EPSTEIN-HUBBELL ELLIPTICTYPE INTEGRAL AND ITS GENERALIZATIONS

A survey of the evaluation, series expansions, properties, and approxi mation of the Epstein-Hubbell elliptictype integral Omega(j) = integra l(0)(pi)(1 - k(2) cos theta)(-j-1/2) d theta, 0 less than or equal to k < 1, j = 0,1,2,..., is considered. We review different generalizatio ns of this integral ((R(mu)(k,alpha, gamma), K-mu(k, m), S-mu(k, v)... etc.) and examine some of their important properties, including asymp totic expansions in the neighborhood of k(2) = 1. We express Omega(j)( k) and its generalizations in terms of hypergeometric series of argume nt k(4) Furthermore, we show that a new infinite series of Epstein-Hub bell integral obtained recently by some authors, using the residue the ory of complex variables, can be easily deduced from known transformat ions. It is shown that the elliptictype integrals can be expressed as the differintegral of elementary functions.