The Laplace's and steepest descents methods revisited

Abstract

We revise Laplace’s and Steepest Descents methods of asymptotic expansions of integrals. The main difficulties in these methods are originated by a change of variables and an eventual deformation of the integration contour. We present a simplification of these methods that only requires an expansion of the integrand at the critical point(s). In this way, the calculation of the coefficients of the asymptotic expansion is simpler. The simplification in the case of several relevant critical points is even more significant and requires multi-point Taylor expansions. The new method that we present here unifies Laplace’s and Steepest Descents methods in one unique formulation. Uniformity properties of the method are discussed. Asymptotic expansions of the Bernoulli and Jacobi polynomials and of a generalized confluent hypergeometric function are given as illustration. Mathematics Subject Classification: 41A60, 41A58, 33C65