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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 Clausen function ： ウィキペディア英語版 Clausen function In mathematics, the Clausen function - introduced by - is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other special functions. It is intimately connected with the Polylogarithm, Inverse tangent integral, Polygamma function, Riemann Zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 - often referred to as ''the'' Clausen function, despite being but one of a class of many - is given by the integral: :$\backslash operatorname\_2(\backslash varphi)=-\backslash int\_0^\; \backslash log\backslash Bigg|2\backslash sin\backslash frac\; \backslash Bigg|\backslash ,\; dx:$ In the range : the Sine function inside the absolute value sign remains strictly positive, so the absolute value signs may be omitted. The Clausen function also has the Fourier series representation: :$\backslash operatorname\_2(\backslash varphi)=\backslash sum\_^\backslash frac\; =\; \backslash sin\backslash varphi\; +\backslash frac+\backslash frac+\backslash frac+\; \backslash ,\; \backslash cdots$ The Clausen functions - as a class of functions - feature extensively in many areas of modern mathematical research, particularly in relation to the evaluation of many classes of logarithmic and Polylogarithmic integrals, both definite and indefinite. They also have numerous applications with regard to the summation of Hypergeometric series, summations involving the inverse of the central binomial coefficient, sums of the Polygamma function, and Dirichlet L-series. ==Basic properties== The Clausen function (of order 2) has simple zeros at all (integer) multiples of :$\backslash pi,\; \backslash ,$ since if :$k\backslash in\; \backslash mathbb\; \backslash ,$ is an integer, :$\backslash sin\; k\backslash pi=0$ :$\backslash text\_2(m\backslash pi)\; =0,\; \backslash quad\; m=\; 0,\backslash ,\; \backslash pm\; 1,\backslash ,\; \backslash pm\; 2,\backslash ,\; \backslash pm\; 3,\backslash ,\; \backslash cdots$ It has maxima at :$\backslash theta\; =\; \backslash frac+2m\backslash pi\; \backslash quad()$ :$\backslash text\_2\backslash left(\backslash frac+2m\backslash pi\; \backslash right)\; =1.01494160\; \backslash cdots$ and minima at :$\backslash theta\; =\; -\backslash frac+2m\backslash pi\; \backslash quad()$ :$\backslash text\_2\backslash left(-\backslash frac+2m\backslash pi\; \backslash right)\; =-1.01494160\; \backslash cdots$ The following properties are immediate consequences of the series definition: :$\backslash text\_2(\backslash theta+2m\backslash pi)\; =\; \backslash text\_2(\backslash theta)$ :$\backslash text\_2(-\backslash theta)\; =\; -\backslash text\_2(\backslash theta)$ (Ref: See Lu and Perez, 1992, below for these results - although no proofs are given). 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「Clausen 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.