Logarithmic-exponential power series

Lou Van Den Dries; Angus Macintyre; David Marker

doi:10.1112/S0024610797005437

Logarithmic-exponential power series

Lou Van Den Dries, Angus Macintyre, David Marker

Research output: Contribution to journal › Article › peer-review

Abstract

We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x)(log log x) is not asymptotic as x → + ∞ to a composition of semialgebraic functions, log and exp.

Original language	English (US)
Pages (from-to)	417-434
Number of pages	18
Journal	Journal of the London Mathematical Society
Volume	56
Issue number	3
DOIs	https://doi.org/10.1112/S0024610797005437
State	Published - Dec 1997

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Mathematics(all)

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10.1112/S0024610797005437

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abstract = "We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x)(log log x) is not asymptotic as x → + ∞ to a composition of semialgebraic functions, log and exp.",

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note = "Funding Information: The first author was partially supported by NSF grants DMS-9202833 and INT-9224546, the second author by an EPSRC Senior Research Fellowship, and the third author by NSF grants DMS-9306159 and INT-9224546 and an AMS Centennial Fellowship.",

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AU - Macintyre, Angus

AU - Marker, David

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N2 - We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x)(log log x) is not asymptotic as x → + ∞ to a composition of semialgebraic functions, log and exp.

AB - We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x)(log log x) is not asymptotic as x → + ∞ to a composition of semialgebraic functions, log and exp.

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Logarithmic-exponential power series

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