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

Abstract

We study linear ordinary differential equations near singular points of higher Poincare 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 rolutions 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 Nzr 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 coefficients 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.