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^[1]^[2]^[3]^[4]^[5]. 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 $z>0$ .

References

↑ Schlomilch, G. (1857). On Bessel's function. Zeitschrift fur Math, and Pkys., 2, 155-158.
↑ Whittaker, E. T., & Watson, G. N. (1996). A course of modern analysis. Cambridge university press.
↑ Rayleigh, L. (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.
↑ Watson, G. N. (1995). A treatise on the theory of Bessel functions. Cambridge university press.
↑ Chapman, S. (1911). On the general theory of summability, with application to Fourier's and other series. Quarterly Journal, 43, 1-52.

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[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] Rayleigh, L. (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.