Hankel Transform

The Hankel transform (of order zero) is an integral transform equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel and also called the Fourier-Bessel transform. It is defined as

			(1)
			(2)

Let

			(3)
			(4)

so that

			(5)
			(6)
			(7)

			(8)
			(9)
			(10)

Then

			(11)
			(12)
			(13)
			(14)
			(15)
			(16)

where is a zeroth order Bessel function of the first kind.

Therefore, the Hankel transform pairs are

			(17)
			(18)

A slightly differently normalized Hankel transform and its inverse are implemented in the Wolfram Language as HankelTransform[expr, r, s] and InverseHankelTransform[expr, s, r], respectively.

The following table gives Hankel transforms for a number of common functions (Bracewell 1999, p. 249). Here, is a Bessel function of the first kind and is a rectangle function equal to 1 for and 0 otherwise, and

			(19)
			(20)

where is a Bessel function of the first kind, is a Struve function and is a modified Struve function.

	1

The Hankel transform of order is defined by

(21)

(Bronshtein et al. 2004, p. 706).

A different kind of Hankel transform can also be defined for integer sequences (Layman 2001).