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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Numerical evaluation of the Gauss hypergeometric function by power summations
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by Jurgen A. Doornik PDF
Math. Comp. 84 (2015), 1813-1833 Request permission


Numerical evaluation of the Gauss hypergeometric function ${}_2F_1(a,b;c;z)$, with complex parameters $a,b,c$ and complex argument $z$ is notoriously difficult. Carrying out the summation that defines the function may fail, even for moderate values of $z$. Formulae are available to transform the effective argument in the series, potentially leading to a numerically successful summation. Unfortunately, these transformations have a singularity when $b-a$ or $c-a-b$ is an integer, and suffer numerical instability near that. This singularity has to be removed analytically after collecting powers in $z$.

The contributions in this paper are fourfold. First, analytical expressions are provided that remove the singularity from Bühring’s $1/(z-z_0)$ transformation. This is more difficult, because the singularity occurs twice, and it is necessary to collect powers of $z_0$, as well as $z$. The resulting expression has a three-term recursion, like the original. Next, improved expressions are derived for the cases that have been addressed before. We study a transformation that converges outside $|z-0.32| > 0.32$ for ${\mathcal {R}}z>0$, which is tighter than the $|z-0.5| > 0.5$ which is normally considered. Finally, we derive an improved algorithm for the numerical evaluation of ${}_2F_1$.

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Additional Information
  • Jurgen A. Doornik
  • Affiliation: Institute for New Economic Thinking at the Oxford Martin School, University of Oxford
  • Email:
  • Received by editor(s): May 1, 2013
  • Received by editor(s) in revised form: September 25, 2013, and October 25, 2013
  • Published electronically: December 3, 2014
  • Additional Notes: This research was supported in part by grants from the Open Society Foundations and the Oxford Martin School.
  • © Copyright 2014 American Mathematical Society
  • Journal: Math. Comp. 84 (2015), 1813-1833
  • MSC (2010): Primary 33C05, 65D20
  • DOI:
  • MathSciNet review: 3335893