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 αs = 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 τ decay width. By taking the average upon this set, with a conservative treatment of the errors, we obtain αs(M 2 τ ) = 0.3195 +0.0189 −0.0138 . PACS numbers: 12.38.Bx, 12.38.Cy 1 In the so-called "order-dependent" conformal mappings, which were defined also in the coupling plane [12,13], the singularity is shifted away from the origin by a certain amount at each finiteorder, and tends to the origin only when an infinite number of terms are considered.

The technique of conformal mapping is applied to enlarge the convergence domain of the Borel series and to accelerate the convergence of Borel-summed Green functions in perturbative QCD. We use the optimal mapping, which takes into account the location of all the singularities of the Borel transform as well as the present knowledge about its behavior near the first branch points. The determination of ␣ s (m 2 ) from the hadronic decay rate of the lepton is discussed as an illustration of the method.

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