I would like to modify the black curve in order to include the red dashed line that now is left outside. The arc around the origin should instead be around the red point in the lower-half of the complex plane.
\documentclass[8pt,usenames,dvipsnames]{beamer}
\usepackage{tikz}
\usetikzlibrary{calc,decorations.markings}
\begin{document}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.7, every node/.style={scale=0.85}]
% Configurable parameters
\def\gap{0.2}
\def\bigradius{3}
\def\littleradius{0.25}
% Axes
\draw [help lines,->] (-1.25*\bigradius, 0) -- (1.25*\bigradius,0);
\draw [help lines,->] (0, -1.25*\bigradius) -- (0, 1.25*\bigradius);
% Path
\draw[line width=1pt, decoration={ markings,
mark=at position 0.2455 with {\arrow[line width=0.9pt]{>}},
mark=at position 0.765 with {\arrow[line width=0.9pt]{>}},
mark=at position 0.87 with {\arrow[line width=0.9pt]{>}},
mark=at position 0.97 with {\arrow[line width=0.9pt]{>}}},
postaction={decorate}]
let
\n1 = {asin(\gap/2/\bigradius)},
\n2 = {asin(\gap/2/\littleradius)}
in (\n1:\bigradius) arc (\n1:360-\n1:\bigradius)
-- (-\n2:\littleradius) arc (-\n2:-360+\n2:\littleradius)
-- cycle;
\filldraw [red] (1,0) circle (2pt);
\filldraw [red] (0.5,-1) circle (2pt);
\draw[red, dashed] (1,0) -- (1.25*\bigradius,0);
\draw[red, dashed] (1,0) -- (0.5,-1);
%Labels
\node at (3.6,-0.4){$\Re(z)$};
\node at (-0.6,3.53) {$\Im(z)$};
\end{tikzpicture}
\end{figure}
\end{document}