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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 Morera's theorem ： ウィキペディア英語版 Morera's theorem In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic. Morera's theorem states that a continuous, complex-valued function ''ƒ'' defined on a connected open set ''D'' in the complex plane that satisfies :$\backslash oint\_\backslash gamma\; f(z)\backslash ,dz\; =\; 0$ for every closed piecewise ''C''1 curve $\backslash gamma$ in ''D'' must be holomorphic on ''D''. The assumption of Morera's theorem is equivalent to that ''ƒ'' has an antiderivative on ''D''. The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions. The converse does hold e.g. if the domain is simply connected; this is Cauchy's integral theorem, stating that the line integral of a holomorphic function along a closed curve is zero. ==Proof== There is a relatively elementary proof of the theorem. One constructs an anti-derivative for ''ƒ'' explicitly. Without loss of generality, it can be assumed that ''D'' is connected. Fix a point ''z''0 in ''D'', and for any $z\backslash in\; D$, let $\backslash gamma:\; ()\backslash to\; D$ be a piecewise ''C''1 curve such that $\backslash gamma(0)=z\_0$ and $\backslash gamma(1)=z$. Then define the function ''F'' to be :$F(z)\; =\; \backslash int\_\backslash gamma\; f(\backslash zeta)\backslash ,d\backslash zeta.\backslash ,$ To see that the function is well-defined, suppose $\backslash tau:\; ()\backslash to\; D$ is another piecewise ''C''1 curve such that $\backslash tau(0)=z\_0$ and $\backslash tau(1)=z$. The curve $\backslash gamma\; \backslash tau^$ (i.e. the curve combining $\backslash gamma$ with $\backslash tau$ in reverse) is a closed piecewise ''C''1 curve in ''D''. Then, :$\backslash int\_\; f(\backslash zeta)\backslash ,d\backslash zeta\backslash ,\; +\; \backslash int\_\}\; f(\backslash zeta)\backslash ,d\backslash zeta\backslash ,=0$ And it follows that :$\backslash int\_\; f(\backslash zeta)\backslash ,d\backslash zeta\backslash ,\; =\; \backslash int\_\backslash tau\; f(\backslash zeta)\backslash ,d\backslash zeta.\backslash ,$ Then using the continuity of ''ƒ'' to estimate difference quotients, we get that ''F''′(''z'') = ''ƒ''(''z''). Note that we can apply neither the Fundamental theorem of Calculus nor the mean value theorem since they are only true of real-valued functions. Since ''f'' is the derivative of the holomorphic function ''F'', it is holomorphic. The fact that derivatives of holomorphic functions are holomorphic can be proved by using the fact that holomorphic functions are analytic, i.e. can be represented by a convergent power series, and the fact that power series may be differentiated term by term. This completes the proof. 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「Morera's 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.