The Method of Laplace and Watson’s Lemma

in this paper we present a diferent proof of a well known asymptotic estimate for Laplace integrals. The novelty of our approach is that it emphasizes, and rigorously justifies, the appealing heuristic method of Laplace. As a bonus, we also obtain a simple and short proof of Watson’s Lemma. Let a be an element of the extended real number set [−∞,∞]. If lim x−→a f(x)/g(x) = 1 we write f(x) ∼ g(x), x −→ a and say that f is asymptotic to g, or that g is an asymptotic approximation to f . If there is no risk of ambiguity, we may also write f ∼ g for the sake of brevity. Note that f ∼ g if and only if lim(f(x) − g(x))/g(x) = 0. In other words, if and only if the relative error made in approximating f by g tends to zero. In some cases, in particular when the values involved are either very small or very large, it may be more appropriate to estimate the relative rather than the absolute error. In this article we discuss the problem of obtaining asymptotic estimates for integrals of the form (1) I(x) := ∫ J eq(t) dt, where J is a bounded or unbounded interval, p(x) and q(x) are functions satisfying certain properties, and x −→ ∞. Most authors, such as Bender and Orszag [1] use an appealing heuristic method attributed to Laplace to obtain an asymptotic estimate for I(x). A rigorous proof may be found in, for example, Erdélyi [2, §2.4] (see also Olver [3, pp.80–82]). We give another rigorous proof of this estimate in Theorem 1. The novelty of our approach consists in breaking down the proof by means of two preliminary lemmas that highlight and rigorously justify the method of Laplace. Under suitable conditions I(x) has an infinite asymptotic expansion (see for example [3, pp.85–88]). In Theorem 2 we use Lemma 2 to give a simple proof of Watson’s lemma, which gives an infinite asymptotic expansion for I(x) when p(t) = t. We define asymptotic series in the paragraph preceding the statement of Theorem 2. We begin with Lemma 1. Let J be an interval of the form [a,∞) or [a, b], a < b, and assume that the following conditions are satisfied: