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:

:$f(x)\; sim\; sum\_\{n=0\}^infty\; c\_n\; J\_alpha(lambda\_n\; x/b),$

where $lambda\_n$ is the "n"th zero of $J\_alpha(x)$ (i.e. $J\_alpha(lambda\_n)=0$). From the orthogonality relationship:

:$int\_0^1\; J\_alpha(x\; lambda\_m),J\_alpha(x\; lambda\_n),x,dx=\; frac\{delta\_\{mn\{2\}\; [J\_\{alpha+1\}(lambda\_n)]\; ^2$

the coefficients are given by

:$c\_n\; =frac\{int\_\{0\}^b\; J\_alpha(lambda\_n\; x/b),f(x)\; ,x,dx\; \}\{int\_\{0\}^b\; x\; J\_alpha^2\; (lambda\_n\; x/b)\; dx\}=frac\{langle\; f,\; J\_alpha(lambda\_n\; x/b)\; angle\}\{|J\_alpha(lambda\_n\; x/b)|^2\}.$

The lower integral may be evaluated, yielding:

:$c\_n\; =frac\{int\_\{0\}^b\; J\_alpha(lambda\_n\; x/b),f(x)\; ,x,dx\; \}\{b^2\; J\_\{alphapm\; 1\}^2\; (lambda\_n)/2\}$

where the plus or minus sign is equally valid.

ee also

*orthogonal
*Generalized Fourier series

References

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External links

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