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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

References to Literature

[1]Emden, R., 1907. Gaskugeln. Anwendungen der mechanischen Wärmetheorie auf Kosmologie und meteorologischen Probleme. Leipzig: Teubner.Google Scholar
[2]Fermi, E., 1927. Un metodo statistico per la determinazione di alcune proprietà dell'atome. Atti Accad. Naz. Lincei Rc. (CI. Sci. Fis. Mat. Nat.) 6, 602–607.Google Scholar
[3]Hille, E., 1969. Lectures on Ordinary Differential Equations. Reading, Mass: Addison-Wesley.Google Scholar
[4]Hille, E., 1969. On the Thomas-Fermi equation. Proc. Natn. Acad. Sci. U.S.A., 62, 7–10.Google Scholar
[5]Hille, E., 1970. Some aspects of the Thomas-Fermi equation. J. Analyse Math., 23, 147–170.Google Scholar
[6]Hille, E., 1970. Aspects of Emden's equation. J. Fac. Sci. Tokyo Univ. (I), 17, 11–30.Google Scholar
[7]Hille, E., 1972. Pseudo-poles in the theory of Emden's equation. Proc. Natn. Acad. Sci. U.S.A., 69, 1271–1272.CrossRefGoogle Scholar
[8]Ince, E. L., 1927 (1944). Ordinary Differential Equations. New York: Dover reprint.Google Scholar
[9]Thomas, L. H., 1927. The calculation of atomic fields. Proc. Camb. Phil. Soc. Math. Phys. Sci., 23, 542–548.Google Scholar
[10]Malmquist, J., 1921. Sur les points singuliers des équations différentielles, I, II. Ark. Mat. Astr. Fys., 15 (3), (27).Google Scholar
[11]Avakumovic, V. G., 1947–1948. Sur l'équation différentielle de Thomas-Fermi I, II. Pubis Inst. Math. Belgr., 1, 101–113; 2, 223–235.Google Scholar
[12]Avakumovic, V. G., 1948. O diferencijalnim jednačinama Thomas-Fermi-eva tipa. Egzistencija integrala. Glas. Srp. Akad. Nauk, 191, 163–185.Google Scholar
[13]Karamata, J., 1947. Sur l'application des théorèmes de nature tauberienne à l'étude des équations différentielles. Pubis Inst. Math. Belgr., 1, 93–97.Google Scholar
[14]Mihailović, M. V., 1950. Sur l'intégrale de l'équation différentielle de Thomas-Fermi autour du point x = 0, y = 1. Pubis Inst. Math. Belgr., 3, 259–270.Google Scholar
[15]Marić, V., 1955. O asimptotskom ponaišanju integrala jadne klase nelinearnih diferencijalnih jednačina drugog reda. Zborn. Rad. Mat. Inst., 4, 27–40.Google Scholar
[16]Karamata, J. and Marić, V., 1960. On some solutions of the differential equation y″(x) = f(x)y λ(x). Rev. Fac. Arts Nat. Sci., Novi Sad, 5, 415–422.Google Scholar