Asymptotics of a class of multidimensional Laplace–type integrals. II. Treble integrals

Abstract

This paper is an extension to three (and more) dimensions of a technique developed to obtain the algebraic asymptotic behaviour of a class of double Laplace–type integrals. The method proceeds by representing treble Laplace–type integrals as iterated Mellin–Barnes integrals, followed by judicious application of residue theory to determine new asymptotic expansions. The class of ‘phase functions’ of the Laplace integrals to which the analysis applies is restricted to ‘polynomials’ (non–integral powers are permitted) with an isolated, though possible highly degenerate, critical point at the origin. The determination of which residues to use in constructing the expansions is characterized in elementary geometric terms. Numerical examples highlighting the use of the expansions are supplied.