Asymptotics of the first Appell function F 1 with large parameters

Abstract

We consider the asymptotic behaviour of the first Appell function F 1(a, b, b′, c; x, y) when several of the parameters a, b, b′, c are large. We reduce to 23 the possible combinations of large parameters that require an asymptotic analysis. In this paper, we analyse 3 of those 23 asymptotic regions: (i) large positive c, (ii) large positive b and (iii) large positive c and large negative b. We derive complete asymptotic expansions of the F 1 function for every one of these three regions. The starting point is a single integral representation of F 1(a, b, b′, c; x, y). We use three different asymptotic techniques: Laplace's method and two variants of Laplace's method. Analytical and numerical aspects of these formulas are discussed.