翻訳と辞書 Words near each other ・ Humbert Fugazy ・ Humbert I of Viennois ・ Humbert I, Count of Savoy ・ Humbert II ・ Humbert II of Viennois ・ Humbert II, Count of Savoy ・ Humbert III, Count of Savoy ・ Humbert La Moto Du Ciel ・ Humbert Lundén ・ Humbert of Maroilles ・ Humbert of Romans ・ Humbert of Silva Candida ・ Humbert polynomials ・ Humbert Pugliese ・ Humbert Roque Versace ・ Humbert series ・ Humbert surface ・ Humbert Tétras ・ Humbert V de Beaujeu ・ Humbert Wolfe ・ Humbert, bastard of Savoy ・ Humbert, Pas-de-Calais ・ Humberto ・ Humberto Acosta-Rosario ・ Humberto Aguilar Coronado ・ Humberto Ak'ab'al ・ Humberto Albiñana ・ Humberto Albornoz ・ Humberto Alonso Morelli ・ Humberto Alonso Razo 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 Humbert series ： ウィキペディア英語版 Humbert series In mathematics, Humbert series are a set of seven hypergeometric series Φ1, Φ2, Φ3, Ψ1, Ψ2, Ξ1, Ξ2 of two variables that generalize Kummer's confluent hypergeometric series 1''F''1 of one variable and the confluent hypergeometric limit function 0''F''1 of one variable. The first of these double series was introduced by . ==Definitions== The Humbert series Φ1 is defined for |''x''| < 1 by the double series: :$$ \Phi_1(a,b,c;x,y) = F_1(a,b,-,c;x,y) = \sum_^\infty \frac \,x^m y^n ~, where the Pochhammer symbol (''q'')''n'' represents the rising factorial: :$(q)\_n\; =\; q\backslash ,(q+1)\; \backslash cdots\; (q+n-1)\; =\; \backslash frac~,$ where the second equality is true for all complex $q$ except $q=0,-1,-2,\backslash ldots$. For other values of ''x'' the function Φ1 can be defined by analytic continuation. The Humbert series Φ1 can also be written as a one-dimensional Euler-type integral: :$$ \Phi_1(a,b,c;x,y) = \frac \int_0^1 t^ (1-t)^ (1-xt)^ e^ \,\mathrmt, \quad \real \,c > \real \,a > 0 ~. This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration. Similarly, the function Φ2 is defined for all ''x'', ''y'' by the series: :$$ \Phi_2(b_1,b_2,c;x,y) = F_1(-,b_1,b_2,c;x,y) = \sum_^\infty \frac \,x^m y^n ~, the function Φ3 for all ''x'', ''y'' by the series: :$$ \Phi_3(b,c;x,y) = \Phi_2(b,-,c;x,y) = F_1(-,b,-,c;x,y) = \sum_^\infty \frac \,x^m y^n ~, the function Ψ1 for |''x''| < 1 by the series: :$$ \Psi_1(a,b,c_1,c_2;x,y) = F_2(a,b,-,c_1,c_2;x,y) = \sum_^\infty \frac \,x^m y^n ~, the function Ψ2 for all ''x'', ''y'' by the series: :$$ \Psi_2(a,c_1,c_2;x,y) = \Psi_1(a,-,c_1,c_2;x,y) = F_2(a,-,-,c_1,c_2;x,y) = F_4(a,-,c_1,c_2;x,y) = \sum_^\infty \frac \,x^m y^n ~, the function Ξ1 for |''x''| < 1 by the series: :$$ \Xi_1(a_1,a_2,b,c;x,y) = F_3(a_1,a_2,b,-,c;x,y) = \sum_^\infty \frac \,x^m y^n ~, and the function Ξ2 for |''x''| < 1 by the series: :$$ \Xi_2(a,b,c;x,y) = \Xi_1(a,-,b,c;x,y) = F_3(a,-,b,-,c;x,y) = \sum_^\infty \frac \,x^m y^n ~. 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「Humbert series」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.