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-rw-r--r--buch/chapters/060-integral/experiments/rxy.maxima9
-rw-r--r--buch/papers/kugel/images/Makefile15
-rw-r--r--buch/papers/kugel/images/curvature.maxima6
-rw-r--r--buch/papers/kugel/images/curvature.pov139
-rw-r--r--buch/papers/kugel/images/curvgraph.m140
-rw-r--r--vorlesungen/18_hermiteintegrierbar/Makefile33
-rw-r--r--vorlesungen/18_hermiteintegrierbar/MathSem-18-hermiteintegrierbar.tex14
-rw-r--r--vorlesungen/18_hermiteintegrierbar/common.tex17
-rw-r--r--vorlesungen/18_hermiteintegrierbar/hermiteintegrierbar-handout.tex11
-rw-r--r--vorlesungen/18_hermiteintegrierbar/slides.tex11
-rw-r--r--vorlesungen/slides/hermite/Makefile.inc5
-rw-r--r--vorlesungen/slides/hermite/hermiteentwicklung.tex69
-rw-r--r--vorlesungen/slides/hermite/loesung.tex56
-rw-r--r--vorlesungen/slides/hermite/normalhermite.tex88
-rw-r--r--vorlesungen/slides/hermite/normalintegrale.tex54
-rw-r--r--vorlesungen/slides/hermite/skalarprodukt.tex72
-rw-r--r--vorlesungen/slides/test.tex6
17 files changed, 744 insertions, 1 deletions
diff --git a/buch/chapters/060-integral/experiments/rxy.maxima b/buch/chapters/060-integral/experiments/rxy.maxima
new file mode 100644
index 0000000..0d5a56d
--- /dev/null
+++ b/buch/chapters/060-integral/experiments/rxy.maxima
@@ -0,0 +1,9 @@
+y: sqrt(a*x^2+b*x+c);
+
+F: log(x + b/(2 * a) + y/sqrt(a))/sqrt(a);
+
+f: diff(F, x);
+
+ratsimp(f);
+
+ratsimp(y*f);
diff --git a/buch/papers/kugel/images/Makefile b/buch/papers/kugel/images/Makefile
new file mode 100644
index 0000000..e8bf919
--- /dev/null
+++ b/buch/papers/kugel/images/Makefile
@@ -0,0 +1,15 @@
+#
+# Makefile -- build images
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all: curvature.jpg
+
+curvature.inc: curvgraph.m
+ octave curvgraph.m
+
+curvature.png: curvature.pov curvature.inc
+ povray +A0.1 +W1920 +H1080 +Ocurvature.png curvature.pov
+
+curvature.jpg: curvature.png
+ convert curvature.png -density 300 -units PixelsPerInch curvature.jpg
diff --git a/buch/papers/kugel/images/curvature.maxima b/buch/papers/kugel/images/curvature.maxima
new file mode 100644
index 0000000..6313642
--- /dev/null
+++ b/buch/papers/kugel/images/curvature.maxima
@@ -0,0 +1,6 @@
+
+f: exp(-r^2/sigma^2)/sigma;
+laplacef: ratsimp(diff(r * diff(f,r), r) / r);
+f: exp(-r^2/(2*sigma^2))/(sqrt(2)*sigma);
+laplacef: ratsimp(diff(r * diff(f,r), r) / r);
+
diff --git a/buch/papers/kugel/images/curvature.pov b/buch/papers/kugel/images/curvature.pov
new file mode 100644
index 0000000..3b15d77
--- /dev/null
+++ b/buch/papers/kugel/images/curvature.pov
@@ -0,0 +1,139 @@
+//
+// curvature.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+ assumed_gamma 1
+}
+
+#declare imagescale = 0.09;
+
+camera {
+ location <10, 10, -40>
+ look_at <0, 0, 0>
+ right 16/9 * x * imagescale
+ up y * imagescale
+}
+
+light_source {
+ <-10, 10, -40> color White
+ area_light <1,0,0> <0,0,1>, 10, 10
+ adaptive 1
+ jitter
+}
+
+sky_sphere {
+ pigment {
+ color rgb<1,1,1>
+ }
+}
+
+//
+// draw an arrow from <from> to <to> with thickness <arrowthickness> with
+// color <c>
+//
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+ sphere {
+ from, 1.1 * arrowthickness
+ }
+ cylinder {
+ from,
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ arrowthickness
+ }
+ cone {
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ 2 * arrowthickness,
+ to,
+ 0
+ }
+ pigment {
+ color c
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+#end
+
+arrow(<-3.1,0,0>, <3.1,0,0>, 0.01, White)
+arrow(<0,-1,0>, <0,1,0>, 0.01, White)
+arrow(<0,0,-2.1>, <0,0,2.1>, 0.01, White)
+
+#include "curvature.inc"
+
+#declare sigma = 1;
+#declare s = 1.4;
+#declare N0 = 0.4;
+#declare funktion = function(r) {
+ (exp(-r*r/(sigma*sigma)) / sigma
+ -
+ exp(-r*r/(2*sigma*sigma)) / (sqrt(2)*sigma)) / N0
+};
+#declare hypot = function(xx, yy) { sqrt(xx*xx+yy*yy) };
+
+#declare Funktion = function(x,y) { funktion(hypot(x+s,y)) - funktion(hypot(x-s,y)) };
+#macro punkt(xx,yy)
+ <xx, Funktion(xx, yy), yy>
+#end
+
+#declare griddiameter = 0.006;
+union {
+ #declare xmin = -3;
+ #declare xmax = 3;
+ #declare ymin = -2;
+ #declare ymax = 2;
+
+
+ #declare xstep = 0.2;
+ #declare ystep = 0.02;
+ #declare xx = xmin;
+ #while (xx < xmax + xstep/2)
+ #declare yy = ymin;
+ #declare P = punkt(xx, yy);
+ #while (yy < ymax - ystep/2)
+ #declare yy = yy + ystep;
+ #declare Q = punkt(xx, yy);
+ sphere { P, griddiameter }
+ cylinder { P, Q, griddiameter }
+ #declare P = Q;
+ #end
+ sphere { P, griddiameter }
+ #declare xx = xx + xstep;
+ #end
+
+ #declare xstep = 0.02;
+ #declare ystep = 0.2;
+ #declare yy = ymin;
+ #while (yy < ymax + ystep/2)
+ #declare xx = xmin;
+ #declare P = punkt(xx, yy);
+ #while (xx < xmax - xstep/2)
+ #declare xx = xx + xstep;
+ #declare Q = punkt(xx, yy);
+ sphere { P, griddiameter }
+ cylinder { P, Q, griddiameter }
+ #declare P = Q;
+ #end
+ sphere { P, griddiameter }
+ #declare yy = yy + ystep;
+ #end
+
+ pigment {
+ color rgb<0.8,0.8,0.8>
+ }
+ finish {
+ metallic
+ specular 0.8
+ }
+}
+
diff --git a/buch/papers/kugel/images/curvgraph.m b/buch/papers/kugel/images/curvgraph.m
new file mode 100644
index 0000000..75effd6
--- /dev/null
+++ b/buch/papers/kugel/images/curvgraph.m
@@ -0,0 +1,140 @@
+#
+# curvature.m
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+global N;
+N = 10;
+
+global sigma2;
+sigma2 = 1;
+
+global s;
+s = 1.4;
+
+global cmax;
+cmax = 0.9;
+global cmin;
+cmin = -0.9;
+
+global Cmax;
+global Cmin;
+Cmax = 0;
+Cmin = 0;
+
+xmin = -3;
+xmax = 3;
+xsteps = 200;
+hx = (xmax - xmin) / xsteps;
+
+ymin = -2;
+ymax = 2;
+ysteps = 200;
+hy = (ymax - ymin) / ysteps;
+
+function retval = f0(r)
+ global sigma2;
+ retval = exp(-r^2/sigma2)/sqrt(sigma2) - exp(-r^2/(2*sigma2))/(sqrt(2*sigma2));
+end
+
+global N0;
+N0 = f0(0)
+N0 = 0.4;
+
+function retval = f1(x,y)
+ global N0;
+ retval = f0(hypot(x, y)) / N0;
+endfunction
+
+function retval = f(x, y)
+ global s;
+ retval = f1(x+s, y) - f1(x-s, y);
+endfunction
+
+function retval = curvature0(r)
+ global sigma2;
+ retval = (
+ -4*(sigma2-r^2)*exp(-r^2/sigma2)
+ +
+ (2*sigma2-r^2)*exp(-r^2/(2*sigma2))
+ ) / (sigma2^(5/2));
+endfunction
+
+function retval = curvature1(x, y)
+ retval = curvature0(hypot(x, y));
+endfunction
+
+function retval = curvature(x, y)
+ global s;
+ retval = curvature1(x+s, y) - curvature1(x-s, y);
+endfunction
+
+function retval = farbe(x, y)
+ global Cmax;
+ global Cmin;
+ global cmax;
+ global cmin;
+ c = curvature(x, y);
+ if (c < Cmin)
+ Cmin = c
+ endif
+ if (c > Cmax)
+ Cmax = c
+ endif
+ u = (c - cmin) / (cmax - cmin);
+ if (u > 1)
+ u = 1;
+ endif
+ if (u < 0)
+ u = 0;
+ endif
+ color = [ u, 0.5, 1-u ];
+ color = color/max(color);
+ color(1,4) = c/2;
+ retval = color;
+endfunction
+
+function dreieck(fn, A, B, C)
+ fprintf(fn, "\ttriangle {\n");
+ fprintf(fn, "\t <%.4f,%.4f,%.4f>,\n", A(1,1), A(1,3), A(1,2));
+ fprintf(fn, "\t <%.4f,%.4f,%.4f>,\n", B(1,1), B(1,3), B(1,2));
+ fprintf(fn, "\t <%.4f,%.4f,%.4f>\n", C(1,1), C(1,3), C(1,2));
+ fprintf(fn, "\t}\n");
+endfunction
+
+function viereck(fn, punkte)
+ color = farbe(mean(punkte(:,1)), mean(punkte(:,2)));
+ fprintf(fn, " mesh {\n");
+ dreieck(fn, punkte(1,:), punkte(2,:), punkte(3,:));
+ dreieck(fn, punkte(2,:), punkte(3,:), punkte(4,:));
+ fprintf(fn, "\tpigment { color rgb<%.4f,%.4f,%.4f> } // %.4f\n",
+ color(1,1), color(1,2), color(1,3), color(1,4));
+ fprintf(fn, " }\n");
+endfunction
+
+fn = fopen("curvature.inc", "w");
+punkte = zeros(4,3);
+for ix = (0:xsteps-1)
+ x = xmin + ix * hx;
+ punkte(1,1) = x;
+ punkte(2,1) = x;
+ punkte(3,1) = x + hx;
+ punkte(4,1) = x + hx;
+ for iy = (0:ysteps-1)
+ y = ymin + iy * hy;
+ punkte(1,2) = y;
+ punkte(2,2) = y + hy;
+ punkte(3,2) = y;
+ punkte(4,2) = y + hy;
+ for i = (1:4)
+ punkte(i,3) = f(punkte(i,1), punkte(i,2));
+ endfor
+ viereck(fn, punkte);
+ end
+end
+#fprintf(fn, " finish { metallic specular 0.5 }\n");
+fclose(fn);
+
+printf("Cmax = %.4f\n", Cmax);
+printf("Cmin = %.4f\n", Cmin);
diff --git a/vorlesungen/18_hermiteintegrierbar/Makefile b/vorlesungen/18_hermiteintegrierbar/Makefile
new file mode 100644
index 0000000..a2dfb87
--- /dev/null
+++ b/vorlesungen/18_hermiteintegrierbar/Makefile
@@ -0,0 +1,33 @@
+#
+# Makefile -- hermiteintegrierbar
+#
+# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+#
+all: hermiteintegrierbar-handout.pdf MathSem-18-hermiteintegrierbar.pdf
+
+include ../slides/Makefile.inc
+
+SOURCES = common.tex slides.tex $(slides)
+
+MathSem-18-hermiteintegrierbar.pdf: MathSem-18-hermiteintegrierbar.tex $(SOURCES)
+ pdflatex MathSem-18-hermiteintegrierbar.tex
+
+hermiteintegrierbar-handout.pdf: hermiteintegrierbar-handout.tex $(SOURCES)
+ pdflatex hermiteintegrierbar-handout.tex
+
+thumbnail: thumbnail.jpg # fix1.jpg
+
+thumbnail.pdf: MathSem-18-hermiteintegrierbar.pdf
+ pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \
+ MathSem-18-hermiteintegrierbar.pdf 1
+thumbnail.jpg: thumbnail.pdf
+ convert -density 300 thumbnail.pdf \
+ -resize 1920x1080 -units PixelsPerInch thumbnail.jpg
+
+fix1.pdf: MathSem-18-hermiteintegrierbar.pdf
+ pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \
+ MathSem-18-hermiteintegrierbar.pdf 1
+fix1.jpg: fix1.pdf
+ convert -density 300 fix1.pdf \
+ -resize 1920x1080 -units PixelsPerInch fix1.jpg
+
diff --git a/vorlesungen/18_hermiteintegrierbar/MathSem-18-hermiteintegrierbar.tex b/vorlesungen/18_hermiteintegrierbar/MathSem-18-hermiteintegrierbar.tex
new file mode 100644
index 0000000..7a3a647
--- /dev/null
+++ b/vorlesungen/18_hermiteintegrierbar/MathSem-18-hermiteintegrierbar.tex
@@ -0,0 +1,14 @@
+%
+% MathSem-18-hermiteintegrierbar.tex -- Präsentation
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{true}
+\begin{document}
+\begin{frame}
+\titlepage
+\end{frame}
+\input{slides.tex}
+\end{document}
diff --git a/vorlesungen/18_hermiteintegrierbar/common.tex b/vorlesungen/18_hermiteintegrierbar/common.tex
new file mode 100644
index 0000000..8b1c71f
--- /dev/null
+++ b/vorlesungen/18_hermiteintegrierbar/common.tex
@@ -0,0 +1,17 @@
+%
+% common.tex -- gemeinsame definition
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\input{../common/packages.tex}
+\input{../common/common.tex}
+\mode<beamer>{%
+\usetheme[hideothersubsections,hidetitle]{Hannover}
+}
+\beamertemplatenavigationsymbolsempty
+\title[$\int P(t)e^{-t^2}\,dt$]{Elementare Stammfunktion für
+$\displaystyle\int P(t)e^{-t^2}\,dt$?}
+\author[A.~Müller]{Prof. Dr. Andreas Müller}
+\date[]{}
+\newboolean{presentation}
+
diff --git a/vorlesungen/18_hermiteintegrierbar/hermiteintegrierbar-handout.tex b/vorlesungen/18_hermiteintegrierbar/hermiteintegrierbar-handout.tex
new file mode 100644
index 0000000..a466024
--- /dev/null
+++ b/vorlesungen/18_hermiteintegrierbar/hermiteintegrierbar-handout.tex
@@ -0,0 +1,11 @@
+%
+% hermiteintegrierbar-handout.tex -- Handout XXX
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[handout,aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{false}
+\begin{document}
+\input{slides.tex}
+\end{document}
diff --git a/vorlesungen/18_hermiteintegrierbar/slides.tex b/vorlesungen/18_hermiteintegrierbar/slides.tex
new file mode 100644
index 0000000..cb3bbea
--- /dev/null
+++ b/vorlesungen/18_hermiteintegrierbar/slides.tex
@@ -0,0 +1,11 @@
+%
+% slides.tex -- XXX
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\folie{hermite/normalintegrale.tex}
+\folie{hermite/normalhermite.tex}
+\folie{hermite/hermiteentwicklung.tex}
+\folie{hermite/loesung.tex}
+\folie{hermite/skalarprodukt.tex}
+
diff --git a/vorlesungen/slides/hermite/Makefile.inc b/vorlesungen/slides/hermite/Makefile.inc
index 5c55467..58c21f2 100644
--- a/vorlesungen/slides/hermite/Makefile.inc
+++ b/vorlesungen/slides/hermite/Makefile.inc
@@ -4,4 +4,9 @@
# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
#
chapterhermite = \
+ ../slides/hermite/normalintegrale.tex \
+ ../slides/hermite/normalhermite.tex \
+ ../slides/hermite/hermiteentwicklung.tex \
+ ../slides/hermite/loesung.tex \
+ ../slides/hermite/skalarprodukt.tex \
../slides/hermite/test.tex
diff --git a/vorlesungen/slides/hermite/hermiteentwicklung.tex b/vorlesungen/slides/hermite/hermiteentwicklung.tex
new file mode 100644
index 0000000..e1ced30
--- /dev/null
+++ b/vorlesungen/slides/hermite/hermiteentwicklung.tex
@@ -0,0 +1,69 @@
+%
+% hermiteentwicklung.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Beliebige Polynome}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Polynom}
+\[
+P(x)
+=
+p_0 + p_1x + p_2x^2 + \dots + p_nx^n
+\]
+als Linearkombination von Hermite-Polynome schreiben:
+\begin{align*}
+P(x)
+&=
+a_0H_0(x)% + a_1H_1(x)
++ \dots + a_nH_n(x)
+\\
+&=
+a_0\cdot 1
+\\
+&\quad + a_1\cdot 2x
+\\
+&\quad + a_2\cdot(4x^2-2)
+\\
+&\quad + a_3\cdot(8x^3-12x)
+\\
+&\quad + a_4\cdot(16x^4-48x^2+12)
+\\
+&\quad\;\;\vdots
+\\
+&\quad + a_n(2^nx^n + \dots)
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Koeffizientenvergleich}
+führt auf ein Gleichungssystem
+\begin{center}
+\begin{tabular}{|>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+a_0&a_1&a_2&a_3&a_4&\dots&\\
+\hline
+ 1& 0& 0& 0& 0&\dots&p_0\\
+ 0& 2& 0& 0& 0&\dots&p_1\\
+-2& 0& 4& 0& 0&\dots&p_2\\
+ 0&-12& 0& 8& 0&\dots&p_3\\
+12& 0&-48& 0& 16&\dots&p_4\\
+\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\
+\hline
+\end{tabular}
+\end{center}
+Dreiecksmatrix, Diagonalelement
+$\ne 0$
+$\Rightarrow$
+$\exists$ eindeutige Lösung
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/hermite/loesung.tex b/vorlesungen/slides/hermite/loesung.tex
new file mode 100644
index 0000000..7d4741f
--- /dev/null
+++ b/vorlesungen/slides/hermite/loesung.tex
@@ -0,0 +1,56 @@
+%
+% loesung.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lösung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Frage}
+Für welche Polynome $P(t)$ kann man eine Stammfunktion
+\[
+\int
+P(t)e^{-\frac{t^2}2}
+\,dt
+\]
+in geschlossener Form angeben?
+\end{block}
+\begin{block}{``Hermite-Antwort''}
+\[
+\int H_n(x)e^{-x^2}\,dx
+\]
+kann genau für $n>0$ in geschlossener Form angegeben werden.
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Allgemein}
+\begin{align*}
+\int P(x)e^{-x^2}\,dx
+&=
+\int \sum_{k=0}^n a_kH_k(x)e^{-x^2}\,dx
+\\
+&=
+\sum_{k=0}^n
+a_k
+\int
+H_k(x)e^{-x^2}\,dx
+\\
+&=
+a_0\operatorname{erf}(x) + C
+\\
+&\hspace*{2mm} + \sum_{k=1}^n a_k\int H_k(x)e^{-x^2}\,dx
+\end{align*}
+\end{block}
+\begin{theorem}
+Das Integral von $P(x)e^{-x^2}$ ist genau dann elementar darstellbar, wenn
+$a_0=0$
+\end{theorem}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/hermite/normalhermite.tex b/vorlesungen/slides/hermite/normalhermite.tex
new file mode 100644
index 0000000..bcd30f2
--- /dev/null
+++ b/vorlesungen/slides/hermite/normalhermite.tex
@@ -0,0 +1,88 @@
+%
+% normalhermite.tex -- integrability of hermite polynomials
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Hermite-Polynome}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition (Rodrigues-Formel)}
+\[
+H_n(x)
+=
+(-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}
+\]
+\end{block}
+\vspace{-10pt}
+\begin{block}{Orthogonalität}
+$H_n(x)$ sind orthogonale Polynome bezüglich $w(x)=e^{-x^2}$, d.~h.
+\begin{align*}
+\langle H_n,H_m\rangle_w
+&=
+\int H_n(x)H_m(x)e^{-x^2}\,dx
+\\
+&=
+\biggl\{
+\renewcommand{\arraycolsep}{1pt}
+\begin{array}{l@{\quad}l}
+1&\text{falls $n=m$}\\
+0&\text{sonst}
+\end{array}
+\biggr\}
+=
+\delta_{mn}
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\begin{block}{Rekursion: Auf-/Absteigeoperatoren}
+Rekursionsformel:
+\[
+H_n(x)
+=
+2x\cdot H_{n-1}(x) - H_{n-1}'(x)
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Stammfunktion}
+\begin{align*}
+\int H_n(x) e^{-x^2}\,dx
+&=
+\int \bigl({\color{red}2x}H_{n-1}(x)
+\\
+&\qquad -H_{n-1}'(x)\bigr) e^{-x^2}\,dx
+\\
+{\color{gray}(e^{-x^2}=-2x)}
+&=
+{\color{red}-}\int {\color{red}(e^{-x^2})'} H_{n-1}(x)\,dx
+\\
+&\qquad
+-
+\int H_{n-1}'(x) e^{-x^2}\,dx
+\\
+\text{\color{gray}(Produktregel)}
+&=
+\int (e^{-x^2}H_{n-1}(x))'\,dx
+\\
+\text{\color{gray}(Ableitung)}
+&=
+e^{-x^2}H_{n-1}(x)
+\end{align*}
+ausser für $n=0$:
+\[
+\int
+H_0(x)e^{-x^2}\,dx
+=
+\int
+e^{-x^2}\,dx
+\]
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/hermite/normalintegrale.tex b/vorlesungen/slides/hermite/normalintegrale.tex
new file mode 100644
index 0000000..88abbe8
--- /dev/null
+++ b/vorlesungen/slides/hermite/normalintegrale.tex
@@ -0,0 +1,54 @@
+%
+% normalintegrale.tex --
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Integranden $P(t)e^{-t^2}$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Frage}
+Für welche Polynome $P(t)$ kann man eine Stammfunktion
+\[
+\int
+P(t)e^{-t^2}
+\,dt
+\]
+in geschlossener Form angeben?
+\end{block}
+\begin{block}{Allgemeine Antwort}
+Satz von Liouville und
+Risch- Algorithmus können entscheiden, ob es eine elementare Stammfunktion gibt
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Negativbeispiel}
+$P(t) = 1$, das Normalverteilungsintegral
+\[
+F(x)
+=
+\frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2}\,dt
+\]
+ist nicht elementar darstellbar.
+\end{block}
+\begin{block}{Positivbeispiel}
+$P(t)=t$. Wegen
+\begin{align*}
+\frac{d}{dx}e^{-x^2}
+&=
+-xe^{-x^2}
+\intertext{ist}
+\int te^{-t^2}\,dt
+&=
+-e^{-x^2}+C
+\end{align*}
+elementar darstellbar.
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/hermite/skalarprodukt.tex b/vorlesungen/slides/hermite/skalarprodukt.tex
new file mode 100644
index 0000000..32b933f
--- /dev/null
+++ b/vorlesungen/slides/hermite/skalarprodukt.tex
@@ -0,0 +1,72 @@
+%
+% skalarprodukt.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Skalarprodukt}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Orthogonale Zerlegung}
+Orthogonale $H_k$ normalisieren:
+\[
+\tilde{H}_k(x) = \frac{1}{\|H_k\|_w} H_k(x)
+\]
+mit Gewichtsfunktion $w(x)=e^{-x^2}$
+\end{block}
+\begin{block}{``Hermite''-Analyse}
+\begin{align*}
+P(x)
+&=
+\sum_{k=1}^\infty a_k H_k(x)
+=
+\sum_{k=1}^\infty \tilde{a}_k \tilde{H}_k(x)
+\\
+\tilde{a}_k
+&=
+\| H_k\|_w\, a_k
+\\
+a_k
+&=
+\frac{1}{\|H_k\|}
+\langle \tilde{H}_k, P\rangle_w
+=
+\frac{1}{\|H_k\|^2}
+\langle H_k, P\rangle_w
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Integrationsproblem}
+Bedingung:
+\begin{align*}
+a_0=0
+\qquad\Leftrightarrow\qquad
+\langle H_0,P\rangle_w
+&=
+0
+\\
+\int_{-\infty}^\infty
+P(t) w(t) \,dt
+=
+\int_{-\infty}^\infty
+P(t) e^{-t^2} \,dt
+&=
+0
+\end{align*}
+\end{block}
+\begin{theorem}
+Das Integral von $P(t)e^{-t^2}$ ist in geschlossener Form darstellbar
+genau dann, wenn
+\[
+\int_{-\infty}^\infty P(t)e^{-t^2}\,dt = 0
+\]
+\end{theorem}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index 6aa09f8..ca4ccc9 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -3,4 +3,8 @@
%
% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
%
-\folie{0/intro.tex}
+\folie{hermite/normalintegrale.tex}
+\folie{hermite/normalhermite.tex}
+\folie{hermite/hermiteentwicklung.tex}
+\folie{hermite/loesung.tex}
+\folie{hermite/skalarprodukt.tex}
generated by cgit v1.2.1 (git 2.18.0) at 2024-12-24 00:13:45 +0100