翻訳と辞書 Words near each other ・ Stieg Persson ・ Stieg Trenter ・ Stiege ・ Stiegel-Coleman House ・ Stieger Lake ・ Stiegler ・ Stieglitz ・ Stieglitz (surname) ・ Stieglitz Lecture ・ Stieglitz Museum of Applied Arts ・ Stieglitz rearrangement ・ Stieler ・ Stielers Handatlas ・ Stielgranate 41 ・ Stieller v Porirua City Council ・ Stieltjes constants ・ Stieltjes matrix ・ Stieltjes moment problem ・ Stieltjes polynomials ・ Stieltjes transformation ・ Stieltjeskanaal ・ Stieltjeskerk ・ Stieltjes–Wigert polynomials ・ Stien Kaiser ・ Stieng ・ Stieng language ・ Stieng people ・ Stiens ・ Stienta ・ Stientje van Veldhoven 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 Stieltjes constants ： ウィキペディア英語版 Stieltjes constants In mathematics, the Stieltjes constants are the numbers $\backslash gamma\_k$ that occur in the Laurent series expansion of the Riemann zeta function: :$\backslash zeta(s)=\backslash frac+\backslash sum\_^\backslash infty\; \backslash frac\; \backslash gamma\_n\; \backslash ;\; (s-1)^n.$ The zero'th constant $\backslash gamma\_0\; =\; \backslash gamma\; =\; 0.577\backslash dots$ is known as the Euler–Mascheroni constant. ==Representations== The Stieltjes constants are given by the limit : $\backslash gamma\_n\; =\; \backslash lim\_$ - \frac\right\}}. (In the case ''n'' = 0, the first summand requires evaluation of 00, which is taken to be 1.) Cauchy's differentiation formula leads to the integral representation :$\backslash gamma\_n\; =\; \backslash frac\; \backslash int\_0^\; e^\; \backslash zeta\backslash left(e^+1\backslash right)\; dx.$ Various representations in terms of integrals and infinite series are given in works of Jensen, Franel, Hermite, Hardy, Ramanujan, Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine and some other authors.〔Marc-Antoine Coppo. ''Nouvelles expressions des constantes de Stieltjes''. Expositiones Mathematicae, vol. 17, pp. 349-358, 1999.〕〔Mark W. Coffey. ''Series representations for the Stieltjes constants'', (arXiv:0905.1111 )〕〔(Mark W. Coffey. ''Addison-type series representation for the Stieltjes constants''. J. Number Theory, vol. 130, pp. 2049-2064, 2010. )〕〔Junesang Choi. ''Certain integral representations of Stieltjes constants'', Journal of Inequalities and Applications, 2013:532, pp. 1-10〕〔〔 In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that : $$ \gamma_n \,=\,\frac\delta_+\,\frac\!\int\limits_0^\infty \! \frac \left\ - \frac \right\}\,, \qquad\quad n=0, 1, 2,\ldots where δ''n,k'' is the Kronecker symbol (Kronecker delta).〔〔 Among other formulae, we find : $$ \gamma_n \,=\,-\frac\! \int\limits_^ \frac\pm ix\big)}\, dx \qquad\qquad\qquad\qquad\qquad\qquad n=0, 1, 2,\ldots : $$ \begin \displaystyle \gamma_1 =-\left(-\frac\right )\ln2+\,i\!\int\limits_0^\infty \! \frac \left\ - \frac \right\}\, \\() \displaystyle \gamma_1 = -\gamma^2 - \int\limits_0^\infty \left() e^\ln x \, dx \end see.〔〔〔(Math StackExchange: A couple of definite integrals related to Stieltjes constants )〕 As concerns series representations, a famous series implying an integer part of a logarithm was given by Hardy in 1912〔G. H. Hardy. ''Note on Dr. Vacca's series for γ'', Q. J. Pure Appl. Math. 43, pp. 215–216, 2012.〕 : $$ \gamma_1\,=\, \frac\sum_^\infty \frac \,\lfloor \log_2\rfloor\cdot \big(2\log_2 - \lfloor \log_2\rfloor\big) Israilov〔M. I. Israilov. ''On the Laurent decomposition of Riemann's zeta function (Russian )''. Trudy Mat. Inst. Akad. Nauk. SSSR, vol. 158, pp. 98-103, 1981.〕 gave semi-convergent series in terms of Bernoulli numbers $B\_$ : $$ \gamma_m\,=\,\sum_^n \frac - \frac - \frac - \sum_^ \frac\left()^_ \,,\qquad 0<\theta<1 Connon,〔Donal F. Connon ''Some applications of the Stieltjes constants'', (arXiv:0901.2083 )〕 Blagouchine〔 and Coppo〔 gave several series with the binomial coefficients : $$ \begin \displaystyle \gamma_m\,=\,-\frac\sum_^\infty\frac\sum_^n (-1)^k \binom\ln^(k+1) \\() \displaystyle \gamma_m\,=\,-\frac\sum_^\infty\frac\sum_^ (-1)^k \binom\frac \\() \displaystyle \gamma_m\,=\sum_^\infty\big| G_\big|\sum_^ (-1)^k \binom\frac \end where ''G''''n'' are Gregory's coefficients, also known as (reciprocal logarithmic numbers ) (''G''1=+1/2, ''G''2=−1/12, ''G''3=+1/24, ''G''4=−19/720,... ). Oloa and Tauraso〔(Math StackExchange: A closed form for the series ... )〕 showed that series with harmonic numbers may lead to Stieltjes constants : $$ \begin \displaystyle \sum_^\infty \frac \,=\, \,-\gamma_1 -\frac\gamma^2+\frac\pi^2 \\() \displaystyle \sum_^\infty \frac \,=\, \,-\gamma_2 -2\gamma\gamma_1 -\frac\gamma^3+\frac\zeta(3) \end Blagouchine〔 obtained slowly-convergent series involving unsigned Stirling numbers of the first kind as well as semi-convergent series with rational terms only : $$ \gamma_m\, =\,\frac\delta_+(-1)^ m!\cdot\!\sum_^\frac\,} \,+\, \theta\cdot\frac\right )\cdot B_\,},\qquad 0<\theta<1, where ''m''=0,1,2,... In particular, series for the first Stieltjes constant has a surprisingly simple form : $$ \gamma_1\, =\,-\frac\sum_^\frac\cdot H_\,} +\, \theta\cdot\frac\cdot H_\,\,},\qquad 0<\theta<1, where ''H''_''n'' is the ''n''th harmonic number.〔 More complicated series for Stieltjes constants are given in works of Lehmer, Liang, Todd, Lavrik, Israilov, Stankus, Keiper, Nan-You, Williams, Coffey.〔〔〔 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「Stieltjes constants」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.