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Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank. (English) Zbl 0899.34034

The authors develop a hyperasymptotic expansion for solutions to general linear homogeneous differential equations of second order,

$\frac{{d}^{2}W}{d{z}^{2}}+f\left(z\right)\frac{dW}{dz}+g\left(z\right)W=0·$

The problem studied has an irregular singularity of rank $r$ at infinity, whereby the functions $f$ and $g$ can be expanded in power series about infinity of the form

$f\left(z\right)={z}^{r-1}\sum _{s=0}^{\infty }\frac{{f}_{s}}{{z}^{s}}\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}g\left(z\right)={z}^{2r-2}\sum _{s=0}^{\infty }\frac{{g}_{s}}{{z}^{s}},$

which converge in an open annulus $|z|>a$. The techniques implemented are very clever transformations applied at various stages throughout the process. The expansions are in terms of certain integrals which are generalizations of the hyperterminant integrals (developed in other papers) for the rank one case. The transformations implemented result in solutions having remainders which are further studied in greater depth. These remainders are minimized by determining the optimal number of terms needed in their expansions to be used with the single standard Poincaré asymptotic series. The paper offers the reader a very concise and clear treatment for this problem.

MSC:
 34E05 Asymptotic expansions (ODE) 34M99 Differential equations in the complex domain