We investigate expansions of the ordered field of real numbers equipped with a family of real power functions. We show in particular that the theory of the ordered field of real numbers augmented by all restricted analytic functions and all real power functions admits elimination of quantifiers and has a universal axiomatization. We derive that every function of one variable definable in this structure, not ultimately identically 0, is asymptotic at + ∞ to a real function of the form x (...) cxr, c ≠ 0; in particular, this structure is polynomially bounded. Furthermore, given any definable function f:U → with U open in n, if α ε U and f is infinitely differentiable at α, then f is real analytic in a neighborhood of α. (shrink)

We prove that no restriction of the sine function to any (open and nonempty) interval is definable in $\langle\mathbf{R}, +, \cdot, , and that no restriction of the exponential function to an (open and nonempty) interval is definable in $\langle \mathbf{R}, +, \cdot, , where $\sin_0(x) = \sin(x)$ for x ∈ [ -π,π], and $\sin_0(x) = 0$ for all $x \not\in\lbrack -\pi,\pi\rbrack$.