克劳森函数

克劳森函数是丹麦数学家托马斯·克劳森最先研究的特殊函数，定义如下：

:\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

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

极大值点 :\theta = \frac{\pi}{3}+2m\pi \quad[m\in\mathbb{Z}]

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

极小值点 :\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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与伯努利多项式的关系
: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_jx^{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}

与多重对数函数的关系
Cl2 := -(1/2I)(polylog(2, exp(I\phi))-polylog(2, exp(-I\phi)))

参考文献
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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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