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 in the n-dimensional complex space is called a domain of holomorphy if there do not exist non-empty open sets and where is connected, and such that for every holomorphic function on there exists a holomorphic function on with on
In the 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 inverse. For this is no longer true, as it follows from Hartogs' lemma.
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