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]