  Employing the linear response Kubo formalism as implemented in a fully
relativistic multiple-scattering Korringa-Kohn-Rostoker Green function method a
systematic first-principles study based on density-functional theory (DFT) of
the anomalous Hall conductivity (AHC) of the 3$d$-transition-metals Fe, Co and
Ni is presented. To account for the temperature dependence of the AHC an
alloy-analogy for a set of thermal lattice displacements acting as a scattering
mechanism is used which is subsequently solved using the coherent potential
approximation. Further, impurity scattering has been considered to elucidate
the importance of an additional possible contribution to the AHC that might be
present in experiment. The impact of correlations beyond the local spin-density
approximation to the exchange-correlation functional in DFT is studied within
the LSDA+$U$ approach. It is shown that both, the inclusion of correlations and
thermal lattice vibrations, is needed to give a material-specific description
of the AHC in transition-metals.
