Conformal mapping

I am trying to draw this Image may be NSFW.
Clik here to view.

So far I have

\documentclass[tikz,border=3mm]{standalone}\usepackage{tikz}\usepackage{pgfplots}\usepackage{amsfonts}\begin{document}    \begin{tikzpicture}    \begin{scope}[local bounding box=L]    \draw[-stealth] (0,-2) -- (0,2)node[anchor=south east] {y};     \draw[-stealth] (-2,-1) -- (2,-1) node[anchor=north west] {x};     \fill[step=0.1cm,black,thick] (-2,-1) grid (1,1);    \path(0.1,1.1) node{$\pi$};    \path(1,-1.2) node{$1$};    \path (1.8,1.8) node{$\mathbb{C}$};    \end{scope}    \begin{scope}[xshift=6cm,local bounding box=R]    \draw[-stealth] (0,-2) -- (0,2)node[anchor=south east] {v};     \draw[-stealth] (-2,-1) -- (2,-1) node[anchor=north west] {u};     \clip[draw](0,-1) circle (1cm);    \draw [step=0.1cm,black,thick] (-2,-1) grid (1,1);    \path (1.8,1.8) node{$\mathbb{C}$};    \end{scope}    %    \draw[-latex] (L.east) -- (L.east-|R.west) node[midway,above]{$f(z)=e^z$};    \end{tikzpicture}\end{document}

which gives the below. Can someone help me? Image may be NSFW.
Clik here to view.