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On the asymptotic expansions of solutions of an nth order linear differential equation

Published online by Cambridge University Press:  14 November 2011

R. B. Paris
R. B. Paris
Affiliation:
Association Euratom — CEA sur la Fusion, Département de Physique du Plasma et de la Fusion Contrôlée, Centre d'Etudes Nucléaires, 92260 Fontenay-aux-Roses, France
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Synopsis

The asymptotic expansions of solutions of a class of linear ordinary differential equations of arbitrary order n are investigated for large values of the independent variable z in the complex plane. Solutions are expressed in terms of Mellin-Barnes integrals and their asymptotic expansions are subsequently determined by means of the asymptotic theory of integral functions of the hypergeometric type. Three classes of solutions are considered: (i) solutions whose behaviour is either exponentially large or algebraic for |z|→∞ in different sectors of the z-plane, (ii) solutions which are even and odd functions of z when the order n of the differential equation is even and (iii) solutions which are exponentially damped as |z|→∞ in a certain sector of the z-plane.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 85 , Issue 1-2 , 1980 , pp. 15 - 57
Copyright
Copyright © Royal Society of Edinburgh 1980

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