Add presentation

14 files changed, 2762 insertions, 18 deletions

diff --git a/buch/papers/laguerre/definition.tex b/buch/papers/laguerre/definition.tex
index 3e5d423..e511f43 100644
--- a/buch/papers/laguerre/definition.tex
+++ b/buch/papers/laguerre/definition.tex
@@ -18,6 +18,9 @@ x \in \mathbb{R}
 .
 \label{laguerre:dgl}
 \end{align}
+Spannenderweise wurde die verallgemeinerte Laguerre-Differentialgleichung
+zuerst von Yacovlevich Sonine (1849 - 1915) beschrieben,
+aber auf Grund ihrer Ähnlichkeit wurde sie nach Laguerre benannt.
 Die klassische Laguerre-Diffentialgleichung erhält man, wenn $\nu = 0$.
 Hier wird die verallgemeinerte Laguerre-Differentialgleichung verwendet,
 weil die Lösung mit der selben Methode berechnet werden kann,
diff --git a/buch/papers/laguerre/gamma.tex b/buch/papers/laguerre/gamma.tex
index da2fa93..a04ec47 100644
--- a/buch/papers/laguerre/gamma.tex
+++ b/buch/papers/laguerre/gamma.tex
@@ -295,7 +295,7 @@ m^*
 
 \begin{figure}
 \centering
-\input{papers/laguerre/images/schaetzung.pgf}
+\input{papers/laguerre/images/estimate.pgf}
 \caption{Schätzung Mittelwert von $m$ und Fehler}
 \label{laguerre:fig:schaetzung}
 \end{figure}
diff --git a/buch/papers/laguerre/images/schaetzung.pgf b/buch/papers/laguerre/images/estimate.pgf
index 873a10c..3d11371 100644
--- a/buch/papers/laguerre/images/schaetzung.pgf
+++ b/buch/papers/laguerre/images/estimate.pgf
@@ -557,7 +557,7 @@
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@@ -1153,7 +1153,7 @@
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diff --git a/buch/papers/laguerre/images/gammaplot.pdf b/buch/papers/laguerre/images/gammaplot.pdf
new file mode 100644
index 0000000..92e9261
--- /dev/null
+++ b/buch/papers/laguerre/images/gammaplot.pdf
diff --git a/buch/papers/laguerre/images/rel_error_range.pgf b/buch/papers/laguerre/images/rel_error_range.pgf
new file mode 100644
index 0000000..ff73501
--- /dev/null
+++ b/buch/papers/laguerre/images/rel_error_range.pgf
@@ -0,0 +1,887 @@
+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
+%%   \input{<filename>.pgf}
+%%
+%% Make sure the required packages are loaded in your preamble
+%%   \usepackage{pgf}
+%%
+%% Also ensure that all the required font packages are loaded; for instance,
+%% the lmodern package is sometimes necessary when using math font.
+%%   \usepackage{lmodern}
+%%
+%% Figures using additional raster images can only be included by \input if
+%% they are in the same directory as the main LaTeX file. For loading figures
+%% from other directories you can use the `import` package
+%%   \usepackage{import}
+%%
+%% and then include the figures with
+%%   \import{<path to file>}{<filename>.pgf}
+%%
+%% Matplotlib used the following preamble
+%%   \usepackage{fontspec}
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diff --git a/buch/papers/laguerre/images/rel_error_shifted.pgf b/buch/papers/laguerre/images/rel_error_shifted.pgf
new file mode 100644
index 0000000..c11b676
--- /dev/null
+++ b/buch/papers/laguerre/images/rel_error_shifted.pgf
@@ -0,0 +1,1329 @@
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diff --git a/buch/papers/laguerre/images/targets.pdf b/buch/papers/laguerre/images/targets.pdf
index 22c2c5a..adaeeef 100644
--- a/buch/papers/laguerre/images/targets.pdf
+++ b/buch/papers/laguerre/images/targets.pdf
diff --git a/buch/papers/laguerre/main.tex b/buch/papers/laguerre/main.tex
index 9f836ef..f4263de 100644
--- a/buch/papers/laguerre/main.tex
+++ b/buch/papers/laguerre/main.tex
@@ -12,7 +12,7 @@
 benannt nach Edmond Laguerre (1834 - 1886),
 sind Lösungen der ebenfalls nach Laguerre benannten Differentialgleichung.
 Laguerre entdeckte diese Polynome als er Approximationsmethoden 
-für das Integral $\int_0^\infty exp(-x)\, dx$ suchte.
+für das Integral $\int_0^\infty \exp(-x) / x \, dx$ suchte.
 Darum möchten wir in diesem Kapitel uns, 
 ganz im Sinne des Entdeckers,
 den Laguerre-Polynomen für Approximationen von Integralen mit 
diff --git a/buch/papers/laguerre/presentation/presentation.tex b/buch/papers/laguerre/presentation/presentation.tex
new file mode 100644
index 0000000..f49cf1e
--- /dev/null
+++ b/buch/papers/laguerre/presentation/presentation.tex
@@ -0,0 +1,134 @@
+\documentclass[ngerman, aspectratio=169, xcolor={rgb}]{beamer}
+
+% style
+\mode<presentation>{
+	\usetheme{Frankfurt}
+}
+%packages
+\usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-}
+\usepackage[ngerman]{babel}
+\usepackage{graphicx}
+\usepackage{array}
+
+\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
+\usepackage{ragged2e}
+
+\usepackage{bm} % bold math
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{mathtools}
+\usepackage{amsmath}
+\usepackage{multirow} % multi row in tables
+\usepackage{booktabs} %toprule midrule bottomrue in tables
+\usepackage{scrextend}
+\usepackage{textgreek}
+\usepackage[rgb]{xcolor}
+
+\usepackage{ marvosym } % \Lightning
+
+\usepackage{multimedia} % embedded videos
+
+\usepackage{tikz}
+\usepackage{pgf}
+\usepackage{pgfplots}
+
+\usepackage{algorithmic}
+
+%citations
+\usepackage[style=verbose,backend=biber]{biblatex}
+\addbibresource{references.bib}
+
+
+%math font
+\usefonttheme[onlymath]{serif}
+
+%Beamer Template modifications
+%\definecolor{mainColor}{HTML}{0065A3} % HSR blue
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+
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+\setbeamercolor{structure}{fg=black} % itemize, enumerate, etc (bullet points)
+\setbeamercolor{section in toc}{fg=black} % TOC sections
+\setbeamertemplate{section in toc}[sections numbered]
+\setbeamertemplate{subsection in toc}{%
+	\hspace{1.2em}{$\bullet$}~\inserttocsubsection\par}
+
+\setbeamertemplate{itemize items}[circle]
+\setbeamertemplate{description item}[circle]
+\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true]
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+\setbeamercolor{footline}{fg=gray}
+\setbeamertemplate{footline}{%
+	\hfill\usebeamertemplate***{navigation symbols}
+	\hspace{0.5cm}
+	\insertframenumber{}\hspace{0.2cm}\vspace{0.2cm}
+}
+
+\usepackage{caption}
+\captionsetup{labelformat=empty}
+
+%Title Page
+\title{Laguerre-Polynome}
+\subtitle{Anwendung: Approximation der Gamma-Funktion}
+\author{Patrik Müller}
+% \institute{OST Ostschweizer Fachhochschule}
+% \institute{\includegraphics[scale=0.3]{../img/ost_logo.png}}
+\date{\today}
+
+\input{../packages.tex}
+
+\newcommand*{\QED}{\hfill\ensuremath{\blacksquare}}%
+
+\newcommand*{\HL}{\textcolor{mainColor}}
+\newcommand*{\RD}{\textcolor{red}}
+\newcommand*{\BL}{\textcolor{blue}}
+\newcommand*{\GN}{\textcolor{dgreen}}
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+
+\makeatletter
+\newcount\my@repeat@count
+\newcommand{\myrepeat}[2]{%
+	\begingroup
+	\my@repeat@count=\z@
+	\@whilenum\my@repeat@count<#1\do{#2\advance\my@repeat@count\@ne}%
+	\endgroup
+}
+\makeatother
+
+\usetikzlibrary{automata,arrows,positioning,calc,shapes.geometric, fadings}
+
+\begin{document}
+
+\begin{frame}
+	\titlepage
+\end{frame}
+
+\begin{frame}{Inhaltsverzeichnis}
+	\tableofcontents
+\end{frame}
+
+\input{sections/laguerre}
+
+\input{sections/gaussquad}
+
+\input{sections/gamma}
+
+\input{sections/gamma_approx}
+
+\appendix
+\begin{frame}
+	\centering
+	\Large
+	\textbf{Vielen Dank für die Aufmerksamkeit}
+\end{frame}
+
+\end{document}
diff --git a/buch/papers/laguerre/presentation/sections/gamma.tex b/buch/papers/laguerre/presentation/sections/gamma.tex
new file mode 100644
index 0000000..37f4a0b
--- /dev/null
+++ b/buch/papers/laguerre/presentation/sections/gamma.tex
@@ -0,0 +1,50 @@
+\section{Gamma-Funktion}
+
+\begin{frame}{Gamma-Funktion}
+\begin{columns}
+
+\begin{column}{0.48\textwidth}
+\begin{figure}[h]
+\centering
+% \scalebox{0.51}{\input{../images/gammaplot.pdf}}
+\includegraphics[width=1\textwidth]{../images/gammaplot.pdf}
+% \caption{Gamma-Funktion}
+\end{figure}
+\end{column}
+
+\begin{column}{0.52\textwidth}
+Verallgemeinerung der Fakultät
+\begin{align*}
+\Gamma(n) = (n-1)!    
+\end{align*}
+
+Integralformel
+\begin{align*}
+\Gamma(z) 
+=
+\int_0^\infty x^{z-1} e^{-x} \, dx
+,\quad
+\operatorname{Re} z > 0
+\end{align*}
+
+Funktionalgleichung
+\begin{align*}
+z \Gamma(z)
+=
+\Gamma(z + 1)
+\end{align*}
+
+Reflektionsformel
+\begin{align*}
+\Gamma(z) \Gamma(1 - z)
+=
+\frac{\pi}{\sin \pi z}
+, \quad
+\text{für }
+z \notin \mathbb{Z}
+\end{align*}
+
+\end{column}
+\end{columns}
+
+\end{frame}
\ No newline at end of file
diff --git a/buch/papers/laguerre/presentation/sections/gamma_approx.tex b/buch/papers/laguerre/presentation/sections/gamma_approx.tex
new file mode 100644
index 0000000..f5f889e
--- /dev/null
+++ b/buch/papers/laguerre/presentation/sections/gamma_approx.tex
@@ -0,0 +1,176 @@
+\section{Approximieren der Gamma-Funktion}
+
+\begin{frame}{Anwenden der Gauss-Laguerre-Quadratur auf $\Gamma(z)$}
+
+\begin{align*}
+\Gamma(z)
+ & =
+\int_0^\infty x^{z-1} e^{-x} \, dx
+\approx
+\sum_{i=1}^{n} f(x_i) A_i
+=
+\sum_{i=1}^{n} x^{z-1} A_i
+\\\\
+ & \text{wobei }
+A_i = \frac{x_i}{(n+1)^2 \left[ L_{n+1}(x_i) \right]^2}
+\text{ und $x_i$ die Nullstellen von $L_n(x)$}
+\end{align*}
+
+\end{frame}
+
+\begin{frame}{Fehlerabschätzung}
+\begin{align*}
+R_n(\xi)
+ & =
+\frac{(n!)^2}{(2n)!} f^{(2n)}(\xi)
+\\
+ & =
+(z - 2n)_{2n} \frac{(n!)^2}{(2n)!} \xi^{z - 2n - 1}
+,\quad
+0 < \xi < \infty
+\end{align*}
+
+% \textbf{Probleme:}
+\begin{itemize}
+\item Funktion ist unbeschränkt
+\item Maximum von $R_n$ gibt oberes Limit des Fehlers an
+\uncover<2->{\item[$\Rightarrow$] Schwierig ein Maximum von $R_n(\xi)$ zu finden}
+\end{itemize}
+\end{frame}
+
+\begin{frame}{Einfacher Ansatz}
+
+\begin{figure}[h]
+\centering
+\scalebox{0.91}{\input{../images/rel_error_simple.pgf}}
+\caption{Relativer Fehler des einfachen Ansatzes für verschiedene reele Werte
+von $z$ und Grade $n$ der Laguerre-Polynome}
+\end{figure}
+
+\end{frame}
+
+\begin{frame}{Wieso sind die Resultate so schlecht?}
+
+\textbf{Beobachtungen}
+\begin{itemize}
+\item Wenn $z \in \mathbb{Z}$ relativer Fehler $\rightarrow 0$
+\item Gewisse Periodizität zu erkennen
+\item Für grosse und kleine $z$ ergibt sich ein schlechter relativer Fehler
+\item Es gibt Intervalle $[a,a+1]$ mit minimalem relativem Fehler
+\item $a$ ist abhängig von $n$
+\end{itemize}
+
+\uncover<2->{
+\textbf{Ursache?}
+\begin{itemize}
+\item Vermutung: Integrand ist problematisch
+}
+\uncover<3->{
+\item[$\Rightarrow$] Analysieren des Integranden
+}
+\end{itemize}
+\end{frame}
+
+\begin{frame}{$f(x) = x^z$}
+\begin{figure}[h]
+\centering
+\scalebox{0.91}{\input{../images/integrands.pgf}}
+% \caption{Integrand $x^z$ mit unterschiedlichen Werten für $z$}
+\end{figure}
+\end{frame}
+
+\begin{frame}{Integrand $x^z e^{-x}$}
+\begin{figure}[h]
+\centering
+\scalebox{0.91}{\input{../images/integrands_exp.pgf}}
+% \caption{Integrand $x^z$ mit unterschiedlichen Werten für $z$}
+\end{figure}
+\end{frame}
+
+\begin{frame}{Neuer Ansatz?}
+
+\textbf{Vermutung}
+\begin{itemize}
+\item Es gibt Intervalle $[a(n), a(n+1)]$ in denen der relative Fehler minimal
+ist
+\item $a(n) > 0$
+\end{itemize}
+
+\uncover<2->{
+\textbf{Idee}
+\begin{itemize}
+\item[$\Rightarrow$] Berechnen von $\Gamma(z)$ im geeigneten Intervall und dann
+mit Funktionalgleichung zurückverschieben
+\end{itemize}
+}
+
+\uncover<3->{
+\textbf{Wie finden wir $\boldsymbol{a(n)}$?}
+\begin{itemize}
+\item Minimieren des Fehlerterms mit zusätzlichem Verschiebungsterm
+}
+\uncover<4->{$\Rightarrow$ Schwierig das Maximum des Fehlerterms zu bestimmen}
+\uncover<5->{\item Emprisch $a(n)$ bestimmen}
+\uncover<6->{$\Rightarrow$ Sinnvoll,
+da Gauss-Quadratur nur für kleine $n$ praktischen Nutzen hat}
+\end{itemize}
+\end{frame}
+
+\begin{frame}{Verschiebungsterm}
+\begin{align*}
+\Gamma(z)
+\approx
+\frac{1}{(z-m)_m} \sum_{i=1}^{n} x_i^{z + m - 1} A_i
+\end{align*}
+
+\begin{figure}[h]
+\centering
+\includegraphics[width=0.5\textwidth]{../images/targets.pdf}
+\caption{Verschiebungsterm $m$ in Abhängigkeit von $z$ und $n$}
+\end{figure}
+\end{frame}
+
+\begin{frame}{Schätzen von $m^*$}
+\begin{columns}
+\begin{column}{0.6\textwidth}
+\begin{figure}
+\centering
+\vspace{-24pt}
+\scalebox{0.7}{\input{../images/estimate.pgf}}
+% \caption{Integrand $x^z$ mit unterschiedlichen Werten für $z$}
+\end{figure}
+\end{column}
+\begin{column}{0.39\textwidth}
+\begin{align*}
+m^*
+=
+\lceil \hat{m} - \operatorname{Re}z \rceil
+\end{align*}
+\end{column}
+\end{columns}
+
+\end{frame}
+
+\begin{frame}{}
+\begin{figure}[h]
+\centering
+\scalebox{0.6}{\input{../images/rel_error_shifted.pgf}}
+\caption{Relativer Fehler mit $n=8$, unterschiedlichen Verschiebungstermen $m$ und $z\in(0, 1)$}
+\end{figure}
+\end{frame}
+
+\begin{frame}{}
+\begin{figure}[h]
+\centering
+\scalebox{0.6}{\input{../images/rel_error_range.pgf}}
+\caption{Relativer Fehler mit $n=8$, Verschiebungsterm $m^*$ und $z\in(-5, 5)$}
+\end{figure}
+\end{frame}
+
+\begin{frame}{Vergleich mit Lanczos-Methode}
+Maximaler relativer Fehler für $n=6$
+\begin{itemize}
+    \item Lanczos-Methode $< 10^{-12}$
+    \item Unsere Methode $\approx 10^{-6}$ 
+\end{itemize}
+\end{frame}
\ No newline at end of file
diff --git a/buch/papers/laguerre/presentation/sections/gaussquad.tex b/buch/papers/laguerre/presentation/sections/gaussquad.tex
new file mode 100644
index 0000000..4d973b8
--- /dev/null
+++ b/buch/papers/laguerre/presentation/sections/gaussquad.tex
@@ -0,0 +1,67 @@
+\section{Gauss-Quadratur}
+
+\begin{frame}{Gauss-Quadratur}
+\textbf{Idee}
+\begin{itemize}[<+->]
+\item Polynome können viele Funktionen approximieren
+\item Wenn Verfahren gut für Polynome funktioniert,
+sollte es auch für andere Funktionen funktionieren
+\item Integrieren eines Interpolationspolynom
+\item Interpolationspolynom ist durch Funktionswerte $f(x_i)$ bestimmt
+$\Rightarrow$ Integral kann durch Funktionswerte berechnet werden
+\item Evaluation der Funktionswerte an geeigneten Stellen
+\end{itemize}
+\end{frame}
+
+\begin{frame}{Gauss-Quadratur}
+\begin{align*}
+\int_{-1}^{1} f(x) \, dx
+\approx
+\sum_{i=1}^n f(x_i) A_i
+\end{align*}
+
+\begin{itemize}[<+->]
+\item Exakt für Polynome mit Grad $2n-1$
+\item Interpolationspolynome müssen orthogonal sein
+\item Stützstellen $x_i$ sind Nullstellen des Polynoms
+\item Fehler:
+\begin{align*}
+E
+=
+\frac{f^{(2n)}(\xi)}{(2n)!} \int_{-1}^{1} l(x)^2 \, dx
+,\quad
+\text{wobei }
+l(x) = \prod_{i=1}^n (x-x_i)
+\end{align*}
+\end{itemize}
+\end{frame}
+
+\begin{frame}{Gauss-Laguerre-Quadratur}
+\begin{itemize}[<+->]
+\item Erweiterung des Integrationsintervall von $[-1, 1]$ auf $(a, b)$
+\item Hinzufügen einer Gewichtsfunktion
+\item Bei uneigentlichen Integralen muss Gewichtsfunktion schneller als jedes
+Integrationspolynom gegen $0$ gehen
+\item[$\Rightarrow$] Für Laguerre-Polynome haben wir den Definitionsbereich
+$(0, \infty)$ und die Gewichtsfunktion $w(x) = e^{-x}$
+\begin{align*}
+\int_0^\infty & f(x) e^{-x} \, dx
+\approx
+\sum_{i=1}^n f(x_i) A_i
+\\
+                & \text{wobei }
+A_i = \frac{x_i}{(n+1)^2 \left[ L_{n+1}(x_i) \right]^2}
+\text{ und $x_i$ die Nullstellen von $L_n(x)$}
+\end{align*}
+\end{itemize}
+\end{frame}
+
+\begin{frame}{Fehler der Gauss-Laguerre-Quadratur}
+\begin{align*}
+R_n
+=
+\frac{(n!)^2}{(2n)!} f^{(2n)}(\xi)
+,\quad
+0 < \xi < \infty
+\end{align*}
+\end{frame}
\ No newline at end of file
diff --git a/buch/papers/laguerre/presentation/sections/laguerre.tex b/buch/papers/laguerre/presentation/sections/laguerre.tex
new file mode 100644
index 0000000..cba9ffb
--- /dev/null
+++ b/buch/papers/laguerre/presentation/sections/laguerre.tex
@@ -0,0 +1,88 @@
+\section{Laguerre-Polynome}
+
+\begin{frame}{Laguerre-Differentialgleichung}
+
+\begin{itemize}
+\item Benannt nach Edmond Nicolas Laguerre (1834-1886)
+\item Aus Artikel von 1879,
+in dem er $\int_0^\infty \exp(-x)/x \, dx$ analysierte
+\end{itemize}
+
+\begin{align*}
+x y''(x) + (1 - x) y'(x) + n y(x)
+ & =
+0
+, \quad
+n \in \mathbb{N}_0
+, \quad
+x \in \mathbb{R}
+\end{align*}
+
+\end{frame}
+
+\begin{frame}{Lösen der Differentialgleichung}
+
+\begin{align*}
+x y''(x) + (1 - x) y'(x) + n y(x)
+ & =
+0
+\\
+\end{align*}
+
+\uncover<2->{
+\centering
+\begin{tikzpicture}[remember picture,overlay]
+%% use here too
+\path[draw=mainColor, very thick,->](0, 1.1) to
+node[anchor=west]{Potenreihenansatz} (0, -0.8);
+\end{tikzpicture}
+}
+
+\begin{align*}
+\uncover<3->{
+L_n(x)
+ & =
+\sum_{k=0}^{n} \frac{(-1)^k}{k!} \binom{n}{k} x^k
+}
+\end{align*}
+\uncover<4->{
+\begin{itemize}
+    \item Die Lösungen der DGL sind die Laguerre-Polynome
+\end{itemize}
+}
+\end{frame}
+
+\begin{frame}
+\begin{figure}[h]
+\centering
+\scalebox{0.66}{\input{../images/laguerre_polynomes.pgf}}
+\caption{Laguerre-Polynome vom Grad $0$ bis $7$}
+\end{figure}
+\end{frame}
+
+\begin{frame}{Orthogonalität}
+\begin{itemize}[<+->]
+\item Beweis: Umformen in Sturm-Liouville-Problem (siehe Paper)
+\begin{alignat*}{5}
+((p(x) &y'(x)))' + q(x) &y(x) 
+&=
+\lambda &w(x) &y(x)
+\\
+((x e^{-x} &y'(x)))' + 0 &y(x)
+&=
+n &e^{-x} &y(x)
+\end{alignat*}
+\item Definitionsbereich $(0, \infty)$
+\item Gewichtsfunktion $w(x) = e^{-x}$
+\end{itemize}
+
+\only<4>{
+\begin{align*}
+\int_0^\infty e^{-x} L_n(x) L_m(x) \, dx
+=
+0
+,\quad
+n, m \in \mathbb{N}
+\end{align*}
+}
+\end{frame}
\ No newline at end of file
diff --git a/buch/papers/laguerre/scripts/gamma_approx.py b/buch/papers/laguerre/scripts/gamma_approx.py
index 9c8f3ee..dd50d92 100644
--- a/buch/papers/laguerre/scripts/gamma_approx.py
+++ b/buch/papers/laguerre/scripts/gamma_approx.py
@@ -39,7 +39,7 @@ def find_shift(z, target):
 
 def find_optimal_shift(z, n):
     mhat = 1.34093 * n + 0.854093
-    steps = int(np.ceil(mhat - np.real(z))) - 1
+    steps = int(np.floor(mhat - np.real(z)))
     return steps
 
 
@@ -136,7 +136,7 @@ ax.set_xlabel(r"$z$")
 ax.set_ylabel("Relativer Fehler")
 ax.legend(ncol=3, fontsize="small")
 ax.grid(1, "both")
-# fig.savefig(f"{img_path}/rel_error_simple.pgf")
+fig.savefig(f"{img_path}/rel_error_simple.pgf")
 
 
 # Mirrored
@@ -162,7 +162,7 @@ ax2.set_xlabel(r"$z$")
 ax2.set_ylabel("Relativer Fehler")
 ax2.legend(ncol=1, loc="upper left", fontsize="small")
 ax2.grid(1, "both")
-# fig2.savefig(f"{img_path}/rel_error_mirror.pgf")
+fig2.savefig(f"{img_path}/rel_error_mirror.pgf")
 
 
 # Move to target
@@ -202,7 +202,7 @@ ax3.set_yticks(np.arange(len(ns)))
 ax3.set_yticklabels(ns)
 ax3.set_xlabel(r"$z$")
 ax3.set_ylabel(r"$n$")
-# fig3.savefig(f"{img_path}/targets.pdf")
+fig3.savefig(f"{img_path}/targets.pdf")
 
 targets = np.mean(bests, -1)
 intercept, bias = np.polyfit(ns, targets, 1)
@@ -211,16 +211,16 @@ fig4, axs4 = plt.subplots(
 )
 xl = np.array([ns[0] - 0.5, ns[-1] + 0.5])
 axs4[0].plot(xl, intercept * xl + bias, label=r"$\hat{m}$")
-axs4[0].plot(ns, targets, "x", label=r"$\bar{m}$")
-axs4[1].plot(ns, ((intercept * ns + bias) - targets), "-x", label=r"$\hat{m} - \bar{m}$")
+axs4[0].plot(ns, targets, "x", label=r"$\overline{m}$")
+axs4[1].plot(ns, ((intercept * ns + bias) - targets), "-x", label=r"$\hat{m} - \overline{m}$")
 axs4[0].set_xlim(*xl)
 # axs4[0].set_title("Schätzung von Mittelwert")
 # axs4[1].set_title("Fehler")
-axs4[-1].set_xlabel(r"$z$")
+axs4[-1].set_xlabel(r"$n$")
 for ax in axs4:
     ax.grid(1)
     ax.legend()
-# fig4.savefig(f"{img_path}/schaetzung.pgf")
+fig4.savefig(f"{img_path}/estimate.pgf")
 
 print(f"Intercept={intercept:.6g}, Bias={bias:.6g}")
 predicts = np.ceil(intercept * ns[:, None] + bias - x)
@@ -234,14 +234,14 @@ gamma = scipy.special.gamma(x)[:, None]
 n = 8
 targets = np.arange(10, 14)
 gamma = scipy.special.gamma(x)
-fig5, ax5 = plt.subplots(num=1, clear=True, constrained_layout=True)
+fig5, ax5 = plt.subplots(num=5, clear=True, constrained_layout=True)
 for target in targets:
     gamma_lag = eval_laguerre_gamma(x, target=target, n=n, func="shifted")
     rel_error = np.abs(calc_rel_error(gamma, gamma_lag))
-    ax5.semilogy(x, rel_error, label=f"$m={target}$")
+    ax5.semilogy(x, rel_error, label=f"$m={target}$", linewidth=3)
 gamma_lgo = eval_laguerre_gamma(x, n=n, func="optimal_shifted")
 rel_error = np.abs(calc_rel_error(gamma, gamma_lgo))
-ax5.semilogy(x, rel_error, label="$m^*$")
+ax5.semilogy(x, rel_error, "c", linestyle="dotted", label="$m^*$", linewidth=3)
 ax5.set_xlim(x[0], x[-1])
 ax5.set_ylim(5e-9, 5e-8)
 ax5.set_xlabel(r"$z$")
@@ -254,10 +254,10 @@ x = np.linspace(-5+ EPSILON, 5-EPSILON, N)
 gamma = scipy.special.gamma(x)[:, None]
 n = 8
 gamma = scipy.special.gamma(x)
-fig6, ax6 = plt.subplots(num=1, clear=True, constrained_layout=True)
+fig6, ax6 = plt.subplots(num=6, clear=True, constrained_layout=True)
 gamma_lgo = eval_laguerre_gamma(x, n=n, func="optimal_shifted")
 rel_error = np.abs(calc_rel_error(gamma, gamma_lgo))
-ax6.semilogy(x, rel_error, label="$m^*$")
+ax6.semilogy(x, rel_error, label="$m^*$", linewidth=3)
 ax6.set_xlim(x[0], x[-1])
 ax6.set_ylim(5e-9, 5e-8)
 ax6.set_xlabel(r"$z$")
@@ -265,4 +265,14 @@ ax6.grid(1, "both")
 ax6.legend()
 fig6.savefig(f"{img_path}/rel_error_range.pgf")
 
+N = 2001
+x = np.linspace(-5, 5, N)
+gamma = scipy.special.gamma(x)
+fig7, ax7 = plt.subplots(num=7, clear=True, constrained_layout=True)
+ax7.plot(x, gamma)
+ax7.set_xlim(x[0], x[-1])
+ax7.set_ylim(-7.5, 25)
+ax7.grid(1, "both")
+fig7.savefig(f"{img_path}/gamma.pgf")
+
 # plt.show()