Schlömilch's Series

A Fourier Series-like expansion of a twice continuously differentiable function

$$f(x)={\textstyle{1\over 2}}a_0+\sum_{n=1}^\infty a_nJ_0(nx)$$

for $0<x<\pi$, where $J_0(x)$ is a zeroth order Bessel Function of the First Kind and

$$a_0 \equiv 2f(0)+{2\over\pi}\int_0^\pi du\int_0^{\pi/2} f'(u\sin\phi)\,d\phi$$

$$a_n \equiv {2\over\pi}\int_0^\pi du \int_0^{\pi/2} uf'(u\sin\phi)\cos(n\pi)\,d\phi.$$

A special case gives the amazing identity

$$1=J_0(z)+2\sum_{n=1}^\infty J_{2n}(z)=[J_0(z)]^2+2\sum_{n=1}^\infty [J_n(z)]^2.$$

See also Bessel Function of the First Kind, Bessel Function Fourier Expansion, Fourier Series

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1473, 1980.