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Hyperasymptotic solutions of second-order linear differential equations. I. (English) Zbl 0847.34060
The authors consider linear differential equations of second order, ${W}^{\text{'}\text{'}}+{a}_{1}{W}^{\text{'}}+{a}_{0}W=0$, where the coefficients ${a}_{0}$, ${a}_{1}$ are holomorphic in $\left\{z\in ℂ:|z|>ℝ\right\}$ and the point infinity is an irregular singularity of Poincaré rank one. Using the well-known Poincaré asymptotic expansions of the solutions for large $|z|$ they re-expand the remainder terms in series of generalized exponential integrals. Those re-expansions are called exponentially-improved or superasymptotic expansions. Further re-expansions are called hyperasymptotic expansions. The purpose of these papers is to improve the accuracy and extend the region of applicability of the Poincaré asymptotic expansions and to show how to achieve additional improvement by further re-expansions of remainder terms. They show that each step of the process reduces the estimate of the error term by the same exponentially-small factor. They also show how to ensure that the process is numerically stable. In the second part the first author gives alternative forms of these expansions that have some advantages, as well as disadvantages over the expansions of part one. They also include a numerical example and in a concluding section they make comparisons with earlier results. For details we have to refer to the paper.
Reviewer: G.Frank (Berlin)

##### MSC:
 34E05 Asymptotic expansions (ODE) 34M99 Differential equations in the complex domain 33B20 Incomplete beta and gamma functions 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 34A30 Linear ODE and systems, general