Fourier–Bessel series

In mathematics, Fourier–Bessel series is a particular kind of generalized Fourier series (an infinite series expansion on a finite interval) based on Bessel functions.

Fourier–Bessel series are used in the solution to partial differential equations, particularly in cylindrical coordinate systems.

The Fourier–Bessel series of a function f(x) with a domain of [0,b]

is the notation of that function as a linear combination of many orthogonal versions of the same Bessel function of the first kind J_α, where the argument to each version n is differently scaled, according to

where u_α,n is a root, numbered n associated with the Bessel function J_α and c_n are the assigned coefficients:

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.

As said, differently scaled Bessel Functions are orthogonal with respect to the inner product

according to

(where: ${\displaystyle {\delta _{mn}}}$ is the Kronecker delta). The coefficients can be obtained from projecting the function f(x) onto the respective Bessel functions:

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