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 Oskar Schlömilch, who derived the series in 1857. The real-valued function $f(x)$ has the following expansion:

:$f(x) = a_0 + \sum_^\infty a_n J_0(nx),$

where

:$\begin a_0 &= f(0) + \frac \int_0^\pi \int_0^ u f'(u\sin\theta)\ d\theta\ du, \\ a_n &= \frac \int_0^\pi \int_0^ u\cos nu \ f'(u\sin\theta)\ d\theta\ du. \end$

Examples

Some examples of Schlömilch's series are the following:
*Null functions in the interval $(0,\pi)$ can be expressed by Schlömilch's Series, $0 = \frac+\sum_^\infty (-1)^n J_0(nx)$, 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 0; the series oscillates at $x=0$ and diverges at $x=\pi$. This theorem is generalized so that $0 = \frac+\sum_^\infty (-1)^n J_0(nx)/(nx/2)^\nu$ when $-1/2<\nu\leq 1/2$ and 0 and also when $\nu> 1/2$ and 0. These properties were identified by Niels Nielsen.Nielsen, N. (1904). Handbuch der theorie der cylinderfunktionen. BG Teubner.
*x = \frac-2\sum_^\infty \frac, \quad 0
*x^2 = \frac + 8 \sum_^\infty \fracJ_0(nx), \quad -\pi
*\frac + \sum_^k\frac = \frac + \sum_^\infty J_0(nx), \quad 2k\pi
* If $(r,z)$ are the cylindrical polar coordinates, then the series $1+\sum_^\infty e^J_0(nr)$ is a solution of Laplace equation for $z>0$.

See also

* Kapteyn series

References

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Series expansions