Transformation to canonical form for uniform asymptotic expansions (Q913989)

Uniform asymptotic expansions of contour integrals are often obtained by transforming the dominant part of the integrand into a canonical form, for instance into a phase function in the form of a polynomial of low degree. The cubic polynomial is used for obtaining an asymptotic expansion in terms of Airy functions. The present paper generalizes the existing results in two directions: the phase function may have a pole in the domain of interest and a logarithmic singularity; the singularities may collide, under the influence of one or two parameters, with saddle points. The mapping behind this transformation is investigated by a process of successive iteration. This general result is quite important in the study of uniform expansions of integrals.