克劳森函数

克劳森函数是丹麦数学家 Template:Tsl最先研究的特殊函数，定义如下：

{Cl}_{2} (φ) = - \int_{0}^{φ} \log | 2 \sin \frac{x}{2} | d x :

克劳森函数的傅立叶级数为

{Cl}_{2} (φ) = \sum_{k = 1}^{\infty} \frac{\sin k φ}{k^{2}} = \sin φ + \frac{\sin 2 φ}{2^{2}} + \frac{\sin 3 φ}{3^{2}} + \frac{\sin 4 φ}{4^{2}} + \dots

基本性质

{Cl}_{2} (m π) = 0, m = 0, \pm 1, \pm 2, \pm 3, \dots

极大值点 : $θ = \frac{π}{3} + 2 m π [m \in ℤ]$

{Cl}_{2} (\frac{π}{3} + 2 m π) = 1.01494160 \dots

极小值点 : $θ = - \frac{π}{3} + 2 m π [m \in ℤ]$

{Cl}_{2} (- \frac{π}{3} + 2 m π) = - 1.01494160 \dots

{Cl}_{2} (θ + 2 m π) = {Cl}_{2} (θ)

{Cl}_{2} (- θ) = - {Cl}_{2} (θ)

B_{2 n - 1} (x) = \frac{2 (- 1)^{n} (2 n - 1)!}{(2 π)^{2 n - 1}} \sum_{k = 1}^{\infty} \frac{\sin 2 π k x}{k^{2 n - 1}}

B_{2 n} (x) = \frac{2 (- 1)^{n - 1} (2 n)!}{(2 π)^{2 n}} \sum_{k = 1}^{\infty} \frac{\cos 2 π k x}{k^{2 n}}

{Sl}_{2 m} (θ) = \frac{(- 1)^{m - 1} (2 π)^{2 m}}{2 (2 m)!} B_{2 m} (\frac{θ}{2 π})

{Sl}_{2 m - 1} (θ) = \frac{(- 1)^{m} (2 π)^{2 m - 1}}{2 (2 m - 1)!} B_{2 m - 1} (\frac{θ}{2 π})

其中：

B_{n} (x) = \sum_{j = 0}^{n} (\binom{n}{j}) B_{j} x^{n - j}

{Sl}_{1} (θ) = \frac{π}{2} - \frac{θ}{2}

{Sl}_{2} (θ) = \frac{π^{2}}{6} - \frac{π θ}{2} + \frac{θ^{2}}{4}

{Sl}_{3} (θ) = \frac{π^{2} θ}{6} - \frac{π θ^{2}}{4} + \frac{θ^{3}}{12}

{Sl}_{4} (θ) = \frac{π^{4}}{90} - \frac{π^{2} θ^{2}}{12} + \frac{π θ^{3}}{12} - \frac{θ^{4}}{48}

$C l 2 : = - (1 / 2 * I) * (p o l y l o g (2, e x p (I * ϕ)) - p o l y l o g (2, e x p (- I * ϕ)))$