  Angle-resolved photoemission spectroscopy (ARPES) is used to study the
spectral function of the optimally doped high-T$_c$ superconductor
(Bi,Pb)$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ in the vicinity of the antinodal point
in the superconducting state. Using a parameterized self-energy function, it
was possible to describe both the coherent and the incoherent spectral weight
of the bonding and the antibonding band. The renormalization effects can be
assigned to a very strong coupling to the magnetic resonance mode and at higher
energies to a bandwidth renormalization by a factor of two, probably caused by
a coupling to a continuum. The present reevaluation of the ARPES data allows to
come to a more reliable determination of the value of the coupling strength of
the charge carriers to the mode. The experimental results for the dressing of
the charge carriers are compared to theoretical models.
