ANALYTIC CONTINUATIONS OF log-exp-ANALYTIC GERMS

We describe maximal, in a sense made precise, IL-analytic continuations of germs at +infinity of unary functions definable in the o-minimal structure R-an,R-exp on the Riemann surface L of the logarithm. As one application, we give an upper bound on the logarithmic-exponential complexity of the compositional inverse of an infinitely increasing such germ, in terms of its own logarithmic-exponential complexity and its level. As a second application, we strengthen Wilkie's theorem on definable complex analytic continuations of germs belonging to the residue field R-po(ly) of the valuation ring of all polynomially bounded definable germs.