Schlömilch's series

Schlömilch's Series is a Fourier series type expansion of twice continuously differentiable function in the interval $(0,\pi )$ 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 $f(x)$ has the following expansion,

$f(x)=a_{0}+\sum _{n=1}^{\infty }a_{n}J_{0}(nx)$

where

${\begin{aligned}a_{0}&=f(0)+{\frac {1}{\pi }}\int _{0}^{\pi }\int _{0}^{\pi /2}uf'(u\sin \theta )\ d\theta \ du,\\a_{n}&={\frac {2}{\pi }}\int _{0}^{\pi }\int _{0}^{\pi /2}u\cos nu\ f'(u\sin \theta )\ d\theta \ du.\end{aligned}}$

Examples and Properties

Some examples and properties of Schlömilch's Series are following

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