add gamma function section

4 files changed, 153 insertions, 0 deletions

diff --git a/buch/chapters/040-rekursion/images/Makefile b/buch/chapters/040-rekursion/images/Makefile
new file mode 100644
index 0000000..58f79b8
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/Makefile
@@ -0,0 +1,12 @@
+#
+# Makefile
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	gammaplot.pdf
+
+gammaplot.pdf:	gammaplot.tex gammapaths.tex
+	pdflatex gammaplot.tex
+
+gammapaths.tex:	gammaplot.m
+	octave gammaplot.m
diff --git a/buch/chapters/040-rekursion/images/gammaplot.m b/buch/chapters/040-rekursion/images/gammaplot.m
new file mode 100644
index 0000000..3b1d23d
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/gammaplot.m
@@ -0,0 +1,52 @@
+#
+# gammaplot.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+fn = fopen("gammapaths.tex", "w");
+
+function finterval(f, fn, from, to, name, delta)
+	fprintf(fn, "\\def\\gamma%s{", name);
+	x = from + delta;
+	fprintf(fn, "({\\dx*%.4f},{\\dy*%.4f})", x, f(x));
+	x = from + 0.02;
+	for x = (from+0.02:0.02:to-0.02)
+		fprintf(fn, "\n -- ");
+		fprintf(fn, "({\\dx*%.4f},{\\dy*%.4f})", x, f(x));
+	endfor
+	x = to - delta;
+	fprintf(fn, "\n -- ");
+	fprintf(fn, "({\\dx*%.4f},{\\dy*%.4f})", x, f(x));
+	fprintf(fn, "}\n");
+endfunction
+
+function gammainterval(fn, from, to, name, delta)
+	finterval(@gamma, fn, from, to, name, delta)
+endfunction
+
+function retval = gammasin(x) 
+	retval = gamma(x) + sin(x * pi);
+endfunction
+
+function gammasininterval(fn, from, to, name, delta)
+	finterval(@gammasin, fn, from, to, name, delta)
+endfunction
+
+gammainterval(fn,  0,  4.1, "plus",  0.019);
+gammainterval(fn, -1,  0,   "one",   0.019);
+gammainterval(fn, -2, -1,   "two",   0.019);
+gammainterval(fn, -3, -2,   "three", 0.019);
+gammainterval(fn, -4, -3,   "four",  0.005);
+gammainterval(fn, -5, -4,   "five",  0.001);
+gammainterval(fn, -6, -5,   "six",  0.0002);
+
+gammasininterval(fn,  0,  4.1, "sinplus",  0.019);
+gammasininterval(fn, -1,  0,   "sinone",   0.019);
+gammasininterval(fn, -2, -1,   "sintwo",   0.019);
+gammasininterval(fn, -3, -2,   "sinthree", 0.019);
+gammasininterval(fn, -4, -3,   "sinfour",  0.005);
+gammasininterval(fn, -5, -4,   "sinfive",  0.001);
+gammasininterval(fn, -6, -5,   "sinsix",  0.0002);
+
+fclose(fn);
diff --git a/buch/chapters/040-rekursion/images/gammaplot.pdf b/buch/chapters/040-rekursion/images/gammaplot.pdf
new file mode 100644
index 0000000..d0a766e
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/gammaplot.pdf
diff --git a/buch/chapters/040-rekursion/images/gammaplot.tex b/buch/chapters/040-rekursion/images/gammaplot.tex
new file mode 100644
index 0000000..4e11d32
--- /dev/null
+++ b/buch/chapters/040-rekursion/images/gammaplot.tex
@@ -0,0 +1,89 @@
+%
+% gammaplot.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}
+\input{gammapaths.tex}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkgreen}{rgb}{0,0.8,0}
+
+\draw[->] (-6.1,0) -- (5.3,0) coordinate[label={$z$}];
+\draw[->] (0,-5.1) -- (0,6.4) coordinate[label={right:$\Gamma(z)$}];
+
+\foreach \x in {-1,-2,-3,-4,-5,-6}{
+	\draw (\x,-0.1) -- (\x,0.1);
+	\draw[line width=0.1pt] (\x,-5) -- (\x,6.2);
+}
+\foreach \x in {1,2,3,4,5}{
+	\draw (\x,-0.1) -- (\x,0.1);
+	\node at (\x,0) [below] {$\x$};
+}
+\foreach \y in {-5,-4,-3,-2,-1,1,2,3,4,5,6}{
+	\draw (-0.1,\y) -- (0.1,\y);
+}
+\foreach \y in {1,2,3,4,5,6}{
+	\node at (0,\y) [left] {$\y$};
+}
+\foreach \y in {-1,-2,-3,-4,-5}{
+	\node at (0,\y) [right] {$\y$};
+}
+\foreach \x in {-1,-3,-5}{
+	\node at (\x,0) [below left] {$\x$};
+}
+\foreach \x in {-2,-4,-6}{
+	\node at (\x,0) [above left] {$\x$};
+}
+
+\def\dx{1}
+\def\dy{1}
+
+\begin{scope}
+\clip (-6.1,-5) rectangle (4.3,6.2);
+
+\draw[color=darkgreen,line width=1.4pt] \gammasinplus;
+\draw[color=darkgreen,line width=1.4pt] \gammasinone;
+\draw[color=darkgreen,line width=1.4pt] \gammasintwo;
+\draw[color=darkgreen,line width=1.4pt] \gammasinthree;
+\draw[color=darkgreen,line width=1.4pt] \gammasinfour;
+\draw[color=darkgreen,line width=1.4pt] \gammasinfive;
+\draw[color=darkgreen,line width=1.4pt] \gammasinsix;
+
+\draw[color=red,line width=1.4pt] \gammaplus;
+\draw[color=red,line width=1.4pt] \gammaone;
+\draw[color=red,line width=1.4pt] \gammatwo;
+\draw[color=red,line width=1.4pt] \gammathree;
+\draw[color=red,line width=1.4pt] \gammafour;
+\draw[color=red,line width=1.4pt] \gammafive;
+\draw[color=red,line width=1.4pt] \gammasix;
+
+\end{scope}
+
+\fill[color=blue] (1,1) circle[radius=0.08];
+\node[color=blue] at (1,1-0.2) [below] {$\Gamma(1)=0!$};
+\fill[color=blue] (2,1) circle[radius=0.08];
+\node[color=blue] at (2,1) [below right] {$\Gamma(2)=1!$};
+\fill[color=blue] (3,2) circle[radius=0.08];
+\node[color=blue] at (3,2) [right] {$\Gamma(3)=2!$};
+\fill[color=blue] (4,6) circle[radius=0.08];
+\node[color=blue] at (4,6) [right] {$\Gamma(4)=3!$};
+
+\begin{scope}[xshift=3cm,yshift=-2.4cm]
+	\draw[color=red,line width=1.4pt] (-1,0) -- (0,0);
+	\draw[color=darkgreen,line width=1.4pt] (-1,-0.7) -- (0,-0.7);
+	\node at (0.1,0) [right] {$\Gamma(z)$};
+	\node at (0.1,-0.7) [right] {$\Gamma(z)+\sin\pi z$};
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+