Justification of the stationary phase approximation in time-domain asymptotics

A rigorous proof is supplied for the validity of an asymptotic approximation to the integral I(λ) = ∫_a^b g(x)p{λf(x)}dx, where f(x) and g(x) are sufficiently smooth functions on [a, b] and p(x) is a piece-wise smooth periodic function with mean zero. In addition, a two-dimensional generalization is given. Problems concerning coalescence of two stationary points and a stationary point near an end point are also considered. © 1997 The Royal Society.