- Fourier–Bessel series
mathematics, Fourier–Bessel series are a particular kind of infinite seriesexpansion 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 coordinatesystems.
The Fourier–Bessel series may be thought of as a Fourier expansion in the ρ coordinate of
cylindrical coordinates. Just as the Fourier seriesis defined for a finite interval and has a counterpart, the continuous Fourier transformover an infinite interval, so the Fourier–Bessel series has a counterpart over an infinite interval, namely the Hankel transform
Bessel functions are orthogonal with respect to a weight functionon the interval [0, "b"] they can be expanded in a Fourier–Bessel series defined by:
where is the "n"th zero of (i.e. ). From the orthogonality relationship:
the coefficients are given by
The lower integral may be evaluated, yielding:
where the plus or minus sign is equally valid.
Generalized Fourier series
* 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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