  The explicit expression for the R-ratio of electron-positron annihilation
into hadrons, which properly accounts for all the effects due to continuation
of the spacelike perturbative results into the timelike domain, is obtained at
an arbitrary loop level. Several equivalent ways to derive a commonly employed
approximation of the R-ratio are recapped and the impact of discarded in the
latter higher-order pi2-terms on the evaluation of the strong running coupling
is elucidated. The obtained results substantially facilitate the theoretical
study of electron-positron annihilation into hadrons and the related strong
interaction processes.
