The aims of this book are twofold. The first is to present a detailed account of the classical method of steepest descents applied to the asymptotic evaluation of Laplace-type integrals containing a large parameter, and the second is to give a coherent account of the theory of Hadamard expansions. This latter topic, which has been developed during the past decade, extends the method of steepest descents and effectively ‘exactifies’ the procedure since, in theory, the Hadamard expansion of a Laplace or Laplace-type integral can produce unlimited accuracy.

Many texts deal with the method of steepest descents, some in more detail than others. The well-known books by Copson Asymptotic Expansions (1965), Olver Asymptotics and Special Functions (1997), Bleistein and Handelsman Asymptotic Expansion of Integrals (1975), Wong Asymptotic Approximations of Integrals (1989) and Bender and Orszag Advanced Mathematical Methods for Scientists and Engineers (1978) are all good examples. It is our aim in the first chapter to give a comprehensive account of the method of steepest descents accompanied by a set of illustrative examples of increasing complexity. We also consider the common causes of non-uniformity in the asymptotic expansions of Laplace-type integrals and conclude the first chapter with a discussion of the Stokes phenomenon and hyperasymptotics.

The next two chapters present the Hadamard expansion theory of Laplace and of Laplace-type integrals possessing saddle points. A study of these chapters makes it apparent how this theory builds upon and extends the method of steepest descents.