Numerical Evaluation of Integrals With Infinite Limits and Oscillating Integrands

This paper is in the nature of an addendum to an earlier paper in this journal dealing with the numerical evaluation of definite integrals. 1 Integrals with infinite limits and oscillating integrands often arise in technical problems. In this paper, we assume that the integrand is an analytic function of the variable of integration u (in a suitable region) and tends to oscillate at a regular rate as u --* °°. The main concern here is how to deal with cases in which the rate of convergence is slow. A number of ways have been proposed to handle the slow convergence. One is to use integration formulas of Filon's type. In particular, E. O. Tuck 2 has given a formula of this type suited to an infinite range of integration. Another method is to evaluate the contributions of positive and negative loops of the integrand and apply Euler's summation formula for slowly converging alternating series. Still another is to (i) write the integrand in terms of complex exponentials, (ii) deform the path of integration so that it becomes a path (or portions of paths) of steepest descent in the complex u-plane, and then (Hi) integrate along straight-line segments that approximate the path. A closely related procedure is to tilt the path of integration, if possible, so that the integrand will decrease exponentially. 155