Hankel transform

In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind J_ν(kr). The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r-axis. The necessary coefficient F_ν of each Bessel function in the sum, as a function of the scaling factor k constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval.

The Hankel transform of order ν of a function f(r) is given by:

where ${\displaystyle J_{\nu }}$ is the Bessel function of the first kind of order ${\displaystyle \nu }$ with ${\displaystyle \nu \geq -{\frac {1}{2}}}$. The inverse Hankel transform of F_ν(k) is defined as:

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