Domain of holomorphy

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The sets in the definition.

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

When $n = 1$ , then every open set is a domain of holomorphy: we can define a holomorphic function which has zeros which accumulate everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its inverse. For $n \geq 2$ this is no longer true, as it follows from Hartogs' lemma.