Olde Daalhuis, A. B. Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. (English) Zbl 0919.34012 Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 454, No. 1968, 1-29 (1998). “A sequence of re-expansions is developed for the remainder terms in the well-known Poincaré series expansions of the solutions to homogeneous linear differential equations of higher order in the neighborhood of an irregular singularity of rank one. These re-expansions are a series whose terms are a product of Stokes multipliers, coefficients of the original Poincaré series expansions, and certain multiple integrals, the so-called hyperterminants. Each step of the process reduces the estimate of the error term by an exponentially small factor.The method is based on the Borel-Laplace transform, which makes it applicable to other problems. The method is applied to integrals with saddles. A powerful new method is presented to compute the Stokes multipliers. A numerical example is included.” The example demonstrates the power of the procedures developed within the paper. The basic example given is the equation as follows: \[ w^{(4)}(z)-3w^{(3)}(z)+ \Bigl(\textstyle{9\over 4} +\textstyle{1\over 2} z^{-2}\Bigr)w^{(2)}(z)-\Bigl(3 +\textstyle{3\over 4} z^{-2}\Bigr)w'(z)+\Bigl(\textstyle{5\over 4}+\textstyle {9\over 16}z^{-2}\Bigr) w(z)=0. \] As seen by the computations the results require the implementation of 47 terms of the asymptotic expansion of \(w_3 (z,n)\). The paper offers a significant new insight into this type of problem. Reviewer: J.Schmeelk (Richmond) Cited in 17 Documents MSC: 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 34M99 Ordinary differential equations in the complex domain Keywords:asymptotic expansion; exponential improvement; Borel-Laplace transform; hyperasymptotics; differential equations; Stokes multiplier PDFBibTeX XMLCite \textit{A. B. Olde Daalhuis}, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 454, No. 1968, 1--29 (1998; Zbl 0919.34012) Full Text: DOI Digital Library of Mathematical Functions: §2.11(v) Exponentially-Improved Expansions (continued) ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations