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

Abstract

SynopsisThis paper deals with the nature of movable singularities of solutions of Emden's equationat 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 asBy 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 formwhere 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.