On the remainders of asymptotic expansions of solutions of linear differential equations near irregular singular points of higher rank

We study linear ordinary differential equations near singular points of hig her Poincare rank r under the condition that the leading matrix has distinc t eigenvalues. It is well known that there are fundamental systems of forma l 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 poin t: N and the independent variable z are coupled such that Nz(r) is approxim ately constant. We show that the remainders are exponentially small under t his condition and how to choose the constant optimally.: Furthermore, we ob tain:precise asymptotic expansions for these remainders. As corollaries, we obtain well known asymptotic expansions for the coeffici ents in the formal solutions and limit formulas for Stokes' multipliers. The method of proof only uses functions in the original z - plane and its m ain tools are the Cauchy-Heine theorem and the saddle-point method.