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Rosenlicht fields


Author: John Shackell
Journal: Trans. Amer. Math. Soc. 335 (1993), 579-595
MSC: Primary 12H05; Secondary 26A12, 26E99, 34E99
DOI: https://doi.org/10.1090/S0002-9947-1993-1085945-5
MathSciNet review: 1085945
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Abstract: Let $ \phi $ satisfy an algebraic differential equation over $ {\mathbf{R}}$. We show that if $ \phi $ also belongs to a Hardy field, it possesses an asymptotic form which must be one of a restricted number of types. The types depend only on the order of the differential equation. For a particular equation the types are still more restricted. In some cases one can conclude that no solution of the given equation lies in a Hardy field, and in others that a particular asymptotic form is the only possibility for such solutions. This therefore gives a new method for obtaining asymptotic solutions of nonlinear differential equations. The techniques used are in part derived from the work of Rosenlicht in Hardy fields.


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  • [1] M. Boshernitzan, An extension of Hardy's class $ L$ of 'orders of infinity', J. Analyse Math. 39 (1981), 235-255. MR 632463 (82m:26002)
  • [2] -, Hardy fields, existence of transexponential functions and the hypertranscendence of solution to $ g(g(x)) = {e^x}$, Aequationes Math. 30 (1986), 258-280. MR 843667 (88b:26003)
  • [3] N. Bourbaki, Fonctions d'une variable réele, Ch. V, 2nd ed. Hermann, Paris, 1961.
  • [4] P. du Bois-Reymond, Ueber asymptotische Werthe, infinitäre Approximationen und infinitäre Auflösung von Gleichungen, Math. Ann. 8 (1875), 362-414. MR 1509850
  • [5] G. H. Hardy, Orders of infinity, Cambridge Univ. Press, Cambridge, 1910.
  • [6] M. Rosenlicht, Growth properties of functions in Hardy fields, Trans. Amer. Math. Soc. 299 (1987), 261-272. MR 869411 (88b:12010)
  • [7] -, Hardy fields, J. Math. Anal. Appl. 93 (1983), 297-311. MR 700146 (85d:12001)
  • [8] -, Rank change on adjoining real powers to Hardy fields, Trans. Amer. Math. Soc. 284 (1984), 829-836. MR 743747 (85i:12008)
  • [9] -, The rank of a Hardy field, Trans. Amer. Math. Soc. 280 (1983), 659-671. MR 716843 (85d:12002)
  • [10] J. R. Shackell, Growth estimates for $ \operatorname{Exp}$-$ \operatorname{Log}$ functions, J. Symbolic Comp. 10 (1990), 611-632. MR 1087982 (92c:26001)
  • [11] -, Rosenlicht fields and asymptotic forms, Technical Report, University of Kent at Canterbury, 1990.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1085945-5
Keywords: Hardy fields, asymptotic expansions, orders of growth
Article copyright: © Copyright 1993 American Mathematical Society

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