Laplace’s method for integrals

We have already seen in §1.1 that integration by parts is one way of finding asymptotic approximations to integrals. It is one of the simplest procedures but it is rather limited in its applicability. The procedure is essentially to integrate by parts and then show that the resulting series is asymptotic by estimating the remainder which is in the form of an integral: this is exactly what was done in §1.1 to obtain (1.11) for Ei(x) as x → ∞. We include in this procedure the technique where the integrand is expanded as a series and the asymptotic series is obtained by integrating term by term: the asymptotic nature of the resulting series again depends on the estimation of an integral remainder (see, for example, §1.1 exercise 9). A survey of these methods, presented by way of specific examples, is given in the book by Copson (1965).