Analytic properties of the QCD running coupling constant and τ decay

A non-perturbative expansion method which gives a well-defined analytic continuation of the running coupling constant from the spacelike to the timelike region is applied to the inclusive semileptonic decay of the τ-lepton. The method allows us to evaluate Rτ by integration over the non-perturbative physical region, rather than by using Cauchyʹs theorem, and hence to incorporate threshold effects in a very direct way. Within our framework the difference between the effective coupling constants in the timelike and spacelike domains can be substantial and is not simply a matter of the standard π2 terms.