Error bounds for the method of steepest descents

Abstract

New error bounds are found for the asymptotic expansions which arise in the method of steepest descents. These bounds are found by exploiting the reformulation of the method of steepest descents given in recent work by Berry & Howls. A major difference between this and previous work is that the global properties of the integrand (and not just those on the path of steepest descent) play an essential role. The bounds are significantly easier to compute than those in the current literature. Applications of the error bounds are given for asymptotic expansions arising from the Pearcey integral, the Airy function, the modified Bessel function, and the integral of the Airy function.