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Fourier-Bessel Series

Let n>=0 and alpha_1, alpha_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)^inftyA_rJ_n(xalpha_r),
(1)

has coefficients found as follows:

int_0^1xf(x)J_n(xalpha_l)dx=sum_(r=1)^inftyA_rint_0^1xJ_n(xalpha_r)J_n(xalpha_l)dx.
(2)

But orthogonality of Bessel function roots gives

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

(Bowman 1958, p. 108), so

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

and the coefficients are given by

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.

Referenced on Wolfram|Alpha

Fourier-Bessel Series

Cite this as:

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

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