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In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain.
Formally, an open set Ω {\displaystyle \Omega } in the n-dimensional complex space C n {\displaystyle {\mathbb {C} }^{n}} is called a domain of holomorphy if there do not exist non-empty open sets U ⊂ Ω {\displaystyle U\subset \Omega } and V ⊂ C n {\displaystyle V\subset {\mathbb {C} }^{n}} where V {\displaystyle V} is connected, V ⊄ Ω {\displaystyle V\not \subset \Omega } and U ⊂ Ω ∩ V {\displaystyle U\subset \Omega \cap V} such that for every holomorphic function f {\displaystyle f} on Ω {\displaystyle \Omega } there exists a holomorphic function g {\displaystyle g} on V {\displaystyle V} with f = g {\displaystyle f=g} on U {\displaystyle U}
In the n = 1 {\displaystyle n=1} case, every open set is a domain of holomorphy: we can define a holomorphic function with zeros accumulating everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its reciprocal. For n ≥ 2 {\displaystyle n\geq 2} this is no longer true, as it follows from Hartogs' lemma.