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)
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(2)
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Let
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so that
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(8)
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(9)
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(10)
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Then
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where is a zeroth order Bessel function of the first kind.
Therefore, the Hankel transform pairs are
(17)
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(18)
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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)
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(20)
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where is a Bessel function of the first kind, is a Struve function and is a modified Struve function.
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The Hankel transform of order is defined by
(21)
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(Bronshtein et al. 2004, p. 706).
A different kind of Hankel transform can also be defined for integer sequences (Layman 2001).