Uniform Asymptotic Solutions of Second-Order Linear Differential Equations Having a Double Pole with Complex Exponent and a Coalescing Turning Point

Abstract

Second-order linear differential equations having a turning point and double pole with complex exponent are examined. The turning point is assumed to be a real continuous function of a parameter $\alpha $, and coalesces with the pole at the origin when $\alpha \to 0$. Asymptotic expansions for solutions, as a second parameter $u \to \infty$, are constructed in terms of Bessel functions of purely imaginary order. The asymptotic solutions are uniformly valid for the argument lying in both real and complex regions that include both the coalescing turning point and the pole. The theory is then applied to obtain uniform asymptotic expansions for Legendre functions of large real degree and purely imaginary order.