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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Domain of holomorphy ： ウィキペディア英語版 Domain of holomorphy 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 $\backslash Omega$ in the ''n''-dimensional complex space $\}^n$ where $V$ is connected, $V\; \backslash not\backslash subset\; \backslash Omega$ and $U\; \backslash subset\; \backslash Omega\; \backslash cap\; V$ such that for every holomorphic function $f$ on $\backslash Omega$ 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 inverse. For $n\; \backslash geq\; 2$ this is no longer true, as it follows from Hartogs' lemma. == Equivalent conditions == For a domain $\backslash Omega$ the following conditions are equivalent: # $\backslash Omega$ is a domain of holomorphy # $\backslash Omega$ is holomorphically convex # $\backslash Omega$ is pseudoconvex # $\backslash Omega$ is Levi convex - for every sequence $S\_\; \backslash subseteq\; \backslash Omega$ of analytic compact surfaces such that $S\_\; \backslash rightarrow\; S,\; \backslash partial\; S\_\; \backslash rightarrow\; \backslash Gamma$ for some set $\backslash Gamma$ we have $S\; \backslash subseteq\; \backslash Omega$ ($\backslash partial\; \backslash Omega$ cannot be "touched from inside" by a sequence of analytic surfaces) # $\backslash Omega$ has local Levi property - for every point $x\; \backslash in\; \backslash partial\; \backslash Omega$ there exist a neighbourhood $U$ of $x$ and $f$ holomorphic on $U\; \backslash cap\; \backslash Omega$ such that $f$ cannot be extended to any neighbourhood of $x$ Implications $1\; \backslash Leftrightarrow\; 2,\; 3\; \backslash Leftrightarrow\; 4,\; 1\; \backslash Rightarrow\; 4,\; 3\; \backslash Rightarrow\; 5$ are standard results (for $1\backslash Rightarrow\; 3$, see Oka's lemma). The main difficulty lies in proving $5\; \backslash Rightarrow\; 1$, 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 $\backslash bar$-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 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.