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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 Parseval–Gutzmer formula ： ウィキペディア英語版 Parseval–Gutzmer formula In mathematics, the Parseval–Gutzmer formula states that, if ''ƒ'' is an analytic function on a closed disk of radius ''r'' with Taylor series :$f(z)\; =\; \backslash sum^\backslash infty\_\; a\_k\; z^k,$ then for ''z'' = ''re''''iθ'' on the boundary of the disk, :$\backslash int^\_0\; |f(re^)\; |^2\; \backslash ,\; \backslash mathrm\backslash vartheta\; =\; 2\backslash pi\; \backslash sum^\backslash infty\_\; |a\_k|^2r^.$ == Proof == The Cauchy Integral Formula for coefficients states that for the above conditions: :$a\_n\; =\; \backslash frac\; \backslash int^\backslash \; z$ where γ is defined to be the circular path around 0 of radius r. We also have that, for x in the complex plane C, :$\backslash overline\; =\; |x|^2$ We can apply both of these facts to the problem. Using the second fact, :$\backslash int^\_0\; |f(re^)\; |^2\; \backslash ,\; \backslash mathrm\backslash vartheta\; =\; \backslash int^\_0\; \backslash overline\; \backslash ,\; \backslash mathrm\backslash vartheta$ Now, using our Taylor Expansion on the conjugate, :$=\; \backslash int^\_0\; )^k\}\}\; \backslash ,\; \backslash mathrm\backslash vartheta$ Using the uniform convergence of the Taylor Series and the properties of integrals, we can rearrange this to be :$=\; \backslash sum^\backslash infty\_\; \backslash int^\_0\; \backslash frac)(\backslash frac\backslash int^\_0\; \backslash frac\}\; \backslash vartheta$ Now, applying the Cauchy Integral Formula, we get :$=\; \backslash sum^\backslash infty\_\; (\})\; =\; \backslash sum^\backslash infty\_\; \{|a\_k|^2\; r^\{2k\}\}$ 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「Parseval–Gutzmer formula」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.