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Asymptotic expansions of symmetric standard elliptic integrals. (English) Zbl 0994.33010
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.

33E05Elliptic functions and integrals
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33F10Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
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