Domain of holomorphy

In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a set which is maximal in the sense that there exists a holomorphic function on this set which cannot be extended to a bigger set.

Formally, an open set ${\displaystyle \Omega }$ in the n-dimensional complex space ${\displaystyle {\mathbb {C} }^{n}}$ is called a domain of holomorphy if there do not exist non-empty open sets ${\displaystyle U\subset \Omega }$ and ${\displaystyle V\subset {\mathbb {C} }^{n}}$ where ${\displaystyle V}$ is connected, ${\displaystyle V\not \subset \Omega }$ and ${\displaystyle U\subset \Omega \cap V}$ such that for every holomorphic function ${\displaystyle f}$ on ${\displaystyle \Omega }$ there exists a holomorphic function ${\displaystyle g}$ on ${\displaystyle V}$ with ${\displaystyle f=g}$ on ${\displaystyle U}$

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