Uniform asymptotic solutions of second-order linear ordinary differential equations with singular points. I. Formal theory

Abstract

A formal uniform asymptotic solution of the differential equation d 2u dz 2 + λ 2 R ̂ (z, λ) u = 0, z ϵ D, for ¦λ¦ large, when D contains any number and order of poles and turning points, is constructed. The development is a generalization of Langer's method. An asymptotic relation, between the Wronskian of any pair of linearly independent solutions of the differential equation and the Wronskian of the pair of corresponding solutions of the comparison equation, is derived. A formal uniform asymptotic solution of a system of two first-order ordinary differential equation is also obtained.