Undefinability results in o-minimal expansions of the real numbers

We show that if beta is an element of R is not in the field generated by alpha(1),...,alpha(n), then no restriction of the function x(beta) to an interval is definable in (R, +, -, (.), 0, 1, <, x(alpha 1),..., x(alpha n)). We also prove that if the real and imaginary parts of a complex analytic function are definable in R-exp or in the expansion of (R) over bar (definitions in the text) by functions x(alpha), for irrational a, then they are already definable in (R) over bar. We conclude with some conjectures and open questions. (c) 2004 Elsevier B.V. All rights reserved.