wertebereich elliptischer integrale

5 files changed, 441 insertions, 32 deletions

diff --git a/buch/chapters/110-elliptisch/ellintegral.tex b/buch/chapters/110-elliptisch/ellintegral.tex
index 7ac09ca..b6f35fc 100644
--- a/buch/chapters/110-elliptisch/ellintegral.tex
+++ b/buch/chapters/110-elliptisch/ellintegral.tex
@@ -329,6 +329,16 @@ XXX Parameterkonventionen \\
 XXX Wertebereich (Rechtecke) \\
 XXX Komplementäre Integrale \\
 
+\begin{figure}
+\centering
+\includegraphics{chapters/110-elliptisch/images/rechteck.pdf}
+\caption{Der Wertebereich der Funktion $F(k,z)$ ist ein Rechteck
+der Breite $2K(k)$ und $2K(k')$.
+Die obere Halbebene wird in das rote Rechteck abgebildet, die unter
+in das blaue.
+\label{buch:elliptisch:fig:rechteck}}
+\end{figure}
+
 \subsection{Potenzreihe}
 XXX Potenzreihen \\
 XXX Als hypergeometrische Funktionen \\
diff --git a/buch/chapters/110-elliptisch/images/Makefile b/buch/chapters/110-elliptisch/images/Makefile
index 4912cca..6e3a5ce 100644
--- a/buch/chapters/110-elliptisch/images/Makefile
+++ b/buch/chapters/110-elliptisch/images/Makefile
@@ -3,7 +3,7 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-all:	lemniskate.pdf ellipsenumfang.pdf unvollstaendig.pdf
+all:	lemniskate.pdf ellipsenumfang.pdf unvollstaendig.pdf rechteck.pdf
 
 lemniskate.pdf:	lemniskate.tex
 	pdflatex lemniskate.tex
@@ -15,13 +15,19 @@ ekplot.tex:	ellipsenumfang.m
 	octave ellipsenumfang.m
 
 rechteck:	rechteck.cpp
-	g++ -O -Wall -g rechteck.cpp -o rechteck
+	g++ `pkg-config --cflags gsl` `pkg-config --libs gsl` -O -Wall -g -std=c++11 rechteck.cpp -o rechteck
 
-test:	rechteck
-	./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
+
+
+
diff --git a/buch/chapters/110-elliptisch/images/rechteck.cpp b/buch/chapters/110-elliptisch/images/rechteck.cpp
index 6bd6a06..3ece11c 100644
--- a/buch/chapters/110-elliptisch/images/rechteck.cpp
+++ b/buch/chapters/110-elliptisch/images/rechteck.cpp
@@ -8,47 +8,360 @@
 #include <cstdio>
 #include <complex>
 #include <iostream>
+#include <fstream>
+#include <list>
+#include <getopt.h>
+#include <gsl/gsl_sf_ellint.h>
 
 double	ast = 1;
 
-std::complex<double>	integrand(double k, const std::complex<double>& z) {
-	std::complex<double>	s = z * z;
-	std::complex<double>	eins(1);
-	std::complex<double>	result = eins / sqrt((eins - s) * (eins - k * k * s));
-	return ast * ((result.imag() < 0) ? -result : result);
+/**
+ * \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);
 }
 
-std::complex<double>	segment(double k, const std::complex<double>& z1,
-		const std::complex<double>& z2, int n = 100) {
+/**
+ * \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. / n;
-	summe = integrand(k, z1);
-	summe += integrand(k, z2);
-	for (int i = 1; i < n; i++) {
+	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 += 2. * integrand(k, z);
+		summe += _f(z);
 	}
-	return dz * h * summe / 2.;
+	return dz * h * summe * 2.;
 }
 
-int	main(int argc, char *argv[]) {
-	double	k = 0.5;
-	double	y = -0.0001;
-	double	xstep = -0.1;
-	ast = 1;
-	std::complex<double>	z(0, y);
-	std::complex<double>	w = segment(k, std::complex<double>(0), z);
-	std::cout << z << std::endl;
+/**
+ * \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 (counter-- > 0) {
-		std::complex<double>	znext = z + std::complex<double>(xstep);
-		std::complex<double>	incr = segment(k, z, znext);
-		w += incr;
-		std::cout << znext << " -> " << w << ", ";
-		std::cout << integrand(k, znext) << std::endl;
-		z = znext;
+	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' },
+{ NULL,		0,			NULL,	 0  }
+};
+
+/**
+ * \brief Main function
+ */
+int	main(int argc, char *argv[]) {
+	double	k = 0.6;
+
+	int	c;
+	int	longindex;
+	std::string	outfilename;
+	while (EOF != (c = getopt_long(argc, argv, "o:k:", longopts, &longindex)))
+		switch (c) {
+		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 = 0.2; im < 3; im += 0.2) {
+		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));
+	}
+
+	// argument 1/k
+	{
+		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));
+	}
+
+	// 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..5fa07cb
--- /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}
+