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Series expansions of symmetric elliptic integrals

Author: Toshio Fukushima
Journal: Math. Comp. 81 (2012), 957-990
MSC (2010): Primary 33E05
Posted: August 15, 2011
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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.

References

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