Approximation of solutions of y″=ϕ(x,ε)y with several transition points and moving singularities

Given a second-order linear differential equation y″=ϕ(x,ε)y. Let ϕ(x,ε) be a meromorphic real-valued function of two independent variables x and ε. x is allowed to vary on an interval [a,b] and ε varies on (0,ε₀]. The approximation of its solutions, as well as of their derivatives as ε→0⁺ are provided for the entire interval [a,b] in the presence of several-coalescing transition points. Approximations on closed subintervals of [a,b] are provided. The end points of those subintervals are allowed to coincide with turning points.

The approximations given could replace a nonrigorous analysis in the literature which had similar goals and in which an error was found. The formulas could prove to be useful for numerical implementation in problems of wave propagation in inhomogeneous medium.

Our methods avoid analytic continuation, Langer transformations and special functions.