On the Remainders of Asymptotic Expansions of Solutions of Linear Diierential Equations near Irregular Singular Points of Higher Rank Formal Solutions

  • Reinhard Hoeppner
  • Published 1996

Abstract

We study linear ordinary diierential equations near singular points of higher Poincar e rank r under the condition that the leading matrix has distinct eigenvalues. It is well known that there are fundamental systems of formal vector solutions and that in certain sectors, there are actual solutions having those as asymptotic expansions. We study the corresponding remainders, in particular if the truncation point N and the independent variable z are coupled such that Nz r is approximately constant. We show that the remainders are exponentially small under this condition and how to choose the constant optimally. Furthermore , we obtain precise asymptotic expansions for these remainders. As corollaries, we obtain well known asymptotic expansions for the coeecients in the formal solutions and limit formulas for Stokes' multipliers. The method of proof only uses functions in the original z-plane and its main tools are the Cauchy{Heine theorem and the saddle-point method. We consider a system of n (2) homogeneous linear ordinary diierential equations z r+1 y 0 = A(z)y; A(z) = 1 X m=0 A m z m (jzj < R) (0.1) (the A m are n n-matrices) with a singularity of Poincar e rank r 1 at z = 0. We assume that the leading matrix A 0 has distinct eigenvalues 1 ; : : : ; n. It is well-known that there is a formal fundamental matrix of (0.1) of the form ^

Cite this paper

@inproceedings{Hoeppner1996OnTR, title={On the Remainders of Asymptotic Expansions of Solutions of Linear Diierential Equations near Irregular Singular Points of Higher Rank Formal Solutions}, author={Reinhard Hoeppner}, year={1996} }