Exponentially improved asymptotic solutions of ordinary differential equations. II: Irregular singularities of rank one.

*(English)*Zbl 0809.34007The authors consider the asymptotic expansions of solutions of the general homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one. The re- expansions for the optimal remainder terms in these asymptotic series are given. The re-expansions are obtained in the terms of generalized exponential integrals. The explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are also constructed.

{For part I see [the first author, SIAM J. Math. Anal. 24, 756-767 (1993; Zbl 0779.34048)]}.

Reviewer: S.Mazanik (Minsk)

##### MSC:

34M35 | Singularities, monodromy, local behavior of solutions, normal forms |

34A25 | Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) |

34E05 | Asymptotic expansions (ODE) |