On the remainders of the asymptotic expansion of solutions of differential equations near irregular singular points

Abstract

We consider systems of linear differential equations with an irregular singular point of Poincare rank 1 at infinity. It is well known that there is a fundamental system of formal solution vectors and, for each halfplane, a fundamental system of actual solution vectors having the formal ones as asymptotic expansions. These asymptotic expansions (in the sense of Poincare) describe the behavior of the actual solutions as theindependent variable zgrows indefinitely, but give no precise error bounds for a given z, if the asymptotic series are truncated after Nterms. In this paper we show that for large values of |z| the best choice of Nis proportional to |z| and that the resulting error terms are exponentially small.