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Journal of the London Mathematical Society 1997 56(3):417-434; doi:10.1112/S0024610797005437
© 1997 by London Mathematical Society
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© The London Mathematical Society

Logarithmic-Exponential Power Series

Lou van den Dries, Angus Macintyre and David Marker

University of Illinois Urbana, Illinois 61801-2917, USA. E-mail: vddries{at}math.uiuc.edu
Oxford University Oxford. E-mail: ajm{at}vax.ox.ac.uk
Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago Chicago, Illinois 60607-7045, USA. E-mail: marker{at}math.uic.edu

Received 22 February 1995. Revision received 26 July 1995.

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->+{infty} to a composition of semialgebraic functions, log and exp.


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