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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 Mittag-Leffler function ： ウィキペディア英語版 Mittag-Leffler function In mathematics, the Mittag-Leffler function ''E''''α'',''β'' is a special function, a complex function which depends on two complex parameters ''α'' and ''β''. It may be defined by the following series when the real part of α is strictly positive: :$E\_\; (z)\; =\; \backslash sum\_^\backslash infty\; \backslash frac.$ In the case ''α'' and ''β'' are real and positive, the series converges for all values of the argument ''z'', so the Mittag-Leffler function is an entire function. This function is named after Gösta Mittag-Leffler. This class of functions are important in the theory of the fractional calculus. For ''α'' > 0, the Mittag-Leffler function ''E''''α'',1 is an entire function of order 1/''α'', and is in some sense the simplest entire function of its order. ==Special cases== For $\backslash alpha=0,1/2,1,2$ we find The sum of a geometric progression: :$E\_(z)\; =\; \backslash sum\_^\backslash infty\; z^k\; =\; \backslash frac.$ Exponential function: :$E\_(z)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash exp(z).$ Error function: :$E\_(z)\; =\; \backslash exp(z^2)\backslash operatorname(-z).$ Hyperbolic cosine: :$E\_(z)\; =\; \backslash cosh(\backslash sqrt).$ For $\backslash alpha=0,1,2$, the integral :$\backslash int\_0^E\_(-s^2)s$ gives, respectively :$\backslash arctan(z)$, :$\backslash tfrac\backslash operatorname(z)$, :$\backslash sin(z)$. 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「Mittag-Leffler function」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.