Hyperasymptotic solutions of second-order linear differential equations I

Abstract

A sequence of re-expansions is developed for the remainder terms in the well-known Poincare series expansions of the solutions of homogeneous linear differential equations of the second order in the neighborhood of an irregular singularity of rank one. These re-expansions are in series of repeated integrals of the generalized exponential integral, and the coefficients in these series are the same as those of the original Poincare expansions. Each step of the process reduces the estimate of the error term by the same exponentially-small factor, while increasing the region of validity.