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Perform simple and accurate Hankel transformations using the method of Ogata 2005.

Hankel transforms and integrals are commonplace in any area in which Fourier Transforms are required over fields that are radially symmetric (see Wikipedia for a thorough description). They involve integrating an arbitrary function multiplied by a Bessel function of arbitrary order (of the first kind). Typical integration schemes often fall over because of the highly oscillatory nature of the transform. Ogata’s quadrature method used in this package provides a fast and accurate way of performing the integration based on locating the zeros of the Bessel function.


  • Accurate and fast solutions to many Hankel integrals
  • Easy to use and re-use
  • Arbitrary order transforms
  • Built-in support for radially symmetric Fourier Transforms
  • Thoroughly tested.
  • Python 2 and 3 compatible.


Either clone the repository at github.com/steven-murray/hankel and use python setup.py install, or simply install using pip install hankel.

The only dependencies are numpy, scipy and mpmath.


Based on the algorithm provided in

H. Ogata, A Numerical Integration Formula Based on the Bessel Functions, Publications of the Research Institute for Mathematical Sciences, vol. 41, no. 4, pp. 949-970, 2005.

Also draws inspiration from

Fast Edge-corrected Measurement of the Two-Point Correlation Function and the Power Spectrum Szapudi, Istvan; Pan, Jun; Prunet, Simon; Budavari, Tamas (2005) The Astrophysical Journal vol. 631 (1)

Indices and tables