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
Let
so that
Then
where is a zeroth order Bessel function of the first kind.
Therefore, the Hankel transform pairs are
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
where is a Bessel function of the first kind, is a Struve function and is a modified Struve function.
The Hankel transform of order is defined by
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(22)
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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).
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL:
Academic Press, p. 795, 1985.
Bracewell, R. "The Hankel Transform." The Fourier Transform and Its Applications, 3rd ed. New
York: McGraw-Hill, pp. 244-250, 1999.
Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook
of Mathematics, 4th ed. New York: Springer-Verlag, pp. 705-706, 2004.
Layman, J. W. "The Hankel Transform and Some of Its Properties." J. Integer Sequences 4, No. 01.1.5, 2001. http://www.math.uwaterloo.ca/JIS/VOL4/LAYMAN/hankel.
Oberhettinger, F. Tables of Bessel Transforms. New York: Springer-Verlag,
1972.
Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland:
Gordon and Breach, p. 23, 1993.
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