Fourier-Bessel Series

Let and , , ...be the positive roots of , where is a Bessel function of the first kind. An expansion of a function in the interval in terms of Bessel functions of the first kind

(1)

has coefficients found as follows:

(2)

But orthogonality of Bessel function roots gives

(3)

(Bowman 1958, p. 108), so

			(4)
			(5)

and the coefficients are given by

(6)