Fourier–Bessel series

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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 $λ n$ is the nth zero of $J α (x)$ (i.e. $J α (λ n) = 0$ ). 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

$c_n =\frac{\int_{0}^b J_\alpha(\lambda_n x/b)\,f(x) \,x\,dx }{\int_{0}^b x J_\alpha^2 (\lambda_n x/b) dx} =\frac{\langle f, J_\alpha(\lambda_n x/b) \rangle}{\|J_\alpha(\lambda_n x/b)\|^2}.$