Shows up when trying to solve 2D wave equation on a circular domain in polar coordinates with separation of variables, where we have to decompose the initial condition in termes of a fourier-Bessel series, exactly like the Fourier series appears when solving the wave equation in linear coordinates.

For the same fundamental reasons, also appears when calculating the Schrödinger equation solution for the hydrogen atom.

Completeness: math.stackexchange.com/questions/2192665/is-this-set-of-bessel-functions-a-basis-for-all-c10-a-functions TODO

This is the bessel function analogue to Fourier basis is complete for $L_{2}$.