In this opening chapter we present a detailed account, together with a series of examples of increasing complexity, of the classical method of steepest descents applied to Laplace-type integrals. Consideration is also given to the common causes of non-uniformity in the asymptotic expansions so produced due to a variety of coalescence phenomena. The chapter concludes with a brief discussion of the Stokes phenomenon and hyperasymptotics, both of which have undergone intense development during the past two decades. Such a preliminary discussion, as well as hopefully being of general interest in its own right, is necessary for the remaining chapters, since the Hadamard expansion procedure can be viewed as an ‘exactification’ of the method of steepest descents yielding hyperasymptotic levels of accuracy. Considerable space in the later chapters is devoted to showing how the Hadamard expansion procedure can be modified to deal with various coalescence problems.

Historical introduction

One of the most important methods of asymptotic evaluation of certain types of integral is known as the method of steepest descents. This method has its origins in the observation made by Laplace in connection with the estimation of an integral arising in probability theory of the form (Laplace, 1820; Gillespie, 1997).

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.

Introduction

In Chapter 3 we introduced two basic modes of expansion using Hadamard series, namely the forward expansion Scheme A and the forward-reverse expansion Scheme B. The first scheme uses the Hadamard series Sn(z), defined in (3.2.9), and is suitable for isolated saddle points when adjacent saddles or other singularities are sufficiently remote to result in a sequence of well-separated exponential levels. If maximal exponential separation is employed, the resulting convergence of the Hadamard series has to be accelerated through use of the modified form of the series. This involves the computation of coefficients expressed in terms of one-dimensional integrals in (3.2.18) of a common form at each level of the expansion. If one is prepared to accept a reduced exponential separation, however, it is possible, through judicious choice of the expansion points Ωn, to produce Hadamard series that converge rapidly at a geometric rate without the need for the computationally more expensive modified form.

In Scheme B, the zeroth interval is dealt with by forward expansion as in Scheme A, but with forward-reverse expansion about the points Ωn for the intervals with n ≥ 1. This has the advantage of covering a given interval on the integration path with fewer evaluations of the inversion expansions. By careful choice of the Ωn it is similarly possible to arrange for the Hadamard series at all levels to converge at a geometric rate.