Schlömilch's series

Schlömilch's series is a Fourier series type expansion of twice continuously differentiable function in the interval ${\displaystyle (0,\pi )}$ in terms of the Bessel function of the first kind, named after the German mathematician Oskar Schlömilch, who derived the series in 1857.^{[1][2][3][4][5]} The real-valued function ${\displaystyle f(x)}$ has the following expansion:

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

where

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

Some examples of Schlömilch's series are the following:

• Null functions in the interval ${\displaystyle (0,\pi )}$ can be expressed by Schlömilch's Series, ${\displaystyle 0={\frac {1}{2}}+\sum _{n=1}^{\infty }(-1)^{n}J_{0}(nx)}$, which cannot be obtained by Fourier Series. This is particularly interesting because the null function is represented by a series expansion in which not all the coefficients are zero. The series converges only when ${\displaystyle 0<x<\pi }$; the series oscillates at ${\displaystyle x=0}$ and diverges at ${\displaystyle x=\pi }$. This theorem is generalized so that ${\displaystyle 0={\frac {1}{2\Gamma (\nu +1)}}+\sum _{n=1}^{\infty }(-1)^{n}J_{0}(nx)/(nx/2)^{\nu }}$ when ${\displaystyle -1/2<\nu \leq 1/2}$ and ${\displaystyle 0<x<\pi }$ and also when ${\displaystyle \nu >1/2}$ and ${\displaystyle 0<x\leq \pi }$. These properties were identified by Niels Nielsen.^[6]
• ${\displaystyle x={\frac {\pi ^{2}}{4}}-2\sum _{n=1,3,...}^{\infty }{\frac {J_{0}(nx)}{n^{2}}},\quad 0<x<\pi .}$
• ${\displaystyle x^{2}={\frac {2\pi ^{2}}{3}}+8\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}}}J_{0}(nx),\quad -\pi <x<\pi .}$
• ${\displaystyle {\frac {1}{x}}+\sum _{m=1}^{k}{\frac {2}{\sqrt {x^{2}-4m^{2}\pi ^{2}}}}={\frac {1}{2}}+\sum _{n=1}^{\infty }J_{0}(nx),\quad 2k\pi <x<2(k+1)\pi .}$
• If ${\displaystyle (r,z)}$ are the cylindrical polar coordinates, then the series ${\displaystyle 1+\sum _{n=1}^{\infty }e^{-nz}J_{0}(nr)}$ is a solution of Laplace equation for ${\displaystyle z>0}$.

• Kapteyn series