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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. (English) Zbl 0919.34012

“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:

${w}^{\left(4\right)}\left(z\right)-3{w}^{\left(3\right)}\left(z\right)+\left(\frac{9}{4}+\frac{1}{2}{z}^{-2}\right){w}^{\left(2\right)}\left(z\right)-\left(3+\frac{3}{4}{z}^{-2}\right){w}^{\text{'}}\left(z\right)+\left(\frac{5}{4}+\frac{9}{16}{z}^{-2}\right)w\left(z\right)=0·$

As seen by the computations the results require the implementation of 47 terms of the asymptotic expansion of ${w}_{3}\left(z,n\right)$. The paper offers a significant new insight into this type of problem.

##### MSC:
 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) 34M99 Differential equations in the complex domain