Schlömilch's Series

Schlömilch's Series is a Fourier series type expansion of twice continuously differentiable function in the interval [math]\displaystyle{ (0,\pi) }[/math] in terms of the Bessel function of the first kind, named after the German mathematician Oscar Schlömilch, who derived the series in 1857. The real-valued function [math]\displaystyle{ f(x) }[/math] has the following expansion,

[math]\displaystyle{ f(x) = a_0 + \sum_{n=1}^\infty a_n J_0(nx) }[/math]

where

[math]\displaystyle{ \begin{align} a_0 &= f(0) + \frac{1}{\pi} \int_0^\pi \int_0^{\pi/2} u f'(u\sin\theta)\ d\theta\ du, \\ a_n &= \frac{2}{\pi} \int_0^\pi \int_0^{\pi/2} u\cos nu \ f'(u\sin\theta)\ d\theta\ du. \end{align} }[/math]

Examples and Properties

Some examples and properties of Schlömilch's Series are following

• Null functions in the interval [math]\displaystyle{ (0,\pi) }[/math] can be expressed by Schlömilch's Series, [math]\displaystyle{ 0 = \frac{1}{2}+\sum_{n=1}^\infty (-1)^n J_0(nx) }[/math], which cannot be obtained by Fourier Series.
• [math]\displaystyle{ x = \frac{\pi^2}{4}-2\sum_{n=1,3,...}^\infty \frac{J_0(nx)}{n^2}, \quad 0\lt x\lt \pi. }[/math]
• [math]\displaystyle{ x^2 = \frac{2\pi^2}{3} + 8 \sum_{n=1}^\infty \frac{(-1)^n}{n^2}J_0(nx), \quad -\pi\lt x\lt \pi. }[/math]
• [math]\displaystyle{ \frac{1}{x} + \sum_{m=1}^k\frac{2}{\sqrt{x^2-4m^2\pi^2}} = \frac{1}{2} + \sum_{n=1}^\infty J_0(nx), \quad 2k\pi\lt x\lt 2(k+1)\pi. }[/math]
• If [math]\displaystyle{ (r,z) }[/math] are the cylindrical polar coordinates, then the series [math]\displaystyle{ 1+\sum_{n=1}^\infty e^{-nz}J_0(nr) }[/math] is a solution of Laplace equation for [math]\displaystyle{ z\gt 0 }[/math].