On asymptotics of a second order linear ODE with a turning-regular singular point
This paper deals with the equation \[ \varepsilon^2 \frac{d^2y}{dx^2} -\Bigl(x^m- \frac{\varepsilon}{x} \Bigr) y=0 \] in a neighbourhood of the regular singular point \(x=0,\) where \(\varepsilon >0\) is a small parameter and \(m\) is a positive integer. The outer solution for \(K\varepsilon ^{1/(m+1)} \leq |x|\leq x_0\) and the inner solution for \(0<\varepsilon^{-1/(m+1)}|x| <\infty\) are given in a sector. For the inner solution a canonical domain and Stokes curves are discussed, and a matching matrix connecting the outer and inner solutions is obtained.
scientific article; zbMATH DE number 5776104