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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 (4) (z)-3w (3) (z)+9 4 + 1 2 z -2 w (2) (z)-3 + 3 4 z -2 w ' (z)+5 4 + 9 16 z -2 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.

MSC:
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
34M99Differential equations in the complex domain