  The power corrections in the Operator Product Expansion (OPE) of QCD
correlators can be viewed mathematically as an illustration of the transseries
concept, which allows to recover a function from its asymptotic divergent
expansion. Alternatively, starting from the divergent behavior of the
perturbative QCD encoded in the singularities in the Borel plane, a modified
expansion can be defined by means of the conformal mapping of this plane. A
comparison of the two approaches concerning their ability to recover
nonperturbative properties of the true correlator was not explored up to now.
In the present paper, we make a first attempt to investigate this problem. We
use for illustration the Adler function and observables expressed as integrals
of this function along contours in the complex energy plane. We show that the
expansions based on the conformal mapping of the Borel plane go beyond
finite-order perturbation theory, containing an infinite number of terms when
reexpanded in powers of the coupling. Moreover, the expansion functions exhibit
nonperturbative features of the true function, while the expansions have a
tamed behavior at large orders and are expected even to be convergent. Using
these properties, we argue that there are no mathematical reasons for
supplementing the expansions based on the conformal mapping of the Borel plane
by additional arbitrary power corrections. Therefore, we make the conjecture
that they provide an alternative to the standard OPE in approximating the QCD
correlator. This conjecture allows to slightly improve the accuracy of the
strong coupling extracted from the hadronic $\tau$ decay width. Using the
optimal expansions based on conformal mapping and the contour-improved
prescription of renormalization-group resummation, we obtain
$\alpha_s(m_\tau^2)=0.314 \pm 0.006$, which implies $\alpha_s(m_Z^2)=0.1179 \pm
0.0008$.
