Limit at resonances of linearizations of some complex analytic dynamical systems

We consider the behaviour near resonances of linearizations of germs of holomorphic diffeomorphisms of $({\Bbb C},0)$ and of the semi-standard map.We prove that for each resonance there exists a suitable blow-up of the Taylor series of the linearization under which it converges uniformly to an analytic function as the multiplier, or rotation number, tends non-tangentially to the resonance. This limit function is explicitly computed and related to questions of formal classification, both for the case of germs and for the case of the semi-standard map