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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 Hartogs' theorem ： ウィキペディア英語版 Hartogs' theorem In mathematics, Hartogs' theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if $F:\}$ is a function which is analytic in each variable ''z''''i'', 1 ≤ ''i'' ≤ ''n'', while the other variables are held constant, then ''F'' is a continuous function. A corollary of this is that ''F'' is then in fact an analytic function in the ''n''-variable sense (i.e. that locally it has a Taylor expansion). Therefore 'separate analyticity' and 'analyticity' are coincident notions, in the theory of several complex variables. Starting with the extra hypothesis that the function is continuous (or bounded), the theorem is much easier to prove and in this form is known as Osgood's lemma. Note that there is no analogue of this theorem for real variables. If we assume that a function $f\; \backslash colon\; \}$ is differentiable (or even analytic) in each variable separately, it is not true that $f$ will necessarily be continuous. A counterexample in two dimensions is given by :$f(x,y)\; =\; \backslash frac.$ If in addition we define $f(0,0)=0$, this function has well-defined partial derivatives in $x$ and $y$ at the origin, but it is not continuous at origin. (Indeed, the limits along the lines $x=y$ and $x=-y$ are not equal, so there is no way to extend the definition of $f$ to include the origin and have the function be continuous there.) == References == * Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992. 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「Hartogs' theorem」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.