Uniform Asymptotic Solutions of Second-Order Linear Differential Equations Having a Simple Pole and a Coalescing Turning Point in the Complex Plane

Abstract

The asymptotic behavior, as a parameter $u \to \infty $, of solutions of second-order linear differential equations having a simple pole and a coalescing turning point is considered. Uniform asymptotic approximations are constructed in terms of Whittaker’s confluent hypergeometric functions, which are uniformly valid in a complex domain that includes both the pole and the turning point. Explicit error bounds for the difference between the approximations and the exact solutions are established. These results extend previous real-variable results of F. W. J. Olver and J. J. Nestor to the complex plane.