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| [[Image:Domain of holomorphy illustration2.png|thumb|right|The sets in the definition.]]
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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 [[analytic continuation|extended]] to a bigger set.
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| Formally, an [[open set]] <math>\Omega</math> in the ''n''-dimensional complex space <math>{\mathbb{C}}^n</math> is called a ''domain of holomorphy'' if there do not exist non-empty open sets <math>U \subset \Omega</math> and <math>V \subset {\mathbb{C}}^n</math> where <math>V</math> is [[connected space|connected]], <math>V \not\subset \Omega</math> and <math>U \subset \Omega \cap V</math> such that for every [[holomorphic]] function <math>f</math> on <math>\Omega</math> there exists a holomorphic function <math>g</math> on <math>V</math> with <math>f = g</math> on <math>U</math>
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| In the <math>n=1</math> case, every open set is a domain of holomorphy: we can define a holomorphic function with zeros [[accumulation point|accumulating]] everywhere on the [[boundary (topology)|boundary]] of the domain, which must then be a natural boundary for a domain of definition of its inverse. For <math>n \geq 2</math> this is no longer true, as it follows from [[Hartogs' lemma]].
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| == Equivalent conditions ==
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| For a domain <math>\Omega</math> the following conditions are equivalent:
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| # <math>\Omega</math> is a domain of holomorphy
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| # <math>\Omega</math> is [[holomorphically convex hull|holomorphically convex]]
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| # <math>\Omega</math> is [[pseudoconvex]]
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| # <math>\Omega</math> is '''Levi convex''' - for every sequence <math>S_{n} \subseteq \Omega</math> of analytic compact surfaces such that <math>S_{n} \rightarrow S, \partial S_{n} \rightarrow \Gamma</math> for some set <math>\Gamma</math> we have <math>S \subseteq \Omega</math> (<math>\partial \Omega</math> cannot be "touched from inside" by a sequence of analytic surfaces)
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| # <math>\Omega</math> has '''local Levi property''' - for every point <math>x \in \partial \Omega</math> there exist a neighbourhood <math>U</math> of <math>x</math> and <math>f</math> holomorphic on <math>U \cap \Omega</math> such that <math>f</math> cannot be extended to any neighbourhood of <math>x</math>
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| Implications <math>1 \Leftrightarrow 2, 3 \Leftrightarrow 4, 1 \Rightarrow 4, 3 \Rightarrow 5</math> are standard results (for <math>1\Rightarrow 3</math>, see [[Oka's lemma]]). The main difficulty lies in proving <math>5 \Rightarrow 1</math>, i.e. constructing a global holomoprhic function which admits no extension from non-extendable functions defined only locally. This is called the [[Levi problem]] and was first solved by [[Kiyoshi Oka]], and then by [[Lars Hörmander]] using methods from functional analysis and partial differential equations (a consequence of [[D-bar problem|<math>\bar{\partial}</math>-problem]]).
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| == Properties ==
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| * if <math>\Omega_1, \dots, \Omega_{n}</math> are domains of holomorphy, then their intersection <math>\Omega = \bigcap_{j=1}^{n} \Omega_j</math> is also a domain of holomorphy
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| * if <math>\Omega_{1} \subseteq \Omega_{2} \subseteq \dots</math> is an ascending sequence of domains of holomorphy, then their union <math>\Omega = \bigcup_{n=1}^{\infty}\Omega_{n}</math> is also a domain of holomorphy (see [[Behnke-Stein theorem]])
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| * product <math>\Omega = \Omega_{1} \times \Omega_{2}</math> of domains of holomorphy <math>\Omega_{1}, \Omega_{2}</math> is a domain of holomorphy
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| * the first [[Cousin problems|Cousin problem]] is always solvable in a domain of holomorphy; this is also true, with additional topological assumptions, for the second [[Cousin problems|Cousin problem]]
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| == References ==
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| * Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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| * Boris Vladimirovich Shabat, ''Introduction to Complex Analysis'', AMS, 1992
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| ==See also==
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| * [[Behnke–Stein theorem]]
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| * [[Levi pseudoconvex]]
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| * [[solution of the Levi problem]]
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| * [[Stein manifold]]
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| {{PlanetMath attribution|id=6026|title=Domain of holomorphy}}
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| [[Category:Several complex variables]]
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