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Zbl 0919.34012
Olde Daalhuis, A.B.
Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one.
(English)
[J] Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 454, No.1968, 1-29 (1998). ISSN 1364-5021; ISSN 1471-2946/e

A sequence of re-expansions is developed for the remainder terms in the well-known PoincarĂ© 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 PoincarĂ© 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.\par 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.
[J.Schmeelk (Richmond)]
MSC 2000:
*34A25 Analytical theory of ODE
34M99 Differential equations in the complex domain

Keywords: asymptotic expansion; exponential improvement; Borel-Laplace transform; hyperasymptotics; differential equations; Stokes multiplier

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