Numerical methods for highly oscillatory integrals on semi-finite intervals

In highly oscillatory integrals, the integrand fw(x) oscillates rapidly with a frequency ω. For very high values of ω, numerical evaluation of such integrals by Gaussian quadrature rules can be of very low accuracy. In such problems which have many applications in mathematical physics, it is important to devise algorithms with errors which decrease as fast as w−N , for some N > 0. In this paper, we review some existing methods, and particularly, Levintype methods. Then we present a steepest descent method which converts highly oscillatory integrals to nonoscillatory integrals. The idea behind this method is to replace the integration interval with a path in the complex plane, in such a way that the oscillatory factor of integrand decays exponentially along this path. The method is a modification of a 2006-SIAM paper for highly oscillatory integrals on bounded intervals.