  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
$\tau$--lepton. The method allows us to evaluate $R_{\tau}$ by integration over
the non-perturbative physical region, rather than by using Cauchy's 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 $\pi^2$ terms.
