From f07bb53970ee2ecc53f81a206a3d3a67ef665e4a Mon Sep 17 00:00:00 2001
From: Karl Berry <karl@freefriends.org>
Date: Mon, 9 Jan 2006 00:56:57 +0000
Subject: doc 6

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+\documentclass[a4paper,12pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[latin1]{inputenc}
+\usepackage{geometry}
+\usepackage{url}
+\usepackage{amsmath}
+\usepackage{tabularx}
+\usepackage{longtable}
+\usepackage{pstricks}
+\usepackage{pst-func}
+\let\pstFuncFV\fileversion
+\usepackage{pstricks-add}
+\usepackage{pst-example}
+%
+\usepackage{xspace}
+\def\PS{PostScript\xspace}
+%
+\psset{xyLabel=\footnotesize}
+\usepackage[colorlinks,linktocpage]{hyperref}
+%
+\begin{document}
+\title{\texttt{pst-func}\\plotting special mathematical functions\thanks{%
+		This document was written with \texttt{Kile: 1.6a (Qt: 3.1.1; KDE: 3.1.1;}
+		\protect\url{http://sourceforge.net/projects/kile/}) and the PDF output
+		  was build with VTeX/Free (\protect\url{http://www.micropress-inc.com/linux})}\\
+		  \small v.\pstFuncFV}
+\author{Herbert Voß\thanks{%
+%%JF
+%Thanks to: Attila Gati  and to John Frampton.
+Thanks to: Attila Gati, John Frampton and Lars Kotthoff.
+}}
+\date{\today}
+
+\maketitle
+
+\tableofcontents
+
+\clearpage
+
+\section{\texttt{psPolynomial}}
+The polynomial function is defined as
+\begin{align}
+f(x) &= a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots +a_{n-1}x^{n-1} + a_nx^n\\
+f^{\prime}(x) &= a_1 + 2a_2x + 3a_3x^2 + \ldots +(n-1)a_{n-1}x^{n-2} + na_nx^{n-1}\\
+f^{\prime\prime}(x) &= 2a_2 + 6a_3x + \ldots +(n-1)(n-2)a_{n-1}x^{n-3} + n(n-1)a_nx^{n-2}
+\end{align}
+
+\noindent so \verb+pst-func+ needs only the coefficients of the
+polynomial to calculate the function. The syntax is
+\begin{verbatim}
+\psPolynomial[<options>]{xStart}{xEnd}
+\end{verbatim}
+
+There are the following new options:
+
+\noindent\medskip
+\begin{tabularx}{\linewidth}{>{\ttfamily}l|>{\ttfamily}l>{\ttfamily}lX@{}}
+\textrm{Name} & \textrm{Value}  & \textrm{Default}\\\hline
+coeff        & a0 a1 a2 ... & 0 0 1 & The coefficients must have the order $a_0\ a_1\ a_2 \ldots$ and
+be separated by \textbf{spaces}. The number of coefficients
+is limited only by the memory of the computer ... The default
+value of the parameter \verb+coeff+ is \verb+0 0 1+, which gives
+the parabola $y=a_0+a_1x+a_2x^2=x^2$.\\
+Derivation & <number>     & 0     & the default is the function itself\\
+markZeros    & false|true       & false & dotstyle can be changed\\
+epsZero      & <value> &  0.1 & The distance between two zeros, important for
+                                  the iteration function to test, if the zero value still
+				  exists\\
+dZero      & <value> &  0.1 & When searching for all zero values, the function is scanned
+                              with this step\\
+zeroLineTo    & <number>  & false & plots a line from the zero point to the value of the
+                                    zeroLineTo's Derivation of the polynomial function\\
+zeroLineStyle    & <line style>  & dashed & the style is one of the for PSTricks valid styles.\\
+zeroLineColor  & <color>  & black & any valid xolor is possible\\
+zeroLineWidth  & <width> & 0.5\textbackslash pslinewidth & \\
+\end{tabularx}
+
+
+
+
+\bigskip
+The above parameter are only 
+valid for the \verb+\psPolynomial+
+macro, but can also be set in the usual way with \verb+\psset+.
+
+
+
+
+\begin{Beispiel}
+{\psset{yunit=0.5cm,xunit=1cm}
+\begin{pspicture*}(-3,-5)(5,10)
+  \psaxes[Dy=2]{->}(0,0)(-3,-5)(5,10)
+  \psset{linewidth=1.5pt}
+  \psPolynomial[coeff=6 3 -1,linecolor=red]{-3}{5}
+  \psPolynomial[coeff=2 -1 -1 .5 -.1 .025,linecolor=blue]{-2}{4}
+  \psPolynomial[coeff=-2 1 -1 .5 .1 .025 .2 ,linecolor=magenta]{-2}{4}
+  \rput[lb](4,4){\textcolor{red}{$f(x)$}}
+  \rput[lb](4,8){\textcolor{blue}{$g(x)$}}
+  \rput[lb](2,4){\textcolor{magenta}{$h(x)$}}
+\end{pspicture*}
+}
+\end{Beispiel}
+
+The plot is easily clipped using the star version of the
+\verb+pspicture+ environment, so that points whose coordinates
+are outside of the desired range are not plotted.
+The plotted polynomials are:
+\begin{align}
+f(x) & = 6 + 3x -x^2 \\
+g(x) & = 2 -x -x^2 +0.5x^3 -0.1x^4 +0.025x^5\\
+h(x) & = -2 +x -x^2 +0.5x^3 +0.1x^4 +0.025x^5+0.2x^6
+\end{align}
+
+
+
+\bigskip
+\begin{Beispiel}
+\psset{yunit=0.5cm,xunit=2cm}
+\begin{pspicture*}(-3,-5)(3,10)
+  \psaxes[Dy=2]{->}(0,0)(-3,-5)(3,10)
+  \psset{linewidth=1.5pt}
+  \psPolynomial[coeff=-2 1 -1 .5 .1 .025 .2 ,linecolor=magenta]{-2}{4}
+  \psPolynomial[coeff=-2 1 -1 .5 .1 .025 .2 ,linecolor=red,%
+    linestyle=dashed,Derivation=1]{-2}{4}
+  \psPolynomial[coeff=-2 1 -1 .5 .1 .025 .2 ,linecolor=blue,%
+    linestyle=dotted,Derivation=2]{-2}{4}
+  \rput[lb](2,4){\textcolor{magenta}{$h(x)$}}
+  \rput[lb](1,1){\textcolor{red}{$h^{\prime}(x)$}}
+  \rput[lb](-1,6){\textcolor{blue}{$h^{\prime\prime}(x)$}}
+\end{pspicture*}
+\end{Beispiel}
+
+
+\begin{Beispiel}
+\psset{yunit=0.5cm,xunit=2cm}
+\begin{pspicture*}(-3,-5)(3,10)
+  \psaxes[Dy=2]{->}(0,0)(-3,-5)(3,10)
+  \psset{linewidth=1.5pt}
+  \psPolynomial[coeff=0 0 0 1,linecolor=blue]{-2}{4}
+  \psPolynomial[coeff=0 0 0 1,linecolor=red,%
+    linestyle=dashed,Derivation=2]{-2}{4}
+  \psPolynomial[coeff=0 0 0 1,linecolor=cyan,%
+    linestyle=dotted,Derivation=3]{-2}{4}
+  \rput[lb](1.8,4){\textcolor{blue}{$f(x)=x^3$}}
+  \rput[lb](0.2,8){\textcolor{red}{$f^{\prime}(x)=6x$}}
+  \rput[lb](-2,5.5){\textcolor{magenta}{$f^{\prime\prime}(x)=6$}}
+\end{pspicture*}
+\end{Beispiel}
+
+
+\begin{Beispiel}
+\begin{pspicture*}(-5,-5)(5,5)
+  \psaxes{->}(0,0)(-5,-5)(5,5)%
+  \psset{dotscale=2}
+  \psPolynomial[markZeros,linecolor=red,linewidth=2pt,coeff=-1 1 -1 0 0.15]{-4}{3}%
+  \psPolynomial[markZeros,linecolor=blue,linewidth=1pt,linestyle=dashed,%
+    coeff=-1 1 -1 0 0.15,Derivation=1,zeroLineTo=0]{-4}{3}%
+  \psPolynomial[markZeros,linecolor=magenta,linewidth=1pt,linestyle=dotted,%
+    coeff=-1 1 -1 0 0.15,Derivation=2,zeroLineTo=0]{-4}{3}%
+  \psPolynomial[markZeros,linecolor=magenta,linewidth=1pt,linestyle=dotted,%
+    coeff=-1 1 -1 0 0.15,Derivation=2,zeroLineTo=1]{-4}{3}%
+\end{pspicture*}
+\end{Beispiel}
+
+\begin{Beispiel}
+\psset{xunit=1.5}
+\begin{pspicture*}(-5,-5)(5,5)
+  \psaxes{->}(0,0)(-5,-5)(5,5)%
+  \psset{dotscale=2,dotstyle=x,zeroLineStyle=dotted,zeroLineWidth=1pt}
+  \psPolynomial[markZeros,linecolor=red,linewidth=2pt,coeff=-1 1 -1 0 0.15]{-4}{3}%
+  \psPolynomial[markZeros,linecolor=blue,linewidth=1pt,linestyle=dashed,%
+    coeff=-1 1 -1 0 0.15,Derivation=1,zeroLineTo=0]{-4}{3}%
+  \psPolynomial[markZeros,linecolor=magenta,linewidth=1pt,linestyle=dotted,%
+    coeff=-1 1 -1 0 0.15,Derivation=2,zeroLineTo=0]{-4}{3}%
+  \psPolynomial[markZeros,linecolor=magenta,linewidth=1pt,linestyle=dotted,%
+    coeff=-1 1 -1 0 0.15,Derivation=2,zeroLineTo=1]{-4}{3}%
+\end{pspicture*}
+\end{Beispiel}
+
+
+
+\section{\texttt{psFourier}}
+
+A Fourier sum has the form:
+\begin{align}
+s(x) = \frac{a_0}{2} & + a_1\cos{\omega x} + a_2\cos{2\omega x} +
+   a_3\cos{3\omega x} +
+	\ldots + a_n\cos{n\omega x}\\
+	& + b_1\sin{\omega x} + b_2\sin{2\omega x} + b_3\sin{3\omega x} +
+	\ldots + b_m\sin{m\omega x}
+\end{align}
+
+\noindent The macro \verb+psFourier+ plots Fourier sums. The
+syntax is similiar to \verb+psPolynomial+, except that there are
+two kinds of coefficients:
+\begin{verbatim}
+\psPolynomial[cosCoeff=a0 a1 a2 ..., sinCoeff=b1 b2 ...]{xStart}{xEnd}
+\end{verbatim}
+The coefficients must have the orders $a_0\ a_1\ a_2\ \ldots$
+and $b_1\ b_2\ b_3\ \ldots$ and be separated by
+\textbf{spaces}. The default is \verb+cosCoeff=0,sinCoeff=1+,
+which gives the standard \verb+sin+ function. Note that
+%%JF, I think it is better without the angle brackets, but
+%%you know the conventions used better than I do, so you
+%%may disagree.
+%the constant value can only be set with \verb+cosCoeff=<a0>+.
+the constant value can only be set with \verb+cosCoeff=a0+.
+
+\begin{Beispiel}
+\begin{pspicture}(-5,-3)(5,5.5)
+\psaxes{->}(0,0)(-5,-2)(5,4.5)
+\psset{plotpoints=500,linewidth=1pt}
+\psFourier[cosCoeff=2, linecolor=green]{-4.5}{4.5}
+\psFourier[cosCoeff=0 0 2, linecolor=magenta]{-4.5}{4.5}
+\psFourier[cosCoeff=2 0 2, linecolor=red]{-4.5}{4.5}
+\end{pspicture}
+\end{Beispiel}
+
+\begin{Beispiel}
+\psset{yunit=0.75}
+\begin{pspicture}(-5,-6)(5,7)
+\psaxes{->}(0,0)(-5,-6)(5,7)
+\psset{plotpoints=500}
+\psFourier[linecolor=red,linewidth=1pt]{-4.5}{4.5}
+\psFourier[sinCoeff= -1 1 -1 1 -1 1 -1 1,%
+	linecolor=blue,linewidth=1.5pt]{-4.5}{4.5}
+\end{pspicture}
+\end{Beispiel}
+
+\begin{Beispiel}
+\begin{pspicture}(-5,-5)(5,5.5)
+\psaxes{->}(0,0)(-5,-5)(5,5)
+\psset{plotpoints=500,linewidth=1.5pt}
+\psFourier[sinCoeff=-.5 1 1 1 1 ,sinCoeff=-.5 1 1 1 1 1,%
+	linecolor=blue]{-4.5}{4.5}
+\end{pspicture}
+\end{Beispiel}
+
+\section{\texttt{psBessel}}
+The Bessel function of order $n$ is defined as
+\begin{align}
+J_n(x) &=\frac{1}{\pi}\int_0^\pi\cos(x\sin t-nt)\mathrm{d}t\\
+       &=\sum_{k=0}^{\infty}\frac{(-1)^k \left(\frac{x}{2}\right)^{n+2k}}{k!\Gamma(n+k+1)}
+\end{align}
+
+\noindent The syntax of the macro is
+\begin{verbatim}
+\psBessel[options]{order}{xStart}{xEnd}
+\end{verbatim}
+
+There are two special parameters for the Bessel function, and also the
+settings of many \verb+pst-plot+ or \verb+pstricks+ parameters
+affect the plot. 
+
+\begin{verbatim}
+\def\psset@constI#1{\edef\psk@constI{#1}}
+\def\psset@constII#1{\edef\psk@constII{#1}}
+\psset{constI=1,constII=0}
+\end{verbatim}
+
+These two "'constants"` have the following meaning:
+\[
+f(t) = constI \cdot J_n + constII
+\]
+
+\noindent
+where $constI$ and $constII$ must be real PostScript expressions, e.g.:
+\begin{verbatim}
+\psset{constI=2.3,constII=t k sin 1.2 mul 0.37 add}
+\end{verbatim}
+
+The Bessel function is plotted with the parametricplot macro, this is the
+reason why the variable is named \verb+t+. The internal procedure \verb+k+ 
+converts the value t from radian into degrees. The above setting is
+the same as
+\[
+f(t) = 2.3 \cdot J_n + 1.2\cdot \sin t + 0.37
+\]
+ 
+
+In particular, note that the default for
+\verb+plotpoints+ is $500$. If the plotting computations are too
+time consuming at this setting, it can be decreased in the usual
+way, at the cost of some reduction in graphics resolution.
+
+\begin{Beispiel}
+{
+\psset{xunit=0.25,yunit=5}
+\begin{pspicture}(-13,-.85)(13,1.25)
+\rput(13,0.8){%
+	$\displaystyle J_n(x)=\frac{1}{\pi}\int_0^\pi\cos(x\sin t-nt)\mathrm{d}t$%
+}
+\psaxes[Dy=0.2,Dx=4,xyLabel=\footnotesize]{->}(0,0)(-30,-.8)(30,1.2)
+\psset{linewidth=1pt}
+\psBessel[linecolor=red]{0}{-28}{28}%
+\psBessel[linecolor=blue]{1}{-28}{28}%
+\psBessel[linecolor=green]{2}{-28}{28}%
+\psBessel[linecolor=magenta]{3}{-28}{28}%
+\end{pspicture}
+}
+\end{Beispiel}
+
+
+\begin{Beispiel}
+{
+\psset{xunit=0.25,yunit=2.5}
+\begin{pspicture}(-13,-.85)(13,2)
+\rput(13,0.8){%
+	$\displaystyle f(t) = 2.3 \cdot J_0 + 1.2\cdot \sin t + 0.37$%
+}
+\psaxes[Dy=0.2,Dx=4,xyLabel=\footnotesize]{->}(0,0)(-30,-.8)(30,1.2)
+\psset{linewidth=1pt}
+\psBessel[linecolor=red,constI=2.3,constII={t k sin 1.2 mul 0.37 add}]{0}{-28}{28}%
+\end{pspicture}
+}
+\end{Beispiel}
+
+
+\section{\texttt{psGauss}}
+The Gauss function is defined as
+\begin{align}
+f(x) &= \dfrac{1}{\sigma\sqrt{2\pi}}\,e^{-\dfrac{x^2}{2\sigma{}^2}}
+\end{align}
+
+\noindent The syntax of the macro is
+\begin{verbatim}
+\psGauss[options]{xStart}{xEnd}
+\end{verbatim}
+
+%%JF
+%% comment, the angle brackets below, around "value", make sense
+%% as a convention, so I left them in
+%
+%\noindent where the only new parameter is \verb+sigma=<value>+, with
+%the default of \verb+0.5+ and can also be set in the usual way with
+%\verb+\psset+. It is only valid for the \verb+psGauss+-macro.
+\noindent where the only new parameter is \verb+sigma=<value>+,
+which can also be set in the usual way with \verb+\psset+. It is
+significant only for the \verb+psGauss+-macro. The default is
+\verb+0.5+.
+
+\begin{Beispiel}
+\psset{yunit=4cm,xunit=3}
+\begin{pspicture}(-2,0)(2,1)
+%  \psgrid[griddots=10,gridlabels=0pt, subgriddiv=0]
+  \psaxes[xyLabel=\footnotesize,Dy=0.25]{->}(0,0)(-2,0)(2,1)
+  \uput[-90](6,0){x}\uput[0](0,1){y}
+  \rput[lb](1,0.75){\textcolor{red}{$\sigma =0.5$}}
+  \rput[lb](1,0.5){\textcolor{blue}{$\sigma =1$}}
+  \rput[lb](-2,0.5){$f(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\,e^{-\dfrac{x^2}{2\sigma{}^2}}$}
+  \psGauss[linecolor=red, linewidth=2pt]{-1.75}{1.75}%
+  \psGauss[sigma=1, linecolor=blue, linewidth=2pt]{-1.75}{1.75}
+\end{pspicture}
+\end{Beispiel}
+
+
+\section{Credits}
+Denis Girou |  Manuel Luque | Timothy Van Zandt
+
+
+\nocite{*}
+\bibliographystyle{plain}
+\bibliography{pstricks}
+
+\end{document}
+
+
-- 
cgit v1.2.3

