克劳森函数

Clausen function plot

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：

${\displaystyle {\mathrm{Cl}}_2(\phi )=-{\int }_0^{\phi }\log |2\sin \frac{x}{2}|dx:}$

基本性质

${\displaystyle {\text{Cl}}_2(m\pi )=0,m=0,\pm 1,\pm 2,\pm 3,\cdots }$

${\displaystyle {\text{Cl}}_2\left(\frac{\pi }{3}+2m\pi \right)=1.01494160\cdots }$

${\displaystyle {\text{Cl}}_2\left(-\frac{\pi }{3}+2m\pi \right)=-1.01494160\cdots }$

${\displaystyle {\text{Cl}}_2(-\theta )=-{\text{Cl}}_2(\theta )}$

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

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

${\displaystyle {\text{Sl}}_1(\theta )=\frac{\pi }{2}-\frac{\theta }{2}}$

${\displaystyle {\text{Sl}}_3(\theta )=\frac{{\pi }^2\theta }{6}-\frac{\pi {\theta }^2}{4}+\frac{{\theta }^3}{12}}$

与多重对数函数的关系

${\displaystyle Cl2:=-(1/2\ast I)\ast (polylog(2,exp(I\ast \varphi ))-polylog(2,exp(-I\ast \varphi )))}$