Fourier–Bessel series

Summary

A Fourier–Bessel series expands a function on an interval $[0, b]$ in the functions $J_\nu(j_{\nu,k} x / b)$. Here $J_\nu$ is the Bessel function of the first kind and $j_{\nu,k}$ are its positive zeros. These functions are orthogonal for the weight $x$, so the series is a generalized Fourier series. They are the eigenfunctions of a singular Sturm–Liouville problem coming from Bessel's equation. Fourier–Bessel series solve problems with circular or cylindrical symmetry, such as the vibrations of a circular drum or heat flow in a cylinder.