xmax);
+		curvetracer::curve_t	curve = ct.trace(startz, dir,
+			startw, 1000);
+		cdp->color("red");
+		(*cdp)(curve);
+		(*cdp)(curvetracer::mirrorx(curve));
+		cdp->color("blue");
+		(*cdp)(curvetracer::mirror(curve));
+		(*cdp)(curvetracer::mirrory(curve));
+	}
+
+	// arguments between 1 and 1/k
+	{
+		for (double x0 = 1 + deltax; x0 < 1/k; x0 += deltax) {
+			double	y0 = sqrt(1-1/(x0*x0))/kprime;
+			//std::cout << "y0 = " << y0 << std::endl;
+			double	y = gsl_sf_ellint_F(asin(y0), kprime,
+				GSL_PREC_DOUBLE);
+			std::complex<double>	startz(x0);
+			std::complex<double>	startw(xmax, y);
+			curvetracer::curve_t	curve = ct.trace(startz, dir,
+				startw, 1000);
+			cdp->color("red");
+			(*cdp)(curve);
+			(*cdp)(curvetracer::mirrorx(curve));
+			cdp->color("blue");
+			(*cdp)(curvetracer::mirror(curve));
+			(*cdp)(curvetracer::mirrory(curve));
+		}
+	}
+
+	// argument 1/k
+#if 0
+	{
+		std::complex<double>	startz(1/k);
+		std::complex<double>	startw(xmax, ymax);
+		curvetracer::curve_t	curve = ct.trace(startz, dir,
+			startw, 1000);
+		cdp->color("red");
+		(*cdp)(curve);
+		(*cdp)(curvetracer::mirrorx(curve));
+		cdp->color("blue");
+		(*cdp)(curvetracer::mirror(curve));
+		(*cdp)(curvetracer::mirrory(curve));
+	}
+#endif
+
+	// arguments larger than 1/k
+	{
+		dir = std::complex<double>(0, 0.01);
+		double	x0 = 1;
+		while (x0 <= 1/k + 0.0001) { x0 += deltax; }
+		for (; x0 < 4; x0 += deltax) {
+			std::complex<double>	startz(x0);
+			std::complex<double>	startw(gsl_sf_ellint_F(
+				asin(1/(k*x0)), k, GSL_PREC_DOUBLE), ymax);
+		curvetracer::curve_t	curve = ct.trace(startz, dir,
+			startw, 1000);
+		cdp->color("red");
+		(*cdp)(curve);
+		(*cdp)(curvetracer::mirrorx(curve));
+		cdp->color("blue");
+		(*cdp)(curvetracer::mirror(curve));
+		(*cdp)(curvetracer::mirrory(curve));
+		}
+	}
+
+	// border
+	(*cdp->out()) << "\\def\\xmax{" << xmax << "}" << std::endl;
+	(*cdp->out()) << "\\def\\ymax{" << ymax << "}" << std::endl;
+
+	return EXIT_SUCCESS;
+}
diff --git a/buch/chapters/110-elliptisch/images/rechteck.pdf b/buch/chapters/110-elliptisch/images/rechteck.pdf
new file mode 100644
index 0000000..894091f
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/rechteck.pdf
diff --git a/buch/chapters/110-elliptisch/images/rechteck.tex b/buch/chapters/110-elliptisch/images/rechteck.tex
new file mode 100644
index 0000000..622a9e9
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/rechteck.tex
@@ -0,0 +1,80 @@
+%
+% rechteck.tex -- rechteck für Wertebereich der ell. Integrale
+%
+% (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]
+
+\def\dx{3}
+\def\dy{3}
+
+\input{rechteckpfade.tex}
+
+\begin{scope}
+	\clip ({-\xmax*\dx},{-\ymax*\dy}) rectangle ({\xmax*\dx},{\ymax*\dy});
+	\fill[color=red!10] (0,{\ymax*\dy}) circle[radius=0.45];
+	\fill[color=blue!10] (0,{-\ymax*\dy}) circle[radius=0.45];
+	\draw[color=gray!80] (0,{\ymax*\dy}) -- (0,{\ymax*\dy-0.45});
+	\draw[color=gray!80] (0,{-\ymax*\dy}) -- (0,{-\ymax*\dy+0.45});
+\end{scope}
+
+\draw ({-\xmax*\dx},{-\ymax*\dy}) rectangle ({\xmax*\dx},{\ymax*\dy});
+
+\draw[->] ({-\xmax*\dx-0.3},0) -- ({\xmax*\dx+0.9},0)
+	coordinate[label={$\operatorname{Re}F(k,z)$}];
+
+\draw[<->,line width=0.7pt]
+	({-\xmax*\dx},{\ymax*\dy+0.2})
+	--
+	({\xmax*\dx},{\ymax*\dy+0.2});
+\draw[line width=0.2pt]
+	({-\xmax*\dx},{\ymax*\dy})
+	--
+	({-\xmax*\dx},{\ymax*\dy+0.3});
+\draw[line width=0.2pt]
+	({\xmax*\dx},{\ymax*\dy})
+	--
+	({\xmax*\dx},{\ymax*\dy+0.3});
+
+\node at ({-0.5*\xmax*\dx},{\ymax*\dy+0.2}) [above] {$2K(k)$};
+\draw[->] (0,{\ymax*\dy}) -- (0,{\ymax*\dy+0.7})
+	coordinate[label={right:$\operatorname{Im}F(k,z)$}];
+\draw (0,{-\ymax*\dy}) -- (0,{-\ymax*\dy-0.2});
+
+\draw[line width=0.2pt]
+	({-\xmax*\dx-0.3},{-\ymax*\dy})
+	--
+	({-\xmax*\dx},{-\ymax*\dy});
+\draw[line width=0.2pt]
+	({-\xmax*\dx-0.3},{\ymax*\dy})
+	--
+	({-\xmax*\dx},{\ymax*\dy});
+\draw[<->,line width=0.7pt] 
+	({-\xmax*\dx-0.2},{\ymax*\dy})
+	--
+	({-\xmax*\dx-0.2},0);
+
+\node at ({-\xmax*\dx-0.2},{-0.5*\ymax*\dy}) [rotate=90,above] {$K(k')$};
+\node at ({-\xmax*\dx-0.2},{0.5*\ymax*\dy}) [rotate=90,above] {$K(k')$};
+
+\draw[<->,line width=0.7pt] 
+	({-\xmax*\dx-0.2},{-\ymax*\dy})
+	--
+	({-\xmax*\dx-0.2},0);
+
+\node at ({\xmax*\dx},{\ymax*\dy}) [right] {$K(k)+iK(k')$};
+\fill[color=white] ({\xmax*\dx},{\ymax*\dy}) circle[radius=0.05];
+\draw ({\xmax*\dx},{\ymax*\dy}) circle[radius=0.05];
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/110-elliptisch/images/unvollstaendig.m b/buch/chapters/110-elliptisch/images/unvollstaendig.m
new file mode 100644
index 0000000..25196d0
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/unvollstaendig.m
@@ -0,0 +1,81 @@
+#
+# unvollstaendig.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+global N;
+N = 200;
+global n;
+n = 5;
+
+function retval = integrand_f(t, k)
+	retval = 1 / sqrt((1 - t^2) * (1 - k^2 * t^2));
+endfunction
+
+function retval = integrand_e(t, k)
+	retval = sqrt((1-k^2*t^2)/(1-t^2));
+endfunction
+
+function retval = integrand_pi(n, t, k)
+	retval = 1 / ((1-n*t^2) * sqrt((1-k^2*t^2)*(1-t^2)));
+endfunction
+
+function retval = elliptisch1(f, name, k)
+	global N;
+	global n;
+	s = 0;
+	fprintf(f, "\\def\\ell%s{ (0,0)", name);
+	delta = 1 / N;
+	for x = (0:delta:(1-delta))
+		h = delta / n;
+		for t = (x+h/2:h:x+delta)
+			s = s + integrand_f(t, k) * h;
+		endfor
+		fprintf(f, "\n    -- ({\\dx*%.4f},{\\dy*%.4f})", x + delta, s);
+	endfor
+	fprintf(f, "}\n");
+endfunction
+
+function retval = elliptisch2(f, name, k)
+	global N;
+	global n;
+	s = 0;
+	fprintf(f, "\\def\\ell%s{ (0,0)", name);
+	delta = 1 / N;
+	for x = (0:delta:(1-delta))
+		h = delta / n;
+		for t = (x+h/2:h:x+delta)
+			s = s + integrand_e(t, k) * h;
+		endfor
+		fprintf(f, "\n    -- ({\\dx*%.4f},{\\dy*%.4f})", x + delta, s);
+	endfor
+	fprintf(f, "\n}\n");
+endfunction
+
+fn = fopen("unvollpath.tex", "w");
+
+elliptisch1(fn, "Fzero", 0.0);
+elliptisch1(fn, "Fone", 0.1);
+elliptisch1(fn, "Ftwo", 0.2);
+elliptisch1(fn, "Fthree", 0.3);
+elliptisch1(fn, "Ffour", 0.4);
+elliptisch1(fn, "Ffive", 0.5);
+elliptisch1(fn, "Fsix", 0.6);
+elliptisch1(fn, "Fseven", 0.7);
+elliptisch1(fn, "Feight", 0.8);
+elliptisch1(fn, "Fnine", 0.9);
+elliptisch1(fn, "Ften", 1.0);
+
+elliptisch2(fn, "Ezero", 0.0);
+elliptisch2(fn, "Eone", 0.1);
+elliptisch2(fn, "Etwo", 0.2);
+elliptisch2(fn, "Ethree", 0.3);
+elliptisch2(fn, "Efour", 0.4);
+elliptisch2(fn, "Efive", 0.5);
+elliptisch2(fn, "Esix", 0.6);
+elliptisch2(fn, "Eseven", 0.7);
+elliptisch2(fn, "Eeight", 0.8);
+elliptisch2(fn, "Enine", 0.9);
+elliptisch2(fn, "Eten", 1.0);
+
+fclose(fn);
diff --git a/buch/chapters/110-elliptisch/images/unvollstaendig.pdf b/buch/chapters/110-elliptisch/images/unvollstaendig.pdf
new file mode 100644
index 0000000..4da2c0c
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/unvollstaendig.pdf
diff --git a/buch/chapters/110-elliptisch/images/unvollstaendig.tex b/buch/chapters/110-elliptisch/images/unvollstaendig.tex
new file mode 100644
index 0000000..1a1af13
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/unvollstaendig.tex
@@ -0,0 +1,100 @@
+%
+% unvollstaendig.tex -- Plots der unvollständigen elliptischen integrale
+%
+% (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}
+\input{unvollpath.tex}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\pgfkeys{/pgf/number format/.cd, fixed, fixed zerofill, precision=1}
+
+\def\dx{12.8}
+\def\dy{6}
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{blau}{rgb}{0.3,0.3,1}
+
+\begin{scope}
+\begin{scope}
+\clip (-0.1,-0.1) rectangle ({\dx+0.0},{10.1});
+
+\fill[color=darkgreen!10] \ellEzero -- (\dx,{1.571*\dy}) -- (\dx,0) -- cycle;
+\fill[color=red!10] \ellEzero -- (\dx,{1.571*\dy}) -- (\dx,10.1) -- (0,10.2) -- cycle;
+
+\node[color=red] at ({0.6*\dx},{1.3*\dy}) [scale=2] {$F(x,k)$};
+\node[color=darkgreen] at ({0.6*\dx},{0.3*\dy}) [scale=2] {$E(x,k)$};
+
+
+\draw[color=red!0!blau,line width=1.0pt] \ellFzero;
+\draw[color=red!10!blau,line width=1.0pt] \ellFone;
+\draw[color=red!20!blau,line width=1.0pt] \ellFtwo;
+\draw[color=red!30!blau,line width=1.0pt] \ellFthree;
+\draw[color=red!40!blau,line width=1.0pt] \ellFfour;
+\draw[color=red!50!blau,line width=1.0pt] \ellFfive;
+\draw[color=red!60!blau,line width=1.0pt] \ellFsix;
+\draw[color=red!70!blau,line width=1.0pt] \ellFseven;
+\draw[color=red!80!blau,line width=1.0pt] \ellFeight;
+\draw[color=red!90!blau,line width=1.0pt] \ellFnine;
+\draw[color=red!100!blau,line width=1.0pt] \ellFten;
+
+\draw[color=darkgreen!100!blau,line width=1.0pt] \ellEten;
+\draw[color=darkgreen!90!blau,line width=1.0pt] \ellEnine;
+\draw[color=darkgreen!80!blau,line width=1.0pt] \ellEeight;
+\draw[color=darkgreen!70!blau,line width=1.0pt] \ellEseven;
+\draw[color=darkgreen!60!blau,line width=1.0pt] \ellEsix;
+\draw[color=darkgreen!50!blau,line width=1.0pt] \ellEfive;
+\draw[color=darkgreen!40!blau,line width=1.0pt] \ellEfour;
+\draw[color=darkgreen!30!blau,line width=1.0pt] \ellEthree;
+\draw[color=darkgreen!20!blau,line width=1.0pt] \ellEtwo;
+\draw[color=darkgreen!10!blau,line width=1.0pt] \ellEone;
+\draw[color=darkgreen!0!blau,line width=1.0pt] \ellEzero;
+
+\end{scope}
+
+\draw[line width=0.2pt] (\dx,0) -- (\dx,10.1);
+
+\begin{scope}
+	\clip ({0.7*\dx},0) rectangle (\dx,10.1);
+	\draw[color=white,line width=0.5pt] \ellEzero -- (\dx,{1.571*\dy});
+\end{scope}
+
+\draw[->] ({-0.1},0) -- ({\dx+0.3},0) coordinate[label={$x$}];
+\foreach \x in {0,0.2,...,1.0}{
+	\draw ({\x*\dx},-0.1) -- ({\x*\dx},0.1);
+	\node at ({\x*\dx},0) [below] {$\pgfmathprintnumber{\x}$};
+}
+\draw[->] (0,{-0.1}) -- (0,{10.3}) coordinate[label={right:$y$}];
+\foreach \y in {0.5,1,1.5}{
+	%\draw[line width=0.2pt] (0,{\y*\dy}) -- (\dx,{\y*\dy});
+	\draw (-0.1,{\y*\dy}) -- (0.1,{\y*\dy});
+	\node at (0,{\y*\dy}) [left] {$\pgfmathprintnumber{\y}$};
+}
+\foreach \c in {0,10,...,100}{
+	\pgfmathparse{\c/100}
+	\xdef\k{\pgfmathresult}
+	\node[color=red!\c!blau] at ({0.02*\dx},{0.95*\dy+0.04*\c})
+		[right] {$k=\pgfmathprintnumber{\k}$};
+}
+\foreach \c in {0,10,...,100}{
+	\pgfmathparse{\c/100}
+	\xdef\k{\pgfmathresult}
+	\node[color=darkgreen!\c!blau] at ({0.98*\dx},{0.75*\dy-0.04*\c})
+		[left] {$k=\pgfmathprintnumber{\k}$};
+}
+
+\draw ({\dx-0.1},{1.571*\dy}) -- ({\dx+0.1},{1.571*\dy});
+\node at (\dx,{1.571*\dy}) [right] {$\frac{\pi}2$};
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+