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 \Omega in the n-dimensional complex space is called a domain of holomorphy if there do not exist non-empty open sets and where V is connected, and such that for every holomorphic function f on \Omega there exists a holomorphic function g on V with f = g on U In the 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 \geq 2 this is no longer true, as it follows from Hartogs' lemma.

Equivalent conditions

For a domain \Omega the following conditions are equivalent: Implications are standard results (for, see Oka's lemma). The main difficulty lies in proving, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the Levi problem (after E. E. Levi) and was first solved by Kiyoshi Oka, and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of -problem).

Properties