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Fourier-Bessel Series

A Fourier-Bessel series is an expansion in Bessel functions of the first kind. For a real number n>=0, let alpha_1<alpha_2<... be the positive roots of J_n(x)=0. A function f in L^2((0,1),xdx) has the expansion

f(x)=sum_(r=1)^inftyA_rJ_n(xalpha_r),
(1)

where convergence is in the weighted norm ||g||^2=int_0^1x|g(x)|^2dx. Functions equal almost everywhere are identified. This norm convergence does not assert pointwise convergence at every x (National Institute of Standards and Technology 2026, §10.23(iii)).

The coefficients can be obtained without interchanging an infinite sum and an integral. Let S_N(x)=sum_(r=1)^(N)A_rJ_n(xalpha_r) be the Nth partial sum. Then

int_0^1xS_N(x)J_n(xalpha_l)dx=sum_(r=1)^NA_rint_0^1xJ_n(xalpha_r)J_n(xalpha_l)dx.
(2)

The orthogonality relation is

int_0^1xJ_n(xalpha_l)J_n(xalpha_r)dx=1/2delta_(l,r)[J_(n+1)(alpha_r)]^2,
(3)

where delta_(l,r) is the Kronecker delta (Bowman 1958, p. 108; National Institute of Standards and Technology 2026, Eq. 10.22.37). Thus, for N>=l,

int_0^1xS_N(x)J_n(xalpha_l)dx=1/2sum_(r=1)^(N)A_rdelta_(l,r)[J_(n+1)(alpha_r)]^2
(4)
=1/2A_l[J_(n+1)(alpha_l)]^2.
(5)

By Schwarz's inequality, the weighted inner product with J_n(xalpha_l) is continuous in this norm. Passing to the limit N->infty therefore gives

A_l=2/([J_(n+1)(alpha_l)]^2)int_0^1xf(x)J_n(xalpha_l)dx.
(6)

See also

Bessel Function Neumann Series, Fourier-Legendre Series, Fourier Series, Generalized Fourier Series, Schlömilch's Series

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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.National Institute of Standards and Technology. "Integrals" and "Sums." §§10.22-10.23 in Digital Library of Mathematical Functions. Version 1.2.8, 2026. https://dlmf.nist.gov/10.22 and https://dlmf.nist.gov/10.23.

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Fourier-Bessel Series

Cite this as:

Weisstein, Eric W. "Fourier-Bessel Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Fourier-BesselSeries.html

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