Numerical approximation of highly oscillatory integrals on semi-finite intervals by steepest descent method

A numerical steepest descent method, based on the Laguerre quadrature rule, is developed for integration of one-dimensional highly oscillatory functions on 0,?∞?) of a general class. It is shown that if the integrand is analytic, then in the absence of stationary points, the method is rapidly convergent. The method is extended to the case when there are a finite number of stationary points in 0,?∞?). It can be further extended to the case when the integrand is only smooth to some degree (not necessarily analytic). We illustrate the theoretical results using some numerical experiences.