  This paper is devoted to the study of semigroups of composition operators and
semigroups of holomorphic mappings. We establish conditions under which these
semigroups can be extended in their parameter to sector given a priori. We show
that the size of this sector can be controlled by the image properties of the
infinitesimal generator, or, equivalently, by the geometry of the so-called
associated planar domain. We also give a complete characterization of all
composition operators acting on the Hardy space $H^p$ on the right half-plane.
