克劳森函数

克劳森函数是丹麦数学家托马斯·克劳森（英语：Thomas_Clausen_(mathematician)）最先研究的特殊函数，定义如下： Cl 2 ⁡ ( φ ) = − ∫ 0 φ log ⁡ | 2 sin ⁡ x 2 | d x : {\displaystyle \operatorname {Cl} _{2}(\varphi )=-\int _{0}^{\varphi }\log {\Bigg |}2\sin {\frac {x}{2}}{\Bigg |}\,dx:} 克劳森函数的傅立叶级数为 Cl 2 ⁡ ( φ ) = ∑ k = 1 ∞ sin ⁡ k φ k 2 = sin ⁡ φ + sin ⁡ 2 φ 2 2 + sin ⁡ 3 φ 3 2 + sin ⁡ 4 φ 4 2 + ⋯ {\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 } Cl 2 ( m π ) = 0 , m = 0 , ± 1 , ± 2 , ± 3 , ⋯ {\displaystyle {\text{Cl}}_{2}(m\pi )=0,\quad m=0,\,\pm 1,\,\pm 2,\,\pm 3,\,\cdots } 极大值点 : θ = π 3 + 2 m π [ m ∈ Z ] {\displaystyle \theta ={\frac {\pi }{3}}+2m\pi \quad [m\in \mathbb {Z} ]} Cl 2 ( π 3 + 2 m π ) = 1.01494160 ⋯ {\displaystyle {\text{Cl}}_{2}\left({\frac {\pi }{3}}+2m\pi \right)=1.01494160\cdots } 极小值点 : θ = − π 3 + 2 m π [ m ∈ Z ] {\displaystyle \theta =-{\frac {\pi }{3}}+2m\pi \quad [m\in \mathbb {Z} ]} Cl 2 ( − π 3 + 2 m π ) = − 1.01494160 ⋯ {\displaystyle {\text{Cl}}_{2}\left(-{\frac {\pi }{3}}+2m\pi \right)=-1.01494160\cdots } Cl 2 ( θ + 2 m π ) = Cl 2 ( θ ) {\displaystyle {\text{Cl}}_{2}(\theta +2m\pi )={\text{Cl}}_{2}(\theta )} Cl 2 ( − θ ) = − Cl 2 ( θ ) {\displaystyle {\text{Cl}}_{2}(-\theta )=-{\text{Cl}}_{2}(\theta )} . B 2 n − 1 ( x ) = 2 ( − 1 ) n ( 2 n − 1 ) ! ( 2 π ) 2 n − 1 ∑ k = 1 ∞ sin ⁡ 2 π k x k 2 n − 1 {\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}}}} B 2 n ( x ) = 2 ( − 1 ) n − 1 ( 2 n ) ! ( 2 π ) 2 n ∑ k = 1 ∞ cos ⁡ 2 π k x k 2 n {\displaystyle B_{2n}(x)={\frac {2(-1)^{n-1}(2n)!}{(2\pi )^{2n}}}\,\sum _{k=1}^{\infty }{\frac {\cos 2\pi kx}{k^{2n}}}} Sl 2 m ( θ ) = ( − 1 ) m − 1 ( 2 π ) 2 m 2 ( 2 m ) ! B 2 m ( θ 2 π ) {\displaystyle {\text{Sl}}_{2m}(\theta )={\frac {(-1)^{m-1}(2\pi )^{2m}}{2(2m)!}}B_{2m}\left({\frac {\theta }{2\pi }}\right)} Sl 2 m − 1 ( θ ) = ( − 1 ) m ( 2 π ) 2 m − 1 2 ( 2 m − 1 ) ! B 2 m − 1 ( θ 2 π ) {\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： B n ( x ) = ∑ j = 0 n ( n j ) B j x n − j {\displaystyle B_{n}(x)=\sum _{j=0}^{n}{\binom {n}{j}}B_{j}x^{n-j}} Sl 1 ( θ ) = π 2 − θ 2 {\displaystyle {\text{Sl}}_{1}(\theta )={\frac {\pi }{2}}-{\frac {\theta }{2}}} Sl 2 ( θ ) = π 2 6 − π θ 2 + θ 2 4 {\displaystyle {\text{Sl}}_{2}(\theta )={\frac {\pi ^{2}}{6}}-{\frac {\pi \theta }{2}}+{\frac {\theta ^{2}}{4}}} Sl 3 ( θ ) = π 2 θ 6 − π θ 2 4 + θ 3 12 {\displaystyle {\text{Sl}}_{3}(\theta )={\frac {\pi ^{2}\theta }{6}}-{\frac {\pi \theta ^{2}}{4}}+{\frac {\theta ^{3}}{12}}} Sl 4 ( θ ) = π 4 90 − π 2 θ 2 12 + π θ 3 12 − θ 4 48 {\displaystyle {\text{Sl}}_{4}(\theta )={\frac {\pi ^{4}}{90}}-{\frac {\pi ^{2}\theta ^{2}}{12}}+{\frac {\pi \theta ^{3}}{12}}-{\frac {\theta ^{4}}{48}}} C l 2 := − ( 1 / 2 ∗ I ) ∗ ( p o l y l o g ( 2 , e x p ( I ∗ ϕ ) ) − p o l y l o g ( 2 , e x p ( − I ∗ ϕ ) ) ) {\displaystyle Cl2:=-(1/2*I)*(polylog(2,exp(I*\phi ))-polylog(2,exp(-I*\phi )))} 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. （原始内容存档于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]. （原始内容 (PDF)存档于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. （原始内容存档 (PDF)于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.  arXiv:hep-ph/0411100 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]. （原始内容存档 (PDF)于2015-09-24）.