Langer's method for a second order linear ordinary differential equation with a double pole

Abstract

If a second order ordinary differential equation has a simple or a double pole at a point zfl, then the standard Liouville-Green approximation could sometimes be valid near that point. In this paper we present an asymptotic series solution that is always valid near a double pole. A solution for that of a simple pole is also indicated. Asymptotic validity is proved.