Theorem
Let \(f:G\setminus K\to \mathbb C\) be a holomorphic function, \(G\subseteq \mathbb C^n\) \(n\geq 2\) an open set, and \(K\subset G\) compact. If \(G\setminus K\) is connected, then there exists a unique holomorphic extension of \(f\) to \(G\).
Note that the requirement \(n\geq 2\) is important, when \(n = 1\) the function \(f = 1/z\) on \(\mathbb C\setminus \{0\}\) satisfies all the requirements above but cannot be extended to \(\mathbb C\).