%% 
%%  A DANTE-Edition example
%% 
%% 
%%  Copyright (C) 2011 Herbert Voss
%% 
%%  It may be distributed and/or modified under the conditions
%%  of the LaTeX Project Public License, either version 1.3
%%  of this license or (at your option) any later version.
%% 
%%  See http://www.latex-project.org/lppl.txt for details.
%% 
%% 
%% ==== 
% Show page(s) 1
%% 
\documentclass[]{article}
\pagestyle{empty}
\setlength\textwidth{375.57637pt}
\setlength\parindent{0pt}

\usepackage{pst-plot,amsmath}
\newcommand{\htick}[1]{\psline[linewidth=.1mm](#1,-.75mm)(#1,.75mm)}
\newcommand{\vtick}[1]{\psline[linewidth=.1mm](-.75mm,#1)(.75mm,#1)}
\psset{arrowsize=3pt 3,arrowlength=1.75,arrowinset=.2}

\begin{document}
\footnotesize% Ulrich Dirr
\psset{xunit=15mm,yunit=15mm,curvature=1 .1 .75,plotpoints=200}
\begin{pspicture}(-0.3,-0.3)(4.0,2.5)
\psaxes[Dx=0.5,dx=0.5,Dy=0.5,dy=0.5,labelsep=.75mm,
  showorigin=true,arrowlength=1.75,ticksize=2pt,linewidth=0.17mm]%
{->}(0,0)(-0.3,-0.3)(4.0,2.5)
\uput[90](4,0){$\text{Re}(z)$}
\uput[0](0,2.5){$\text{Im}(z)$}
%
\uput{3mm}[90]{40}(1,.75){$|z|=3$}
\uput[0](2.3,.96){$y$}
\uput[90](1.15,1.92){$x$}
\uput[0](2.85,1){$\sigma=40^\circ$}
\uput[45](2.298133329,1.928362829){$z$}
%
\psline[linewidth=.47mm]{->}(0,0)(2.298133329,1.928362829)
\psline[linewidth=.17mm,linecolor=red,linestyle=dashed,dash=.7mm .5mm]%
       (2.298133329,0)(2.298133329,1.928362829)(0,1.928362829)
\psarc[linewidth=.17mm]{->}{4.5}{0}{40}
\end{pspicture}

\vspace{5mm}
\psset{xunit=8mm,yunit=8mm,curvature=1 .1 .75,plotpoints=200}
\begin{pspicture}(-0.1,-0.35)(8.0,3.5)
\psaxes[labels=none,arrowlength=1.75,ticks=none,linewidth=0.17mm]%
{->}(0,0)(-0.2,-0.35)(8.0,3.5)
\rput[Br](8,-.45){$\text{Re}(z)$}
\rput[tl](.1875,3.5){$\text{Im}(z)$}
\rput[B](1,-.45){$1$}
\rput[r](-.1875,1){$1$}
\htick{1}
\htick{6}
\vtick{1}
%
\uput{1mm}[90]{22.6198}(3,1.25){$|z|$}
\uput{1mm}[0](2,.41){$\sigma$}
\uput{1mm}[0](2.5,.25){$x=|z|\cos\sigma$}
\uput{1mm}[0](6,1.25){$y=|z|\sin\sigma$}
\uput{1mm}[45](6,2.5){$z=(x;y)$}
%
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm](6,0)(6,2.5)
\psline[linewidth=.17mm,linecolor=red]{->}(0,0)(6,2.5)
\psarc[linewidth=.17mm,linecolor=red]{->}{1.6}{0}{22.6198}
\end{pspicture}


\vspace{5mm}
\psset{xunit=10mm,yunit=10mm,curvature=1 .1 .75,plotpoints=200}
\begin{pspicture}(-1.5,-1.7)(7,2.7)
\psaxes[labelsep=.75mm,arrowlength=1.75,ticksize=2pt,linewidth=0.17mm]%
{->}(0,0)(-1.5,-1.5)(7,2)
\rput[Br](7,-.35){$\sigma$}
\rput[tr](-.15,2){$y$}
\rput[Br](-.15,-.35){$0$}
%
\uput{1mm}[45](3.1,.1){$\pi$}
\uput{1mm}[45](2,.9){$y=\sin\sigma$}
\uput{1mm}[135](6,.9){$y=\cos\sigma$}
%
\psline[linewidth=.17mm]{<->}(-1.283185307,-1.5)(5,-1.5)
\psline[linewidth=.35mm](-1.283185307,0)(-1.283185307,.2836621856)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm](-1.283185307,.2836621856)(-0.3,.2836621856)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm](-1.283185307,.2836621856)(-1.283185307,-1.7)
\psline[linewidth=.35mm](5,0)(5,.2836621856)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm](5,.2836621856)(4.5,.2836621856)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm](5,0)(5,-1.7)
%
\psline[linewidth=.17mm]{<->}(-.7831853072,2.5)(5.5,2.5)
\psline[linewidth=.35mm](5.5,0)(5.5,-.7055403256)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm](5.5,2.7)(5.5,1.3)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm](5.5,.5)(5.5,0)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm](5.5,-.7055403256)(4.5,-.7055403256)
\psline[linewidth=.35mm](-.7831853072,-.7055403256)(-.7831853072,0)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm](-.3,-.7055403256)(-.7831853072,-.7055403256)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm](-.7831853072,2.7)(-.7831853072,1.3)
%
\psplot[linewidth=.35mm,linecolor=red]{-1.5}{7}{180 3.141592654 div x mul cos}
\psplot[linewidth=.17mm]{-1.5}{7}{180 3.141592654 div x mul sin}
%
\rput(2.36,2.5){\psframebox*{$2\pi$}}
\rput(1.86,-1.5){\psframebox*{$2\pi$}}
\end{pspicture}

\vspace{5mm}
\psscalebox{0.75}{%
\psset{unit=10mm,curvature=1 .1 .75,plotpoints=200}
\begin{pspicture}(-2.7,-2.7)(3.5,3.5)
\psaxes[labels=none,arrowlength=1.75,ticks=none,linewidth=0.17mm]%
{->}(0,0)(-2.7,-2.7)(3.5,3.5)
\rput[Br](3.5,-.35){$\text{Re}(z)$}
\rput[tl](.15,3.5){$\text{Im}(z)$}
%
\uput{1mm}[45](1.8,1.8){$\left(\frac{1}{2}\sqrt{2};\frac{1}{2}\sqrt{2}\right)$}
\uput{1mm}[45](.8,.2){$45^\circ$}
\uput{1mm}[45](.5,1.4){$90^\circ$}
%
\pscircle[linewidth=.17mm]{2.5}
\psline[linewidth=.35mm,linecolor=red]{->}(0,0)(-2.5,0)
\psline[linewidth=.17mm]{->}(0,0)(2.5,0)
\psline[linewidth=.17mm](0,0)(0,1.5)
\psline[linewidth=.35mm,linecolor=white]{->}(0,1.5)(0,2.5)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm]{->}(0,1.5)(0,2.5)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm](0,1.5)(0,2.5)
\psline[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm]{->}(1.06,1.06)(1.767766,1.767766)
\psline[linewidth=.17mm](0,0)(1.06,1.06)
\psarc[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm]{->}{1.5}{0}{90}
\psarc[linewidth=.17mm,linestyle=dashed,dash=.7mm .5mm]{->}{.8}{0}{45}
%
\uput{1mm}[90](2.5,.1){\psframebox*{$z$}}
\uput{1mm}[90](-2.5,.1){\psframebox*{$z\circ z$}}
\uput{1mm}[270](-2.5,-.1){\psframebox*{$(-1;0)$}}
\uput{1mm}[270](2.4,-.1){\psframebox*{$(1;0)$}}
%
\end{pspicture}}
\end{document}
