Let n>=0 and α_1, α_2, ...be the positive roots of J_n(x) = 0, where J_n(z) is a Bessel function of the first kind. An expansion of a function in the interval (0, 1) in terms of Bessel functions of the first kind f(x) = sum_(r = 1)^∞ A_r J_n(x α_r), has coefficients found as follows: integral_0^1 x f(x) J_n(x α_l) d x = sum_(r = 1)^∞ A_r integral_0^1 x J_n(x α_r) J_n(x α_l) d x. But orthogonality of Bessel function roots gives integral_0^1 x J_n(x α_l) J_n(x α_r) d x = 1/2 δ_(l, r) [J_(n + 1)(α_r)]^2 (Bowman 1958, p.

Bessel function Neumann series | Fourier-Legendre series | Fourier series | generalized Fourier series | Schlömilch's series

Friedrich Wilhelm Bessel | Jean-Baptiste-Joseph Fourier