Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank
type: Item
instance of: scholarly article
title: Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank
zbMATH Open document ID: 0899.34034
author: B. T. M. Murphy
author: A. D. Wood
published in: Methods and Applications of Analysis
publication date: 1998-11-15T00:00:00Z
review text: The authors develop a hyperasymptotic expansion for solutions to general linear homogeneous differential equations of second order, \[ {d^2W\over dz^2}+ f(z) {dW\over dz}+ g(z) W= 0. \] The problem studied has an irregular singularity of rank \(r\) at infinity, whereby the functions \(f\) and \(g\) can be expanded in power series about infinity of the form \[ f(z)= z^{r- 1} \sum^\infty_{s= 0} {f_s\over z^s}\quad\text{and} \quad g(z)= z^{2r- 2} \sum^\infty_{s= 0} {g_s\over z^s}, \] which converge in an open annulus \(| z|> a\). The techniques implemented are very clever transformations applied at various stages throughout the process. The expansions are in terms of certain integrals which are generalizations of the hyperterminant integrals (developed in other papers) for the rank one case. The transformations implemented result in solutions having remainders which are further studied in greater depth. These remainders are minimized by determining the optimal number of terms needed in their expansions to be used with the single standard Poincaré asymptotic series. The paper offers the reader a very concise and clear treatment for this problem.
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34M99
zbMATH DE Number: 1138455
zbMATH Keywords: exponential improvement hyperasymptotics
zbMATH Keywords: irregular singularity
zbMATH Keywords: Stokes multipliers
zbMATH Keywords: remainder terms
zbMATH Keywords: hyperasymptotic expansion
zbMATH Keywords: linear homogeneous differential equations
zbMATH Keywords: transformations
zbMATH Keywords: hyperterminant integrals
reviewed by: John Schmeelk
MaRDI profile type: Publication
full work available at URL: maa.1997.v4.n3.a2
OpenAlex ID: W2650688143
DOI: 10.4310/MAA.1997.V4.N3.A2
description: scientific article; zbMATH DE number 1138455