UNIFORM ASYMPTOTIC SOLUTIONS OF 2ND-ORDER LINEAR-DIFFERENTIAL EQUATIONS HAVING A SIMPLE POLE AND A COALESCING TURNING-POINT IN THE COMPLEX-PLANE

The asymptotic behavior, as a parameter u --> infinity, of solutions o f second-order linear differential equations having a simple pole and a coalescing turning point is considered. Uniform asymptotic approxima tions are constructed in terms of Whittaker's confluent hypergeometric functions, which are uniformly valid in a complex domain that include s both the pole and the turning point. Explicit error bounds for the d ifference between the approximations and the exact solutions are estab lished. These results extend previous real-variable results of F. W. J . Olver and J. J. Nestor to the complex plane.