An improved uniform approximation for diffraction integrals

Abstract

A saddle point formula for integrals whose integrands possess a multi-valued exponent, and one or more poles is derived. The result includes a number of error (or Fresnel) functions, which is equal to the product of the number of poles and the number of sheets possessed by the exponent's Riemann surface. Such formulae have previously been suggested on the basis of known exact results and transformations, both of which are specific to a particular exponent. In general, the multi-valuedness can be taken into account by a straightforward modification to a standard procedure, which would ordinarily yield only a single error function for each pole. In the context of diffraction theory, this type of approximation remains valid in the vicinity of an optical boundary, even in some cases where a wavefield is incident in a direction almost or exactly parallel to a sharp obstacle (grazing incidence).