Epstein–Hubbell integrals regarded as associated Legendre functions of the second kind

Abstract

The Epstein–Hubbell integrals were introduced by Hubbell et al. [1961. Radiation field from a circular disk source. J. Res. Nat. Bur. Stand. 65(C), 249–264] in the context of calculations on radiation from a circular disk source. Papers by Hubbell and Epstein [1963. Evaluation of a generalized elliptic-type integral. J. Res. Nat. Bur. Stand. 67(B), 1–17] and Weiss [1964. A note on a generalized elliptic integral. J. Res. Nat. Bur. Stand. 68(B), 1–2] then examined these integrals from the mathematical point of view. In recent years considerable attention has been given to working out the various mathematical properties of these functions, and to devising various generalizations of them. Here it is shown that apart from a simple algebraic factor, the Epstein–Hubbell integrals are associated Legendre functions of the second kind and half-integral degree, also known as toroidal functions. Hence many of the results developed for Legendre functions can be immediately applied to Epstein–Hubbell integrals, and many of the various properties worked out over the years follow more or less immediately from the Legendre function connection, together with some new results. One of the generalizations devised for the Epstein–Hubbell integrals stands out as most natural in this context.