Abstract

An analytical form that provides a computationally efficient algorithm for numerical evaluation of the Hankel transform of nth order by fast-Fourier-transform techniques is presented and tested with some well-known functions.

© 1999 Optical Society of America

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References

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  1. A. Agnesi, G. C. Reali, G. Patrini, A. Tomaselli, “Numerical evaluation of the Hankel transform: remarks,” J. Opt. Soc. Am. A 10, 1872–1874 (1993).
    [CrossRef]
  2. J. A. Ferrari, “Fast Hankel transform of order zero,” J. Opt. Soc. Am. A 12, 1812–1813 (1995).
    [CrossRef]
  3. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970), Chap. 11, p. 490.
  4. S. Colombo, Les Transformations de Mellin et de Hankel, Monographies du Centre d’Etudes Mathématiques en vue des Applications déjà parues et éditées par le Centre National de la Recherche Scientifique (CNRS) (CNRS, Paris, 1959).
  5. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1996), Chap. 4, p. 132.

1995

1993

Agnesi, A.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970), Chap. 11, p. 490.

Colombo, S.

S. Colombo, Les Transformations de Mellin et de Hankel, Monographies du Centre d’Etudes Mathématiques en vue des Applications déjà parues et éditées par le Centre National de la Recherche Scientifique (CNRS) (CNRS, Paris, 1959).

Ferrari, J. A.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1996), Chap. 4, p. 132.

Patrini, G.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1996), Chap. 4, p. 132.

Reali, G. C.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1996), Chap. 4, p. 132.

Tomaselli, A.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1996), Chap. 4, p. 132.

J. Opt. Soc. Am. A

Other

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970), Chap. 11, p. 490.

S. Colombo, Les Transformations de Mellin et de Hankel, Monographies du Centre d’Etudes Mathématiques en vue des Applications déjà parues et éditées par le Centre National de la Recherche Scientifique (CNRS) (CNRS, Paris, 1959).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1996), Chap. 4, p. 132.

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Equations (5)

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f˜n(ρ)=2π0f(r)Jn(2πρr)rdr,
Φn(x)=2(-i)n0+f((x2+y2)1/2)cos[nθ(x, y)]dy,
F{Φn}=(-i)n--f((x2+y2)1/2)×exp[inθ(x, y)]exp(i2πρx)dydx=(-i)n002πf(r)exp(inθ)×exp[i2πρr cos(θ)]r dθ dr=f˜n(ρ),
Jn(2πρr)=(-i)n2π 02π exp(inθ)exp[i2πρr cos(θ)]dθ.
f˜(ρ)=(π/a)(n+1)ρn exp(-π2ρ2/a).

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