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

摘要

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.