Rosenlicht fields

Shackell, John

doi:10.1090/S0002-9947-1993-1085945-5

Rosenlicht fields
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Trans. Amer. Math. Soc. 335 (1993), 579-595 Request permission

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.

References

Michael Boshernitzan, An extension of Hardy’s class $L$ of “orders of infinity”, J. Analyse Math. 39 (1981), 235–255. MR 632463, DOI 10.1007/BF02803337
Michael Boshernitzan, Hardy fields and existence of transexponential functions, Aequationes Math. 30 (1986), no. 2-3, 258–280. MR 843667, DOI 10.1007/BF02189932

Fonctions d’une variable réele

Paul Du Bois-Reymond, Ueber asymptotische Werthe, infinitäre Approximationen und infinitäre Auflösung von Gleichungen, Math. Ann. 8 (1875), no. 3, 363–414 (German). MR 1509850, DOI 10.1007/BF01443187

Orders of infinity

Maxwell Rosenlicht, Growth properties of functions in Hardy fields, Trans. Amer. Math. Soc. 299 (1987), no. 1, 261–272. MR 869411, DOI 10.1090/S0002-9947-1987-0869411-2
Maxwell Rosenlicht, Hardy fields, J. Math. Anal. Appl. 93 (1983), no. 2, 297–311. MR 700146, DOI 10.1016/0022-247X(83)90175-0
Maxwell Rosenlicht, Rank change on adjoining real powers to Hardy fields, Trans. Amer. Math. Soc. 284 (1984), no. 2, 829–836. MR 743747, DOI 10.1090/S0002-9947-1984-0743747-5
Maxwell Rosenlicht, The rank of a Hardy field, Trans. Amer. Math. Soc. 280 (1983), no. 2, 659–671. MR 716843, DOI 10.1090/S0002-9947-1983-0716843-5
John Shackell, Growth estimates for exp-log functions, J. Symbolic Comput. 10 (1990), no. 6, 611–632. MR 1087982, DOI 10.1016/S0747-7171(08)80161-7

Rosenlicht fields and asymptotic forms