Simplified Asymptotic Solutions of Differential Equations Having Two Turning Points,with an Application to Legendre Functions

Asymptotic approximations of differential equations of the form are obtained, for the case , uniformly valid for real or complex values of x lying in bounded or unbounded intervals or regions. Here, , and have no singularities inside the interval or region under consideration, and does not vanish except at a critical point where it has a double zero. By an appropriate Liouville transformation, along with a perturbation of the new independent variable, uniform asymptotic approximations involving parabolic cylinder functions are obtained. These approximations are accompanied by strict and realistic error bounds, and the new theory is applied to obtain a uniform asymptotic approximation for the associated Legendre function with m and n large, with positive and bounded.