Analytic continuation in several variables

Summary

Analytic continuation in several variables extends a holomorphic function on an open set of $\C^n$ step by step through overlapping domains. It generalises analytic continuation in one variable, and by the identity theorem each step is unique. As in one variable, continuation along different paths can lead to different values. So the result lives on a domain spread over $\C^n$ rather than on a subset of $\C^n$. Unlike in one variable, for $n \ge 2$ every holomorphic function on certain domains continues to a strictly larger domain.