Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one

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 higher order in the neighbourhood of an irregular singularity of rank one. These re–expansions are a series whose terms are a product of Stokes multipliers, coefficients of the original Poincare series expansions, and certain multiple integrals, the so–called hyperterminants. Each step of the process reduces the estimate of the error term by an exponentially small factor. The method of this paper is based on the Borel–Laplace transform, which makes it applicable to other problems. At the end of the paper the method is applied to integrals with saddles. Also, a powerful new method is presented to compute the Stokes multipliers. A numerical example is included.

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