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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The rank of a Hardy field
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by Maxwell Rosenlicht PDF
Trans. Amer. Math. Soc. 280 (1983), 659-671 Request permission


A Hardy field is a field of germs of real-valued functions on positive half-lines that is closed under differentiation. Its rank is the rank of the associated ordered abelian group, the value group of the canonical valuation of the field. The properties of this rank are worked out, its relevance to asymptotic expansions indicated, examples provided, and applications given to the order of growth of solutions of ordinary differential equations.
  • Steven B. Bank and Robert P. Kaufman, A note on Hölder’s theorem concerning the gamma function, Math. Ann. 232 (1978), no. 2, 115–120. MR 477180, DOI 10.1007/BF01421399
  • Michael Boshernitzan, New “orders of infinity”, J. Analyse Math. 41 (1982), 130–167. MR 687948, DOI 10.1007/BF02803397
  • Michael Boshernitzan, “Orders of infinity” generated by difference equations, Amer. J. Math. 106 (1984), no. 5, 1067–1089. MR 761579, DOI 10.2307/2374273
  • N. Bourbaki, Fonctions d’une variable réele. Chapitre VII ("La Fonction Gamma"), 2nd ed., Hermann, Paris, 1961. O. Hölder, Über die Eigenschaft der $\Gamma$-Funktion, keiner algebraischen Differentialgleichung zu genügen, Math. Ann. 28 (1887), 1-13.
  • F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
  • Maxwell Rosenlicht, Differential valuations, Pacific J. Math. 86 (1980), no. 1, 301–319. MR 586879
  • Maxwell Rosenlicht, On the value group of a differential valuation, Amer. J. Math. 101 (1979), no. 1, 258–266. MR 527836, DOI 10.2307/2373949
  • Maxwell Rosenlicht, Hardy fields, J. Math. Anal. Appl. 93 (1983), no. 2, 297–311. MR 700146, DOI 10.1016/0022-247X(83)90175-0
  • M. Singer, Asymptotic behavior of solutions of differential equations and Hardy fields, preliminary report, SUNY at Stony Brook, 1976 (unpublished). L. van den Dries, Bounding the rate of growth of solutions of algebraic differential equations and exponential equations in Hardy fields (to appear).
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 280 (1983), 659-671
  • MSC: Primary 12D15; Secondary 12H05, 13N05, 26A12, 34E05, 41A60
  • DOI:
  • MathSciNet review: 716843