Clausen function plot
**克劳森函数**是丹麦数学家托马斯·克劳森（英语：）最先研究的特殊函数，定义如下：

{\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

{\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 }

基本性质
----

{\displaystyle {\text{Cl}}\_{2}(m\pi )=0,\quad m=0,\,\pm 1,\,\pm 2,\,\pm 3,\,\cdots }
极大值点 :{\displaystyle \theta ={\frac {\pi }{3}}+2m\pi \quad [m\in \mathbb {Z} ]}

{\displaystyle {\text{Cl}}\_{2}\left({\frac {\pi }{3}}+2m\pi \right)=1.01494160\cdots }
极小值点 :{\displaystyle \theta =-{\frac {\pi }{3}}+2m\pi \quad [m\in \mathbb {Z} ]}

{\displaystyle {\text{Cl}}\_{2}\left(-{\frac {\pi }{3}}+2m\pi \right)=-1.01494160\cdots }
{\displaystyle {\text{Cl}}\_{2}(\theta +2m\pi )={\text{Cl}}\_{2}(\theta )}
{\displaystyle {\text{Cl}}\_{2}(-\theta )=-{\text{Cl}}\_{2}(\theta )}
.

与伯努力多项式的关系
----------

{\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}}}}
{\displaystyle B\_{2n}(x)={\frac {2(-1)^{n-1}(2n)!}{(2\pi )^{2n}}}\,\sum \_{k=1}^{\infty }{\frac {\cos 2\pi kx}{k^{2n}}}}

  

{\displaystyle {\text{Sl}}\_{2m}(\theta )={\frac {(-1)^{m-1}(2\pi )^{2m}}{2(2m)!}}B\_{2m}\left({\frac {\theta }{2\pi }}\right)}
{\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：

{\displaystyle B\_{n}(x)=\sum \_{j=0}^{n}{\binom {n}{j}}B\_{j}x^{n-j}}
{\displaystyle {\text{Sl}}\_{1}(\theta )={\frac {\pi }{2}}-{\frac {\theta }{2}}}
{\displaystyle {\text{Sl}}\_{2}(\theta )={\frac {\pi ^{2}}{6}}-{\frac {\pi \theta }{2}}+{\frac {\theta ^{2}}{4}}}
{\displaystyle {\text{Sl}}\_{3}(\theta )={\frac {\pi ^{2}\theta }{6}}-{\frac {\pi \theta ^{2}}{4}}+{\frac {\theta ^{3}}{12}}}
{\displaystyle {\text{Sl}}\_{4}(\theta )={\frac {\pi ^{4}}{90}}-{\frac {\pi ^{2}\theta ^{2}}{12}}+{\frac {\pi \theta ^{3}}{12}}-{\frac {\theta ^{4}}{48}}}

与多重对数函数的关系
----------

{\displaystyle Cl2:=-(1/2\*I)\*(polylog(2,exp(I\*\phi ))-polylog(2,exp(-I\*\phi )))}

参考文献
----

1. ↑  Lu and Perez, 1992,

* Adamchik, Viktor. S.. arXiv:math/0308086v1 可免费查阅.
* Clausen, Thomas.. 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.. 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.. J. Comput. Appl. Math. 1995, **64** (3): 295–297. MR 1365432. doi:10.1016/0377-0427(95)00150-6.
* Borwein, Jonathan M.; Straub, Armin.  (PDF).  [2015-03-07]. （原始内容 (PDF)存档于2013-12-12）.
* Borwein, Jonathan M.; Bradley, David M.; Crandall, Richard E.  (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.. Comput. Phys. Comm. 2005, **172**: 45–59. doi:10.1016/j.cpc.2005.04.013.  arXiv:hep-ph/0411100
* Mathar, R. J.. arXiv:1309.7504 可免费查阅.
* Lu, Hung Jung; Perez, Christopher A.  (PDF). 1992  [2015-03-07]. （原始内容存档 (PDF)于2015-09-24）.