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The difference between fixed-order (FO) and contour-improved (CI)formulations of QCD perturbation theory limits the precision of the strongcoupling determined from the hadronic decay of the $\tau$ lepton. Recently,several attempts to understand the mathematical origin of the difference and tosolve it by subtracting the dominant infrared renormalon divergence have beenmade. Motivated by these studies, we review in this paper an improvedperturbative QCD expansion, defined some time ago, which also exploits therenormalons by means of a suitable conformal mapping of the Borel plane. Inparticular, we revisit the convergence of the new expansion, by completing theproof presented in a previous paper and showing that the domain of convergenceis larger than stated before. We also check the validity of the convergenceconditions for the Adler function and the CI and FO expansions of the $\tau$hadronic spectral function moments, and compare the approach based on conformalmapping with recent solutions to the CIPT-FOPT discrepancy proposed in theliterature.

Revisiting the convergence of the perturbative QCD expansions based on conformal mapping of the Borel plane