Domain of holomorphy

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

Formally, an open set ${\displaystyle \Omega }$ in the ⁠${\displaystyle n}$⁠-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 ⁠${\displaystyle U}$⁠.

Equivalently, for any such ⁠${\displaystyle U}$⁠ and ⁠${\displaystyle V}$⁠, there exists a holomorphic ${\displaystyle f}$ on ⁠${\displaystyle \Omega }$⁠, such that ${\displaystyle f|_{U}}$ cannot be analytically continued to ⁠${\displaystyle V}$⁠.

In the case ⁠${\displaystyle n=1}$⁠, every open set is a domain of holomorphy: we can define a holomorphic function that is not identically zero, but whose zeros accumulate everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its reciprocal. For ${\displaystyle n\geq 2}$ this is no longer true, as it follows from Hartogs's extension theorem.