We consider the asymptotic behavior of the incomplete gamma functions fl(\Gammaa; \Gammaz) and \Gamma(\Gammaa; \Gammaz) as a !1. Uniform expansions are needed to describe the transition area z a, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for fl(a; z) and \Gamma(a; z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final section.

The article surveys the application of complex ray theory to the scalar Helmholtz equation in two dimensions. The first objective is to motivate a framework within which complex rays may be used to make predictions about wavefields in a wide variety of geometrical configurations. A crucial ingredient in this framework is the role played by Stokes' phenomenon in determining the regions of existence of complex rays. The identification of the Stokes surfaces emerges as a key step in the approximation procedure, and this leads to the consideration of the many characterisations of Stokes surfaces, including the adaptation and application of recent developments in exponential asymptotics to the complex wkb expansion of these wavefields. Examples will be given for several cases of physical importance. 1 Introduction The objective of this paper is to lay down a systematic complexified theory of monochromatic high-frequency wave propagation. The computational and analytical usefulness of such ...

A singularly perturbed linear partial differential equation motivated by the geometrical model for crystal growth is considered. A steepest descent analysis of the Fourier transform solution identifies asymptotic contributions from saddle points, end points and poles, and the Stokes lines across which these may be switched on and off. These results are then derived directly from the equation by optimally truncating the naïve perturbation expansion and smoothing the Stokes discontinuities. The analysis reveals two new types of Stokes switching: a higher-order Stokes line which is a Stokes line in the approximation of the late terms of the asymptotic series, and which switches on or off Stokes lines themselves; and a second-generation Stokes line, in which a subdominant exponential switched on at a primary Stokes line is itself responsible for switching on another smaller exponential. The ‘new’ Stokes lines discussed by Berk et al. (1982) are second-generation Stokes lines, while the ‘vanishing ’ Stokes lines discussed by Aoki et al. (1998) are switched off by a higher-order Stokes line.

this paper is organized as follows. We start with the perturbation results of Wadee et al. (1997) and point out the flaw which renders the approach unable to account for all but the primary localized solutions. In x3 we detail how the procedure must be amended in the light of this finding and, crucially, it is shown how in the vicinity of singularities in the wavenumber domain the procedure of matched asymptotics reveals the existence of exponentially-small terms missed by the conventional analysis. The ramifications of such terms are that only symmetric or antisymmetric solutions bifurcate from the point of expansion P = P

The paper discusses asymptotic methods for integrals, in particular uniform approximations. We discuss several examples, where we apply the results to Tricomi's \Psi\Gammafunction, in particular we consider an expansion of Tricomi-Carlitz polynomials in terms of Hermite polynomials. We mention other recent expansions for orthogonal polynomials that do not satisfy a differential equation, and for which methods based on integral representations produce powerful uniform asymptotic expansions. 1991 Mathematics Subject Classification: 41A60, 33B20, 33C10, 33C45, 11B73, 30E15. Keywords and Phrases: uniform asymptotic expansions, Tricomi's \Psi\Gammafunction, Kummer functions, confluent hypergeometric functions, Whittaker functions, Hermite polynomials, Tricomi-Carlitz polynomials. Note: Work carried out under project MAS2.8 Exploratory research. Extended version of a paper presented at the Conference Tricomi's Ideas and Contemporary Applied Mathematics to celebrate the 100th anniversary o...

A stable recurrence for the incomplete

for the degenerated asymptotic solutions of the fourth Painlevé equation