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：

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

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 $

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

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

.



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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* 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

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