克勞森函數

克勞森函數是丹麥數學家托馬斯·克勞森最先研究的特殊函數，定義如下：

Clausen function plot

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

克勞森函數的傅立葉級數為

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

基本性質

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

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

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

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

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

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

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

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與伯努力多項式的關係

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

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

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

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

其中：

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

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

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

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

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

與多重對數函數的關係

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

參考文獻

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• Leonard Lewin, (Ed.). Structural Properties of Polylogarithms (1991) American Mathematical Society, Providence, RI. ISBN 0-8218-4532-2
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