Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point

Abstract

The paper is concerned with an asymptotic behaviour of a contour integral with three critical points: a saddle point, a pole and a branch point. Two leading terms of a uniform asymptotic expansion of the integral have been effectively constructed. The expansion remains valid as its critical points approach one another or coalesce. In the special case that the saddle point is bounded away from one of the remaining critical points, or from both, the expansion reduces to simpler in form quasi-uniform and non-uniform expansions, respectively. With the aid of the non-uniform expansion the integral has been interpreted as describing three different waves, each one associated with a corresponding critical point.