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8.—On a Class of Series Expansions in the Theory of Emden's Equation*

Published online by Cambridge University Press:  14 February 2012

Einar Hille
Einar Hille
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, N.M. 87106, U.S.A.
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Synopsis

This paper deals with the nature of movable singularities of solutions of Emden's equation

at which the solution becomes infinite. If m = 1 + 2/p with p > 1 an integer, then the solution becomes infinite at a given point x = c as

By the general theory of P. Painlevé on movable poles of solutions of non-linear second order differential equations this ‘pseudo-pole’ cannot actually be a pole of order p. Instead of a bona fide Laurent series at x = c we obtain a series expansion of the form

where Pn(t) is a polynomial in t of degree at most [n/(2p + 2)]. The object of this paper is to derive these series and to prove convergence for p = 2. In this case deg [P6m] is strictly equal to m. For other values of p, see Section 8, Addenda.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 71 , Issue 2 , 1973 , pp. 95 - 110
Copyright
Copyright © Royal Society of Edinburgh 1973

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References

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