Aleksandrov-Clark measures and semigroups of analytic functions in the unit disc

In this paper we prove a formula describing the infinitesimal generator of a continuous semigroup (\v_t) of holomorphic self-maps of the unit disc with respect to a boundary regular fixed point. The result is based on Alexandrov-Clark measures techniques. In particular we prove that the Alexandrov-Clark measure of (\v_t) at a boundary regular fixed points is differentiable (in the weak$^\ast$-topology) with respect to $t$.