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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 Catalan's constant ： ウィキペディア英語版 Catalan's constant In mathematics, Catalan's constant ''G'', which occasionally appears in estimates in combinatorics, is defined by :$G\; =\; \backslash beta(2)\; =\; \backslash sum\_^\; \backslash frac\; =\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots\; \backslash !$ where ''β'' is the Dirichlet beta function. Its numerical value () is approximately :''G'' = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 … It is not known whether ''G'' is irrational, let alone transcendental. Catalan's constant was named after Eugène Charles Catalan. ==Integral identities== Some identities include :$G\; =\; \backslash int\_0^1\; \backslash int\_0^1\; \backslash frac\; \backslash ,dx\backslash ,\; dy\; \backslash !$ :$G\; =\; -\backslash int\_0^1\; \backslash frac\; \backslash ,dt\; \backslash !$ :$G\; =\; \backslash int\_^\; \backslash frac\; \backslash ;dt\; \backslash !$ :$G\; =\; \backslash frac\; \backslash int\_^\; \backslash frac\; \backslash ;dt\; \backslash !$ :$G\; =\; \backslash int\_0^\; \backslash ln\; (\; \backslash cot(t)\; )\; \backslash ,dt\; \backslash !$ :$G\; =\; \backslash int\_0^\backslash infty\; \backslash arctan\; (e^)\; \backslash ,dt\; \backslash !$ :$G\; =\; \backslash int\_1^\backslash infty\; \backslash frac\; \backslash ,dt\; \backslash !$ *If K(''t'') is a complete elliptic integral of the first kind, then :$G\; =\; \backslash frac\; \backslash int\_0^1\; \backslash mathrm(t)\backslash ,dt\; \backslash !$ *With Gamma function $\backslash Gamma(x+1)\; =\; x!$ :$G$ = \frac \int_0^1 \Gamma(1+\tfrac)\Gamma(1-\tfrac)\,dx = \frac \int_0^\tfrac12\Gamma(1+y)\Gamma(1-y)\,dy *The integral :$G\; =\; \backslash text\_2(1)=\backslash int\_0^1\; \backslash frac\backslash ,dt.\; \backslash !$ :is a known special function, called the Inverse tangent integral, and was extensively studied by Ramanujan. 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「Catalan's constant」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.