Expansion functions in perturbative QCD and the determination of $\alpha_s(M_\tau^2)$

The conventional series in powers of the coupling in perturbative QCD have zero radius of convergence and fail to reproduce the singularity of the QCD correlators like the Adler function at $\alpha_s=0$. Using the technique of conformal mapping of the Borel plane, combined with the "softening" of the leading singularities, we define a set of new expansion functions that resemble the expanded correlator and share the same singularity at zero coupling. Several different conformal mappings and different ways of implementing the known nature of the first branch-points of the Adler function in the Borel plane are investigated, in both the contour-improved (CI) and fixed-order (FO) versions of renormalization group resummation. We prove the remarkable convergence properties of a set of new CI expansions and use them for a determination of the strong coupling from the hadronic $\tau$ decay width. By taking the average upon this set, with a conservative treatment of the errors, we obtain $\alpha_s(M_\tau^2)= 0.3195^{+ 0.0189}_{- 0.0138}$.