Domain of holomorphy
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 formula_1 in the "n"-dimensional complex space formula_2 is called a "domain of holomorphy" if there do not exist non-empty open sets formula_3 and formula_4 where formula_5 is connected, formula_6 and formula_7 such that for every holomorphic function formula_8 on formula_1 there exists a holomorphic function formula_10 on formula_5 with formula_12  on formula_13 In the formula_14 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 formula_15 this is no longer true, as it follows from Hartogs' lemma. Equivalent conditions. For a domain formula_1 the following conditions are equivalent: Implications formula_34 are standard results (for formula_35, see Oka's lemma). The main difficulty lies in proving formula_36, 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 formula_37-problem).