ABSTRACT
This chapter considers methods of constructing asymptotic approximations of integrals. The crucial idea in the asymptotic approximation of integrals involves identifying the regions of dominant contribution and replacing functions in the integrand by simpler functions which represent their local behavior in this region. The exponential bound condition which was used to prove Watson's lemma is in fact not necessary unless the upper limit of integration is infinite. The gamma function provides an interesting nonstandard application of Laplace's method. The chapter focuses on integrands which rapidly oscillate in sign over the integration range. The contributions to the integral then tend to cancel except at end points where small contributions remain. Contributions also arise from regions where the rate of oscillation of the integrand is reduced. Such regions are called regions of stationary phase. The chapter describes the associated asymptotic approximation technique known as the method of stationary phase.
