Some new methods in asymptotic evaluation of integrals: Application to Airy functions

Abstract

Summary form only given: In various applications involving electromagnetic scattering by smooth convex surfaces, with or without an impedance boundary condition, the appropriate scalar Green's functions can be represented in terms of a integral that involves Fock-type Airy functions. Airy functions by themselves can lead to two different types of asymptotic expansions in the complex plane depending on the ph(z), where z is the complex argument. Crossing the rays in these regions results in huge numerical errors because the dominant and subdominant terms in the asymptotic expansion |z|→ ∞ interchange their respective roles across these rays which are also known as Stokes lines. Addressing the fundamentals of this phenomenon involves a comprehensive review of the basics of the asymptotic expansion of integrals, and is the main subject of this presentation.The general problem is connected to the basics of the traditional saddle point method which is the practical basis of various ray-optic formulations in the asymptotic high-frequency method(s). What is well known is that the first term approximation to the asymptotic evaluation of integrals via saddle-point method can often yield adequate numerical accuracy. Additionally it is also well-known that addition of higher order terms beyond the first can further improve the numerical accuracy of such asymptotic evaluations. The main purpose of this presentation, aside elucidating the more fundamental aspects, is to also focus on the numerical aspects of the asymptotic expansion of integrals since it does not appear that much useful information has been made available for practical use. For the purpose of elaborating the fundamentals of this new approach known as hyperasymptotics, a new result for the Airy function of complex argument obtained by appropriate resummation of the remainder terms will be highlighted. A recent review (D. Chatterjee, Proc. 2013 URSI Comm. B Intl. Symp. EM Theory, pp. 1054-1057, May 2013) of this hyperasymptotic technique was presented earlier. It was included there that the remainder term after optimal truncation was given by |RN(ξ)| ≤ C2e-2|ξ|, where the large parameter is ξ. Hence as ξ→∞ the remainder after optimal truncation is guaranteed to vanish. This feature suggests that the hyperasymptotic technique can yield best numerical accuracy which suggests that significantly more accurate computer simulation codes can be developed for solving electrically large problems.