Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions

The asymptotic behaviour, as a parameter $u \to \infty $, of solutions of second-order linear differential equations with a turning point and a regular (double pole) singularity is considered. It is shown that the solutions can be approximated by expressions involving Bessel functions in a region which includes both the turning point and the singularity. Explicit error bounds for the difference between the approximations and the exact solutions are established. The theory is applied to find uniform asymptotic expansions for Legendre functions.