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)

REFERENCES:

Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958.

Kaplan, W. "Fourier-Bessel Series." §7.15 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 512-518, 1992.