On transasymptotic expansions of o-minimal germs

Given an o-minimal expansion RA of the real ordered field, generated by a generalized quasianalytic class A, we construct an explicit truncation closed ordered differential field embedding of the Hardy field of the expansion RA,exp of RA by the unrestricted exponential function, into the field T of transseries. We use this to prove some non-definability results. In particular, we show that the restriction to the positive half-line of Euler's Gamma function is not definable in the structure R_an*,exp, generated by all convergent generalized power series and the exponential function, thus establishing the non-interdefinability of the restrictions to a neighbourhood of +∞ of Euler's Gamma and of the Riemann Zeta function.