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Series expansions of symmetric elliptic integrals


Author: Toshio Fukushima
Journal: Math. Comp. 81 (2012), 957-990
MSC (2010): Primary 33E05
DOI: https://doi.org/10.1090/S0025-5718-2011-02531-7
Published electronically: August 15, 2011
MathSciNet review: 2869045
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Abstract: Based on general discussion of series expansions of Carlson's symmetric elliptic integrals, we developed fifteen kinds of them including eleven new ones by utilizing the symmetric nature of the integrals. Thanks to the special addition formulas of the integrals, we also obtained their complementary series expansions. By considering the balance between the speed of convergence and the amount of computational labor, we chose four of them as the best series expansions. Practical evaluation of the integrals is conducted by the most suitable one among these four series expansions. Its selection rule was analytically specified in terms of the numerical values of given parameters. As a by-product, we obtained an efficient asymptotic expansion of the integrals around their logarithmic singularities. Numerical experiments confirmed the effectiveness of these new series expansions.


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Additional Information

Toshio Fukushima
Affiliation: National Astronomical Observatory of Japan, 2-21-1, Ohsawa, Mitaka, Tokyo 181-8588, Japan
Email: Toshio.Fukushima@nao.ac.jp

DOI: https://doi.org/10.1090/S0025-5718-2011-02531-7
Keywords: Elliptic integral
Received by editor(s): November 4, 2010
Received by editor(s) in revised form: January 30, 2011
Published electronically: August 15, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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