\documentclass{standalone}

\usepackage{tikz}

\def\pole#1{
  \draw#1++(0.1,0.1)--++(-0.2,-0.2);
  \draw#1++(-0.1,0.1)--++(0.2,-0.2);
  \draw[color=red, line width=1pt]#1circle(0.3);
}

\begin{document}
\begin{tikzpicture}

% branch cut
\fill[color=gray](-3.3,-0.1)--++(3.3,0)--++(0,0.2)--++(-3.3,0)--cycle;

% axes
\draw(-3.3,0)--++(6.6,0);
\draw(0,-3.3)--++(0,6.6);

% -ik0^2
\pole{(0,-2)}

% other poles
\input{poles.tikz}

% contour
\draw[color=red, line width=1pt](-3.3,-0.3)--++(3.3,0);
\draw[color=red, line width=1pt](0,-0.3)..controls(0.3,-0.3)and(0.3,0.3)..(0,0.3);
\draw[color=red, line width=1pt](-3.3,0.3)--++(3.3,0);


\end{tikzpicture}
\end{document}