Clausen function plot 克勞森函數 是丹麥 數學家 托馬斯·克勞森 特殊函數 ，定義如下：
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{\displaystyle \operatorname {Cl} _{2}(\varphi )=-\int _{0}^{\varphi }\log {\Bigg |}2\sin {\frac {x}{2}}{\Bigg |}\,dx:}
克勞森函數的傅立葉級數 為
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{\displaystyle \operatorname {Cl} _{2}(\varphi )=\sum _{k=1}^{\infty }{\frac {\sin k\varphi }{k^{2}}}=\sin \varphi +{\frac {\sin 2\varphi }{2^{2}}}+{\frac {\sin 3\varphi }{3^{2}}}+{\frac {\sin 4\varphi }{4^{2}}}+\,\cdots }
基本性質 [ 編輯 ]
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{\displaystyle {\text{Cl}}_{2}(m\pi )=0,\quad m=0,\,\pm 1,\,\pm 2,\,\pm 3,\,\cdots }
極大值點 :
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{\displaystyle \theta ={\frac {\pi }{3}}+2m\pi \quad [m\in \mathbb {Z} ]}
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{\displaystyle {\text{Cl}}_{2}\left({\frac {\pi }{3}}+2m\pi \right)=1.01494160\cdots }
極小值點 :
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{\displaystyle \theta =-{\frac {\pi }{3}}+2m\pi \quad [m\in \mathbb {Z} ]}
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{\displaystyle {\text{Cl}}_{2}\left(-{\frac {\pi }{3}}+2m\pi \right)=-1.01494160\cdots }
Cl
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{\displaystyle {\text{Cl}}_{2}(\theta +2m\pi )={\text{Cl}}_{2}(\theta )}
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{\displaystyle {\text{Cl}}_{2}(-\theta )=-{\text{Cl}}_{2}(\theta )}
[1]
與伯努力多項式的關係 [ 編輯 ]
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{\displaystyle B_{2n-1}(x)={\frac {2(-1)^{n}(2n-1)!}{(2\pi )^{2n-1}}}\,\sum _{k=1}^{\infty }{\frac {\sin 2\pi kx}{k^{2n-1}}}}
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{\displaystyle B_{2n}(x)={\frac {2(-1)^{n-1}(2n)!}{(2\pi )^{2n}}}\,\sum _{k=1}^{\infty }{\frac {\cos 2\pi kx}{k^{2n}}}}
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{\displaystyle {\text{Sl}}_{2m}(\theta )={\frac {(-1)^{m-1}(2\pi )^{2m}}{2(2m)!}}B_{2m}\left({\frac {\theta }{2\pi }}\right)}
Sl
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{\displaystyle {\text{Sl}}_{2m-1}(\theta )={\frac {(-1)^{m}(2\pi )^{2m-1}}{2(2m-1)!}}B_{2m-1}\left({\frac {\theta }{2\pi }}\right)}
其中：
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{\displaystyle B_{n}(x)=\sum _{j=0}^{n}{\binom {n}{j}}B_{j}x^{n-j}}
Sl
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{\displaystyle {\text{Sl}}_{1}(\theta )={\frac {\pi }{2}}-{\frac {\theta }{2}}}
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{\displaystyle {\text{Sl}}_{2}(\theta )={\frac {\pi ^{2}}{6}}-{\frac {\pi \theta }{2}}+{\frac {\theta ^{2}}{4}}}
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{\displaystyle {\text{Sl}}_{3}(\theta )={\frac {\pi ^{2}\theta }{6}}-{\frac {\pi \theta ^{2}}{4}}+{\frac {\theta ^{3}}{12}}}
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{\displaystyle {\text{Sl}}_{4}(\theta )={\frac {\pi ^{4}}{90}}-{\frac {\pi ^{2}\theta ^{2}}{12}}+{\frac {\pi \theta ^{3}}{12}}-{\frac {\theta ^{4}}{48}}}
與多重對數函數的關係 [ 編輯 ]
C
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{\displaystyle Cl2:=-(1/2*I)*(polylog(2,exp(I*\phi ))-polylog(2,exp(-I*\phi )))}
參考文獻 [ 編輯 ]
^ Lu and Perez, 1992,
Adamchik, Viktor. S. Contributions to the Theory of the Barnes Function. arXiv:math/0308086v1 Clausen, Thomas. Über die Function sin φ + (1/22 ) sin 2φ + (1/32 ) sin 3φ + etc. . Journal für die reine und angewandte Mathematik . 1832, 8 : 298–300 [2015-03-07 ] . ISSN 0075-4102 存檔 於2013-10-04）. Wood, Van E. Efficient calculation of Clausen's integral . Math. Comp. 1968, 22 (104): 883–884. MR 0239733 doi:10.1090/S0025-5718-1968-0239733-9 Leonard Lewin, (Ed.). Structural Properties of Polylogarithms (1991) American Mathematical Society, Providence, RI. ISBN 0-8218-4532-2
Kölbig, Kurt Siegfried. Chebyshev coefficients for the Clausen function Cl2 (x). J. Comput. Appl. Math. 1995, 64 (3): 295–297. MR 1365432 doi:10.1016/0377-0427(95)00150-6 Borwein, Jonathan M.; Straub, Armin. Relations for Nielsen Polylogarithms (PDF) . [2015-03-07 ] . （原始內容 (PDF) 存檔於2013-12-12）. Borwein, Jonathan M.; Bradley, David M.; Crandall, Richard E. Computational Strategies for the Riemann Zeta Function (PDF) . J. Comp. App. Math. 2000, 121 : 247–296 [2015-03-07 ] . MR 1780051 doi:10.1016/s0377-0427(00)00336-8 存檔 (PDF) 於2006-09-25）. Kalmykov, Mikahil Yu.; Sheplyakov, A. LSJK - a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine integral. Comput. Phys. Comm. 2005, 172 : 45–59. doi:10.1016/j.cpc.2005.04.013 Mathar, R. J. A C99 implementation of the Clausen sums. arXiv:1309.7504 Lu, Hung Jung; Perez, Christopher A. Massless one-loop scalar three-point integral and associated Clausen, Glaisher, and L-functions (PDF) . 1992 [2015-03-07 ] . （原始內容存檔 (PDF) 於2015-09-24）. 
![{\displaystyle \theta ={\frac {\pi }{3}}+2m\pi \quad [m\in \mathbb {Z} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/301caa56535304b0db827b753a2cd412f5665c12)
![{\displaystyle \theta =-{\frac {\pi }{3}}+2m\pi \quad [m\in \mathbb {Z} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fae98e863733e316c672bdd9098be49e228b375)
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