The NLO Jet Vertex for Mueller-Navelet and Forward Jets: the Quark Part

We calculate the next-to-leading corrections to the jet vertex which is relevant for the Mueller-Navelet-jets production in p-pbar collisions and for the forward jet cross section in e-p collisions. In this first part we present the results of the vertex for an incoming quark. Particular emphasis is given to the separation of the collinear divergent part and the central region of the produced gluon.