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Zbl 0994.33010
López, José L.
Asymptotic expansions of symmetric standard elliptic integrals. (English)
[J] SIAM J. Math. Anal. 31, No.4, 754-775 (2000). ISSN 0036-1410; ISSN 1095-7154/e

Symmetric standard elliptic integrals are of the form $$\int_0^\infty {dt\over \sqrt{P(t)}}, \quad \text{and} \quad \int_0^\infty {dt \over (t+p) \sqrt{P(t)}}, $$where $P(t)=(t+x)(t+y)(t+z)$, parameters $x$, $y$, $z$ are nonnegative and distinct, and $p>0$ may eventually coincide with $z$. The author constructs the complete asymptotic convergent expansions of these integrals when either $z$ or $p$ are large (tend to $+\infty$). The distributional approach is used for a new derivation of the asymptotic expansion of the generalized Stieltjes transforms, which is applied to the integrals above. Coefficients of these expansions are computed by recurrence. Moreover, accurate error bounds for the remainder are supplied. Numerical results show that the rate of convergence of the asymptotic series increases with the large parameter.
[Andrei Martínez Finkelshtein (Almeria)]

MSC 2000:: *33E05 Elliptic functions and integrals
41A60 Asymptotic problems in approximation
33F10 Symbolic computation of special functions

Keywords: elliptic integrals; asymptotic expansions; distributional approach

Cited in: Zbl 1052.35015

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