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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 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. == Equivalent conditions == For a domain the following conditions are equivalent: # is a domain of holomorphy # is holomorphically convex # is pseudoconvex # is Levi convex - for every sequence of analytic compact surfaces such that for some set we have ( cannot be "touched from inside" by a sequence of analytic surfaces) # has local Levi property - for every point there exist a neighbourhood of and holomorphic on such that cannot be extended to any neighbourhood of 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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「domain of holomorphy」の詳細全文を読む スポンサード リンク
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