New results on the approximations of the generalised elliptic-type integrals

The generalised elliptic-type integral R-mu(k, alpha, gamma) R-mu(k, alpha, gamma) = integral(0)(pi) cos((theta/2))(2 alpha - 1) sin(theta/2))(2 gamma - 2 alpha - 1)/(1 - k(2) cos theta)(mu + 1/2) d theta where 0 less than or equal to k < 1, Re(gamma) > Re(alpha) > 0, Re(mu) > -0.5 has been represented in terms of the Gauss hypergeometric function by Kalla et al. (1986). Furthermore, Kalla et al. (1987) derived a simple-structured single term approximation for this function in the neighbourhood of k(2) = 1 in some range of the parameters alpha, gamma and mu. In this paper, a different technique is used to derive efficient two-term approximations in closed form in the neighbourhood of k(2) = 1 for R-mu(k, alpha, gamma), which may be considered as an extension of the concept of the single term approximation mentioned above. Evidently, a closed form reduces computations considerably, and the improvement in accuracy by having two terms instead of a single one is manifested by the reduction of the error from O(h(2)) to O(h(4)), where h = (1 - k(2))/(2k(2)) much less than 1. The technique used in the approximation may be interpreted as a rational approximation to a function that matches the two rational terms with four terms of the Taylor expansion. Results show that the proposed technique is superior to existing approximations for the same number of terms. The formulation presented in this work has potential application for a wide class of special functions.