Olde Daalhuis, A. B.; Olver, F. W. J. Hyperasymptotic solutions of second-order linear differential equations. I. (English) Zbl 0847.34060 Methods Appl. Anal. 2, No. 2, 173-197 (1995). The authors consider linear differential equations of second order, \(W''+ a_1 W'+ a_0 W=0\), where the coefficients \(a_0\), \(a_1\) are holomorphic in \(\{z\in \mathbb{C}: |z|> \mathbb{R}\}\) 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) Cited in 1 ReviewCited in 13 Documents MSC: 34E05 Asymptotic expansions of solutions to ordinary differential equations 34M99 Ordinary differential equations in the complex domain 33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) 33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\) 34A30 Linear ordinary differential equations and systems Keywords:linear differential equations of the second order; holomorphic; Poincaré asymptotic expansions; series of generalized exponential integrals; superasymptotic expansions PDFBibTeX XMLCite \textit{A. B. Olde Daalhuis} and \textit{F. W. J. Olver}, Methods Appl. Anal. 2, No. 2, 173--197 (1995; Zbl 0847.34060) Full Text: DOI Digital Library of Mathematical Functions: §10.17(v) Exponentially-Improved Expansions ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.40(iv) Exponentially-Improved Expansions ‣ §10.40 Asymptotic Expansions for Large Argument ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §13.29(i) Series Expansions ‣ §13.29 Methods of Computation ‣ Computation ‣ Chapter 13 Confluent Hypergeometric Functions §13.7(iii) Exponentially-Improved Expansion ‣ §13.7 Asymptotic Expansions for Large Argument ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions §2.11(v) Exponentially-Improved Expansions (continued) ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations §9.7(v) Exponentially-Improved Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions