Perturbative expansions in the inclusive decay of the tau-lepton

A comparative analysis is performed for various forms of perturbative expansions in spacelike and timelike regions. As applied to the inclusive decay of the tau-lepton, comparison is given for the results derived within the framework of the standard perturbation theory and the analytic approach that modernizes perturbative expansions so that the new approximations reflect basic principles of the theory: renormalization invariance, spectrality, and causality. Advantages and self-consistency of the analytic approach in describing tau-lepton decay are demonstrated.