Envelope of holomorphy

Summary

The envelope of holomorphy of a domain $D \subseteq \C^n$ is the largest domain to which every holomorphic function on $D$ continues. This domain may be spread over $\C^n$ in several sheets. It exists for every domain and is unique up to isomorphism. In general it is a Riemann domain over $\C^n$ and not a subset of $\C^n$. A domain is a domain of holomorphy exactly when it is its own envelope of holomorphy.