Every punctured neighborhood of an essential singularity has dense image in $\CC$.Suggest a slogan
Let \(\Omega\subseteq\CC\) be open, let \(z_0\in\Omega\), and let \(f\) be holomorphic on \(\Omega\setminus\{z_0\}\) with an essential singularity at \(z_0\). Then for every \(\delta>0\) with \(\{z : 0<\abs{z-z_0}<\delta\}\subseteq\Omega\), the image \(f(\{z : 0<\abs{z-z_0}<\delta\})\) is dense in \(\CC\).