%
% fortsetzreziprok.tex -- analytische Fortsetzung der 1/z-Funktion
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\documentclass[tikz]{standalone}
\usepackage{amsmath}
\usepackage{times}
\usepackage{txfonts}
\usepackage{pgfplots}
\usepackage{csvsimple}
\usetikzlibrary{arrows,intersections,math}
\definecolor{darkgreen}{rgb}{0,0.6,0}
\begin{document}
\def\skala{1}
\def\u{3}
\def\punkt#1#2{
	\fill[color=white] #1 circle[radius=0.08];
	\draw[color=#2] #1 circle[radius=0.08];
}
\begin{tikzpicture}[>=latex,thick,scale=\skala]

\fill[color=gray!70,opacity=0.5] (\u,0) circle[radius=\u];
\fill[color=gray!70,opacity=0.5] (0,\u) circle[radius=\u];
\fill[color=gray!70,opacity=0.5] (-\u,0) circle[radius=\u];
\fill[color=gray!70,opacity=0.5] (0,-\u) circle[radius=\u];

%\draw[line width=0.5pt] (\u,0) circle[radius=\u];
%\draw[line width=0.5pt] (0,\u) circle[radius=\u];
%\draw[line width=0.5pt] (-\u,0) circle[radius=\u];
%\draw[line width=0.5pt] (0,-\u) circle[radius=\u];

\draw[->] ({-0.5-2*\u},0) -- ({2*\u+0.8},0)
	coordinate[label={$\operatorname{Re}z$}];
\draw[->] (0,{-0.5-2*\u}) -- (0,{2*\u+0.8})
	coordinate[label={$\operatorname{Im}z$}];

\draw[color=darkgreen,line width=1.4pt] (0,0) circle[radius=\u];
\node[color=darkgreen] at (70:\u) [above] {$\gamma_+$};
\node[color=darkgreen] at (-70:\u) [below] {$\gamma_-$};
\node at (\u,0) [above right] {$1$};
\node at (0,\u) [above left] {$i$};
\node at (-\u,0) [above left] {$-1$};
\node at (0,-\u) [below left] {$-i$};

\punkt{(\u,0)}{black}
\punkt{(-\u,0)}{black}
\punkt{(0,\u)}{black}
\punkt{(0,-\u)}{black}

\punkt{({0.5*\u},{0.5*\u})}{red}
\punkt{({-0.5*\u},{0.5*\u})}{red}
\punkt{({-0.5*\u},{-0.5*\u})}{red}
\punkt{({0.5*\u},{-0.5*\u})}{red}

\node[color=red] at ({0.5*\u},{0.5*\u}) [above] {$(1+i)\frac{\sqrt{2}}{2}$};
\node[color=red] at ({0.5*\u},{-0.5*\u}) [below] {$(1-i)\frac{\sqrt{2}}{2}$};
\node[color=red] at ({-0.5*\u},{-0.5*\u}) [below] {$(-1-i)\frac{\sqrt{2}}{2}$};
\node[color=red] at ({-0.5*\u},{0.5*\u}) [above] {$(-1+i)\frac{\sqrt{2}}{2}$};

\fill[color=white] (0,0) circle[radius=0.2];

\end{tikzpicture}
\end{document}