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

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

Examples

Some examples of Schlömilch's series are the following:

• ${\displaystyle x={\frac {\pi ^{2}}{4}}-2\sum _{n=1,3,...}^{\infty }{\frac {J_{0}(nx)}{n^{2}}},\quad 0<x<\pi .}$
• ${\displaystyle x^{2}={\frac {2\pi ^{2}}{3}}+8\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}}}J_{0}(nx),\quad -\pi <x<\pi .}$

References

1. Schlomilch, G. (1857). On Bessel's function. Zeitschrift fur Math, and Pkys., 2, 155-158.
2. Whittaker, E. T., & Watson, G. N. (1996). A Course of Modern Analysis. Cambridge university press.
3. Lord Rayleigh (1911). LXII. On a physical interpretation of Schlömilch's theorem in Bessel's functions. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 21(124), 567-571.
4. Watson, G. N. (1995). A treatise on the theory of Bessel functions. Cambridge university press.
5. Chapman, S. (1911). On the general theory of summability, with application to Fourier's and other series. Quarterly Journal, 43, 1-52.