Once a singularity is removable, the holomorphic extension across it is unique.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 a removable singularity at \(z_0\). Then there is a unique holomorphic function \(F\) on \(\Omega\) with \(F=f\) on \(\Omega\setminus\{z_0\}\).