Generalized elliptic-type integrals and their representations

Epstein–Hubbell [L.F. Epstein, J.H. Hubbell, Evaluation of a generalized elliptic-type integral, J. Res. NBS 67B (1963) 1–17] elliptic-type integrals occur in radiation field problems. In this paper, we consider a generalization (10) of the elliptic-type integrals introduced by Kalla and Tuan [S.L. Kalla, Vu Kim Tuan, Asymptotic formulas for generalized elliptic-type integrals, Comput. Math. Appl. 32 (1996) 49–55]. Many generalizations of elliptic-type integrals, studied earlier by several authors, can be derived as particular cases of our unified form. We study the uniform convergence of the integral representation (10). We derive the power series representations which are valid in different domains. Also we obtain some relationships between this generalized form and Laurecella’s hypergeometric function of three variables FD(3), Appell’s hypergeometric functions F1 and F3 and Gauss’ hypergeometric function 2F1. Some important particular cases of these representations are derived.