Some results on generalized elliptic-type integrals

Abstract

In the present paper, we study a family of integrals of the form

where 0<k<1, Re(γ)> Re(α)>0, Re(μ)>−1/2. Special cases of R_μ(k,α,γ) occur in radiation field problems. We obtain a series expansion of R_μ (k,α,γ) and establish its relationship with Gauss' hypergeometric function. Asymptotic expansions valid in the neighborhood of k² = 1, and some recurrence relations are given. Results obtained earlier by Epstein and Hubbell [l],¹ Weiss [4], and Kalla [5] follow as particular cases of our formulae established here. Some numerical values of R_μ(k,α,γ) are computed.

AMS (MOS)::

• 33A25
• 33A30
• 41A60
• 65D20