Method of stationary phase

Abstract

The integrals for which this method was originally developed by Stokes† and Kelvin‡ are of the general form

$$ f(\lambda )\, = \,\int\limits_a^b {g(t)\,e^{i\lambda h(t)} \,dt,} \, $$

((4.1))

where a, b, g(t), h(t), t, and λ are all real. Asymptotic expansions are sought as λ → ∞. It should be said here that if g(t) and h(t) can be suitably analytically continued off the real axis then the class of integrals (4.1) can be treated, as discussed briefly below, by the method of steepest descents in §3.1. However, the original method of stationary phase antecedes the steepest descents one. There are three main reasons for considering it separately. First, the physical idea and motivation behind the original exposition of the method are interesting and instructive. Secondly, we would like to be able to deal with such integrals by considering the integration along the real axis only. Thirdly, the fact that integrals like (4.1) play such a fundamental role in the study of general wave motion, as will be seen in §4.2 below, is in itself a valid reason for obtaining the asymptotic expansion. In §4.2 a brief introduction is given to dispersive wave motion which is of current interest and practical importance: the most exciting developments in the subject have appeared since about 1960.