# Fourier–Bessel series

Fourier–Bessel series

In mathematics, Fourier–Bessel series are a particular kind of infinite series expansion on a finite interval, based on Bessel functions and as such are part of a large class of expansions based on orthogonal functions. Fourier-Bessel series are used in the solution to partial differential equations, particularly in cylindrical coordinate systems.

The Fourier–Bessel series may be thought of as a Fourier expansion in the ρ coordinate of cylindrical coordinates. Just as the Fourier series is defined for a finite interval and has a counterpart, the continuous Fourier transform over an infinite interval, so the Fourier–Bessel series has a counterpart over an infinite interval, namely the Hankel transform

Because Bessel functions are orthogonal with respect to a weight function $x$ on the interval [0, "b"] they can be expanded in a Fourier–Bessel series defined by:

:$f\left(x\right) sim sum_\left\{n=0\right\}^infty c_n J_alpha\left(lambda_n x/b\right),$

where $lambda_n$ is the "n"th zero of $J_alpha\left(x\right)$ (i.e. $J_alpha\left(lambda_n\right)=0$). From the orthogonality relationship:

:$int_0^1 J_alpha\left(x lambda_m\right),J_alpha\left(x lambda_n\right),x,dx= frac\left\{delta_\left\{mn\left\{2\right\} \left[J_\left\{alpha+1\right\}\left(lambda_n\right)\right] ^2$

the coefficients are given by

:$c_n =frac\left\{int_\left\{0\right\}^b J_alpha\left(lambda_n x/b\right),f\left(x\right) ,x,dx \right\}\left\{int_\left\{0\right\}^b x J_alpha^2 \left(lambda_n x/b\right) dx\right\}=frac\left\{langle f, J_alpha\left(lambda_n x/b\right) angle\right\}\left\{|J_alpha\left(lambda_n x/b\right)|^2\right\}.$

The lower integral may be evaluated, yielding:

:$c_n =frac\left\{int_\left\{0\right\}^b J_alpha\left(lambda_n x/b\right),f\left(x\right) ,x,dx \right\}\left\{b^2 J_\left\{alphapm 1\right\}^2 \left(lambda_n\right)/2\right\}$

where the plus or minus sign is equally valid.

ee also

*orthogonal
*Generalized Fourier series

References

*

* Fourier–Bessel series applied to Acoustic Field analysis on [http://www.trinnov.com/research.php#concept Trinnov Audio's research page]

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