# hankel¶

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 fail 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.

## Features¶

• 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.

• only Python 3 compatible.

## Installation¶

Either clone the repository and install locally (best for developer installs):

$git clone https://github.com/steven-murray/hankel.git$ cd hankel/
$pip install -U .  Or install from PyPI: $ pip install hankel


Or install with conda:

\$ conda install -c conda-forge hankel


The only dependencies are numpy, scipy and mpmath. These will be installed automatically if they are not already installed.

Dependencies required purely for development (testing and linting etc.) can be installed via the optional extra pip install hankel[dev]. If using conda, they can still be installed via pip: pip install -r requirements_dev.txt.

For instructions on testing hankel or any other development- or contribution-related issues, see the contributing guide.

## Acknowledging¶

S. G. Murray and F. J. Poulin, “hankel: A Python library for performing simple and accurate Hankel transformations”, Journal of Open Source Software, 4(37), 1397, https://doi.org/10.21105/joss.01397

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## References¶

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. DOI: 10.2977/prims/1145474602

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) DOI: 10.1086/496971