Undefinability results in o-minimal expansions of the real numbers

We show that if β∈R is not in the field generated by α1,…,αn, then no restriction of the function xβ to an interval is definable in 〈R,+,−,⋅,0,1,<,xα1,…,xαn〉. We also prove that if the real and imaginary parts of a complex analytic function are definable in Rexp or in the expansion of R̄ (definitions in the text) by functions xα, for irrational α, then they are already definable in R̄. We conclude with some conjectures and open questions.