Epstein-Hubbell elliptictype integral and its generalizations

摘要

A survey of the evaluation, series expansions, properties, and approximation of the Epstein-Hubbell elliptictype integral is considered. We review different generalizations of this integral (Rμ(k, α, γ), Kμ(k, m), Sμ(k, νv)… etc.) and examine some of their important properties, including asymptotic expansions in the neighborhood of k2 = 1. We express Ωj(k) and its generalizations in terms of hypergeometric series of argument k4. Furthermore, we show that a new infinite series of Epstein-Hubbell integral obtained recently by some authors, using the residue theory of complex variables, can be easily deduced from known transformations. It is shown that the elliptictype integrals can be expressed as the differintegral of elementary functions.