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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 Ramanujan's master theorem ： ウィキペディア英語版 Ramanujan's master theorem In mathematics, Ramanujan's master theorem (named after mathematician Srinivasa Ramanujan〔B. Berndt. Ramanujan’s Notebooks, Part I. Springer-Verlag, New York, 1985.〕) is a technique that provides an analytic expression for the Mellin transform of a function. The result is stated as follows: Assume function $f(x)\; \backslash !$ has an expansion of the form : $f(x)=\backslash sum\_^\backslash infty\; \backslash frac(-x)^k\; \backslash !$ then Mellin transform of $f(x)\; \backslash !$ is given by : $\backslash int\_0^\backslash infty\; x^\; f(x)\; \backslash ,\; dx\; =\; \backslash Gamma(s)\backslash phi(-s)\; \backslash !$ where $\backslash Gamma(s)\; \backslash !$ is the Gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series. Multidimensional versions of this theorem also appear in quantum physics (through Feynman diagrams).〔(A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams by Iv´an Gonz´alez, V. H. Moll and Iv´an Schmidt )〕 A similar result was also obtained by J. W. L. Glaisher.〔J. W. L. Glaisher. A new formula in definite integrals. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 48(315):53–55, Jul 1874.〕 == Alternative formalism == An alternative formulation of Ramanujan's master theorem is as follows: : $\backslash int\_0^\backslash infty\; x^\; ()\; \backslash ,\; dx\; =\; \backslash frac\backslash lambda(-s)$ which gets converted to original form after substituting $\backslash lambda(n)\; =\; \backslash frac\; \backslash !$ and using functional equation for Gamma function. The integral above is convergent for . 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「Ramanujan's master 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.