This is the picture of a graph I would like to plot. This is what I have tried to do, what is missing is how to align all of the points properly so that it looks nice.
\documentclass{article}
\usepackage{tikz}
\usepackage{amssymb}
\usetikzlibrary{calc,decorations.markings}
\usetikzlibrary{decorations.markings,decorations.pathmorphing}
\usetikzlibrary{shapes.geometric}
\begin{document}
\begin{tikzpicture}
\draw [->] (-5.5,0) -- (5.5,0) node [below right] {$\mathfrak{Re}(z)$};
\draw [->] (0,-5.5) -- (0,5.5) node [above ] {$\mathfrak{Im}(z)$};
%Branch cuts
\draw [decorate,decoration={zigzag,segment length=2,amplitude=1,post=lineto,post length=10}] (cos 45,sin 45) -- (5cos 45,5sin 45);
\draw [decorate,decoration={zigzag,segment length=2,amplitude=1,post=lineto,post length=10}] (cos -45,sin -45) -- (5cos 45,5sin -45);
%Path gamma1
\draw [->] (0.7cos 45, 0.3+0.7sin 45 ) -- (5cos 45,0.3+5sin 45);
%Path gamma2
\draw [->] (1.12426cos 45, -0.3+1.12426sin 45 ) -- (5.424264cos 45,-0.3+5.424264sin 45);
%Connecting gamma 1 and gamma 2
\draw [-] (0.7cos 45, 0.3+0.7sin 45 ) -- (1.12426cos 45, -0.3+1.12426sin 45 );
%Path gamma 3
\draw [->] (1.12426cos 45, 0.3+1.12426sin -45 ) -- (1.124265cos 45,0.3+51.12426sin -45);
%Path gamma 4
\draw [->] (0.7cos 45, -0.3+0.7sin -45 ) -- (51.12426cos 45,-0.3+1.124265sin -45);
%Connecting gamma 3 and gamma 4
\draw [-] (1.12426cos 45, 0.3+1.12426sin -45 ) -- (0.7cos 45, -0.3+0.7sin -45 );
%Gamma R
% \draw plot [smooth] coordinates {(5.424264cos 45,-0.3+5.424264sin 45) (5cos 25, 5 sin 25) (5,0 ) (1.124265cos 45,0.3+51.12426sin -45)};
\draw [-] (5.424264cos 45,-0.3+5.424264sin 45) arc (45:-45:51.001cm);
\draw [-] (5cos 45,0.3+5sin 45) arc (45:310.3:51.001cm);
\end{tikzpicture}
\end{document}
