翻訳と辞書 Words near each other ・ Glagolitic Alley ・ Glagolitic alphabet ・ Glagolitic Mass ・ Glaichbea ・ Glaignes ・ Glainach-Ferlach Airport ・ Glaine-Montaigut ・ Glaire ・ Glais ・ Glaisdale ・ Glaisdale railway station ・ Glaisdale School ・ Glaisher ・ Glaisher (crater) ・ Glaisher's theorem ・ Glaisher–Kinkelin constant ・ Glaisnock River ・ Glaisnock Viaduct ・ Glaisnock Water ・ Glaister ・ Glaistig ・ Glaive ・ Glaiza de Castro ・ Glaiza Herradura ・ Glak ・ Glam ・ Glam (album) ・ GLAM (band) ・ GLAM (industry sector) ・ Glam (song) Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Glaisher–Kinkelin constant ： ウィキペディア英語版 Glaisher–Kinkelin constant In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted ''A'', is a mathematical constant, related to the K-function and the Barnes G-function. The constant appears in a number of sums and integrals, especially those involving Gamma functions and zeta functions. It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin. Its approximate value is: :$A\backslash approx1.2824271291\backslash dots$   . The Glaisher–Kinkelin constant $A$ can be given by the limit: :$A=\backslash lim\_\; \backslash frac\}$ where $K(n)=\backslash prod\_^\; k^k$ is the K-function. This formula displays a similarity between ''A'' and which is perhaps best illustrated by noting Stirling's formula: :$\backslash sqrt=\backslash lim\_\; \backslash frac\}\}$ which shows that just as is obtained from approximation of the function $\backslash prod\_^\; k$, ''A'' can also be obtained from a similar approximation to the function $\backslash prod\_^\; k^k$. An equivalent definition for ''A'' involving the Barnes G-function, given by $G(n)=\backslash prod\_^k!=\backslash frac$ where $\backslash Gamma(n)$ is the gamma function is: :$A=\backslash lim\_\; \backslash frac\; e^\}$. The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as: :$\backslash zeta^(-1)=\backslash frac-\backslash ln\; A$ :$\backslash sum\_^\backslash infty\; \backslash frac=-\backslash zeta^(2)=\backslash frac\backslash left(A-\backslash gamma-\backslash ln(2\backslash pi)\backslash right)$ where $\backslash gamma$ is the Euler–Mascheroni constant. The latter formula leads directly to the following product found by Glaisher: :$\backslash prod\_^\; k^\}\}=\backslash left(\backslash frac\}\backslash right)^\}$ The following are some integrals that involve this constant: :$\backslash int\_0^\; \backslash ln\backslash Gamma(x)dx=\backslash frac\; \backslash ln\; A+\backslash frac\; \backslash ln\; 2+\backslash frac\; \backslash ln\; \backslash pi$ :$\backslash int\_0^\backslash infty\; \backslash fracdx=\backslash frac\; \backslash zeta^(-1)=\backslash frac-\backslash frac\backslash ln\; A$ A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse. :$\backslash ln\; A=\backslash frac-\backslash frac\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash sum\_^n\; \backslash left(-1\backslash right)^k\; \backslash binom\; \backslash left(k+1\backslash right)^2\; \backslash ln(k+1)$ ==References== * * (Provides a variety of relationships.) * * 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「Glaisher–Kinkelin 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.