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:


 * $$f(x) = a_0 + \sum_{n=1}^\infty a_n J_0(nx),$$

where


 * $$\begin{align}

a_0 &= f(0) + \frac{1}{\pi} \int_0^\pi \int_0^{\pi/2} u f'(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{align}$$

Examples
Some examples of Schlömilch's series are the following:
 * Null functions in the interval $$(0,\pi)$$ can be expressed by Schlömilch's Series, $$0 = \frac{1}{2}+\sum_{n=1}^\infty (-1)^n J_0(nx)$$, 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 $$0 1/2$$ and $$0<x\leq \pi$$. These properties were identified by Niels Nielsen.
 * $$x = \frac{\pi^2}{4}-2\sum_{n=1,3,...}^\infty \frac{J_0(nx)}{n^2}, \quad 00$$.