Holomorphic extension ($F{|}_U=f,F\in \mathcal{O}(V),U\subseteq V$)

A holomorphic function $F$ on a larger open set $V$ whose restriction to $U$ is the original holomorphic function $f$. A removable singularity is a point across which such an extension exists. On a connected larger domain, a holomorphic extension is unique by the identity theorem, provided $U$ is nonempty.