Evaluation of the optical conductivity tensor in terms of contour integrations

For the case of finite life-time broadening the standard Kubo-formula for the optical conductivity tensor is rederived in terms of Green’s functions by using contour integrations, whereby finite temperatures are accounted for by using the Fermi-Dirac distribution function. For zero life-time broadening, the present formalism is related to expressions well-known in the literature. Numerical aspects of how to calculate the corresponding contour integrals are also outlined.