克勞森函數

克勞森函數是丹麥數學家托馬斯·克勞森（英語：Thomas_Clausen_(mathematician)）最先研究的特殊函數，定義如下：

${\displaystyle \operatorname {Cl} _{2}(\varphi )=-\int _{0}^{\varphi }\log {\Bigg |}2\sin {\frac {x}{2}}{\Bigg |}\,dx:}$

Clausen function plot

克勞森函數的傅立葉級數為

${\displaystyle \operatorname {Cl} _{2}(\varphi )=\sum _{k=1}^{\infty }{\frac {\sin k\varphi }{k^{2}}}=\sin \varphi +{\frac {\sin 2\varphi }{2^{2}}}+{\frac {\sin 3\varphi }{3^{2}}}+{\frac {\sin 4\varphi }{4^{2}}}+\,\cdots }$

基本性質

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

極大值點 :${\displaystyle \theta ={\frac {\pi }{3}}+2m\pi \quad [m\in \mathbb {Z} ]}$

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

極小值點 :${\displaystyle \theta =-{\frac {\pi }{3}}+2m\pi \quad [m\in \mathbb {Z} ]}$

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

${\displaystyle {\text{Cl}}_{2}(\theta +2m\pi )={\text{Cl}}_{2}(\theta )}$

${\displaystyle {\text{Cl}}_{2}(-\theta )=-{\text{Cl}}_{2}(\theta )}$

^[1].

與伯努力多項式的關係

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

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

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

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

其中：

${\displaystyle B_{n}(x)=\sum _{j=0}^{n}{\binom {n}{j}}B_{j}x^{n-j}}$

${\displaystyle {\text{Sl}}_{1}(\theta )={\frac {\pi }{2}}-{\frac {\theta }{2}}}$

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

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

${\displaystyle {\text{Sl}}_{4}(\theta )={\frac {\pi ^{4}}{90}}-{\frac {\pi ^{2}\theta ^{2}}{12}}+{\frac {\pi \theta ^{3}}{12}}-{\frac {\theta ^{4}}{48}}}$

與多重對數函數的關係

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