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Exponentially Improved Asymptotic Solutions of Ordinary Differential Equations. II Irregular Singularities of Rank One

A. B. Olde Daalhuis and F. W. J. Olver
Proceedings: Mathematical and Physical Sciences
Vol. 445, No. 1923 (Apr. 8, 1994), pp. 39-56
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/52551
Page Count: 18
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Exponentially Improved Asymptotic Solutions of Ordinary Differential Equations. II Irregular Singularities of Rank One
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Abstract

Re-expansions are found for the optimal remainder terms in the well-known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one. The re-expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions, as well being considerably more accurate and providing a smooth interpretation of the Stokes phenomenon. They are also of strikingly simple form. In addition, explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained.

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