翻訳と辞書 Words near each other ・ "O" Is for Outlaw ・ "O"-Jung.Ban.Hap. ・ "Ode-to-Napoleon" hexachord ・ "Oh Yeah!" Live ・ "Our Contemporary" regional art exhibition (Leningrad, 1975) ・ "P" Is for Peril ・ "Pimpernel" Smith ・ "Polish death camp" controversy ・ "Pro knigi" ("About books") ・ "Prosopa" Greek Television Awards ・ "Pussy Cats" Starring the Walkmen ・ "Q" Is for Quarry ・ "R" Is for Ricochet ・ "R" The King (2016 film) ・ "Rags" Ragland ・ ! (album) ・ ! (disambiguation) ・ !! ・ !!! ・ !!! (album) ・ !!Destroy-Oh-Boy!! ・ !Action Pact! ・ !Arriba! La Pachanga ・ !Hero ・ !Hero (album) ・ !Kung language ・ !Oka Tokat ・ !PAUS3 ・ !T.O.O.H.! ・ !Women Art Revolution 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 beta function ： ウィキペディア英語版 beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by :$$ \mathrm(x,y) = \int_0^1t^(1-t)^\,\mathrmt \! for $\backslash textrm(x),\; \backslash textrm(y)\; >\; 0.\backslash ,$ The beta function was studied by Euler and Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital β rather than the similar Latin capital B. == Properties == The beta function is symmetric, meaning that :$$ \Beta(x,y) = \Beta(y,x). \!〔Davis (1972) 6.2.2 p.258〕 When ''x'' and ''y'' are positive integers, it follows from the definition of the gamma function $\backslash Gamma\backslash$ that: :$$ \Beta(x,y)=\dfrac \! It has many other forms, including: :$$ \Beta(x,y)=\dfrac \!〔 :$$ \Beta(x,y) = 2\int_0^(\sin\theta)^(\cos\theta)^\,\mathrm\theta, \qquad \mathrm(x)>0,\ \mathrm(y)>0 \!〔 :$$ \Beta(x,y) = \int_0^\infty\dfrac}\,\mathrmt, \qquad \mathrm(x)>0,\ \mathrm(y)>0 \!〔Davis (1972) 6.2.1 p.258〕 :$$ \Beta(x,y) = \sum_^\infty \dfrac , \! :$$ \Beta(x,y) = \frac \prod_^\infty \left( 1+ \dfrac\right)^, \! The Beta function satisfies several interesting identities, including :$$ \Beta(x,y) = \Beta(x, y+1) + \Beta(x+1, y) \! :$$ \Beta(x+1,y) = \Beta(x, y) \cdot \dfrac \! :$$ \Beta(x,y+1) = \Beta(x, y) \cdot \dfrac \! :$$ \Beta(x,y)\cdot(t \mapsto t_+^) = (t \to t_+^) * (t \to t_+^) \qquad x\ge 1, y\ge 1, \! :$$ \Beta(x,y) \cdot \Beta(x+y,1-y) = \dfrac, \! where $t\; \backslash mapsto\; t\_+^x$ is a truncated power function and the star denotes convolution. The lowermost identity above shows in particular $\backslash Gamma(\backslash tfrac12)\; =\; \backslash sqrt\; \backslash pi$. Some of these identities, e.g. the trigonometric formula, can be applied to deriving the volume of an n-ball in Cartesian coordinates. Euler's integral for the beta function may be converted into an integral over the Pochhammer contour ''C'' as :$\backslash displaystyle\; (1-e^)(1-e^)\backslash Beta(\backslash alpha,\backslash beta)\; =\backslash int\_C\; t^(1-t)^\; \backslash ,\; \backslash mathrmt.$ This Pochhammer contour integral converges for all values of ''α'' and ''β'' and so gives the analytic continuation of the beta function. Just as the gamma function for integers describes factorials, the beta function can define a binomial coefficient after adjusting indices: :$=\; \backslash frac1.$ Moreover, for integer ''n'', $\backslash Beta\backslash ,$ can be factored to give a closed form, an interpolation function for continuous values of ''k'': :$=\; (-1)^n\; n!\; \backslash cfrac.$ The beta function was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano. It also occurs in the theory of the preferential attachment process, a type of stochastic urn process. 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「beta 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.