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Transactions of the American Mathematical Society

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

 

 

Fine structure of the integral exponential functions below $ 2\sp {2\sp x}$


Author: Bernd I. Dahn
Journal: Trans. Amer. Math. Soc. 297 (1986), 707-716
MSC: Primary 26A12; Secondary 03D20, 04A99
DOI: https://doi.org/10.1090/S0002-9947-1986-0854094-7
MathSciNet review: 854094
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Abstract: Integral exponential functions are the members of the least class of real functions containing $ 1$, the identity function, and closed under addition, multiplication, and binary exponentiation sending $ f$ and $ g$ to $ {f^g}$. This class is known to be wellordered by the relation of eventual dominance. It is shown that for each natural number $ n$ the order type of the integral exponential functions below $ {2^{{x^n}}}$ (below $ {x^{{x^n}}}$) is exactly $ {\omega ^{{\omega ^{2n - 1}}}}$ ( $ {\omega ^{{\omega ^{2n}}}}$ respectively). The proof, using iterated asymptotic expansions, contains also a new proof that integral exponential functions below $ {2^{{2^x}}}$ are wellordered.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0854094-7
Article copyright: © Copyright 1986 American Mathematical Society