Illustration Jacobi-Gewichtsfunktion

3 files changed, 32 insertions, 4 deletions

diff --git a/buch/chapters/070-orthogonalitaet/images/jacobi.pdf b/buch/chapters/070-orthogonalitaet/images/jacobi.pdf
index 8eb4f09..1504128 100644
--- a/buch/chapters/070-orthogonalitaet/images/jacobi.pdf
+++ b/buch/chapters/070-orthogonalitaet/images/jacobi.pdf
diff --git a/buch/chapters/070-orthogonalitaet/images/jacobi.tex b/buch/chapters/070-orthogonalitaet/images/jacobi.tex
index 7d2853f..2a53e75 100644
--- a/buch/chapters/070-orthogonalitaet/images/jacobi.tex
+++ b/buch/chapters/070-orthogonalitaet/images/jacobi.tex
@@ -11,7 +11,7 @@
 \usepackage{csvsimple}
 \usetikzlibrary{arrows,intersections,math}
 \begin{document}
-\def\skala{1}
+\def\skala{1.07}
 \input{jacobipaths.tex}
 \begin{tikzpicture}[>=latex,thick,scale=\skala]
 
@@ -36,10 +36,14 @@
 		\draw[color=blue] \legendrek;
 		\draw[color=red] \legendreweight;
 	\end{scope}
-	\draw[->] (-6.1,0) -- (6.1,0) coordinate[label={$x$}];
+	\draw[->] (-6.1,0) -- (6.3,0) coordinate[label={$x$}];
 	\draw[->] (0,-2.55) -- (0,2.6) coordinate[label={right:$y$}];
 	\node at (6,2.3) [left] {$a=0$};
 	\node at (-6,2.3) [right] {$b=0$};
+	\draw ({-1*\dx},-0.1) -- ({-1*\dx},0.1);
+	\draw ({1*\dx},-0.1) -- ({1*\dx},0.1);
+	\node at ({1*\dx},-0.1) [below] {$1$};
+	\node at ({-0.97*\dx},-0.1) [below left] {$-1$};
 \end{scope}
 
 \begin{scope}[yshift=-5.6cm]
@@ -63,10 +67,15 @@
 		\draw[color=blue] \aoneo;
 		\draw[color=red] \aoneweight;
 	\end{scope}
-	\draw[->] (-6.1,0) -- (6.1,0) coordinate[label={$x$}];
+	\draw[->] (-6.1,0) -- (6.3,0) coordinate[label={$x$}];
 	\draw[->] (0,-2.55) -- (0,2.6) coordinate[label={right:$y$}];
+	\fill[color=white,opacity=0.8] (4.9,2.1) rectangle (5.9,2.5);
 	\node at (6,2.3) [left] {$a=1$};
 	\node at (-6,2.3) [right] {$b=0$};
+	\draw ({-1*\dx},-0.1) -- ({-1*\dx},0.1);
+	\draw ({1*\dx},-0.1) -- ({1*\dx},0.1);
+	\node at ({1*\dx},-0.1) [below] {$1$};
+	\node at ({-0.97*\dx},-0.1) [below left] {$-1$};
 \end{scope}
 
 \begin{scope}[yshift=-11.2cm]
@@ -91,10 +100,14 @@
 		\draw[color=blue] \aminusoneo;
 		\draw[color=red] \aminusoneweight;
 	\end{scope}
-	\draw[->] (-6.1,0) -- (6.1,0) coordinate[label={$x$}];
+	\draw[->] (-6.1,0) -- (6.3,0) coordinate[label={$x$}];
 	\draw[->] (0,-2.55) -- (0,2.6) coordinate[label={right:$y$}];
 	\node at (6,2.3) [left] {$a=-1$};
 	\node at (-6,2.3) [right] {$b=0$};
+	\draw ({-1*\dx},-0.1) -- ({-1*\dx},0.1);
+	\draw ({1*\dx},-0.1) -- ({1*\dx},0.1);
+	\node at ({1*\dx},-0.1) [below] {$1$};
+	\node at ({-0.97*\dx},-0.1) [below left] {$-1$};
 \end{scope}
 
 
diff --git a/buch/chapters/070-orthogonalitaet/jacobi.tex b/buch/chapters/070-orthogonalitaet/jacobi.tex
index 3dcf523..8ccf6a3 100644
--- a/buch/chapters/070-orthogonalitaet/jacobi.tex
+++ b/buch/chapters/070-orthogonalitaet/jacobi.tex
@@ -177,10 +177,25 @@ Pol vom Grad $-\alpha$ an der Stelle $x=1$.
 Funktionen müssen daher also ein Nullstelle mindestens vom Grad
 $\frac12(1-\alpha)$ haben.
 
+\begin{figure}
+\centering
+\includegraphics[width=\textwidth]{chapters/070-orthogonalitaet/images/jacobi.pdf}
+\caption{Jacobi-Polynome vom Grad $1$ bis $14$ für verschiedene Werte
+der Parameter $a$ und $b$.
+Je grösser $a$, desto weniger Gewicht bekommen die Funktionswerte am
+rechten Rand und desto grösser werden die Funktionswerte.
+Für negative $a$ müssen die Polynome dagegen eine Nullstelle am
+rechten Rand haben.
+\label{buch:orthogonal:fig:jacobi-parameter}}
+\end{figure}
+
 %
 % 
 %
 \subsection{Jacobi-Polynome niedrigen Grades}
+In Abbildung~\ref{buch:orthogonal:fig:jacobi-parameter}
+ist die Abhängigkeit der Jacobi-Polynome von den Parametern $a$ und $b$
+illustriert.
 
 %
 %