  The two-loop invariant (running) coupling of QCD is written in terms of the
Lambert W function. The analyticity structure of the coupling in the complex
Q^2-plane is established. The corresponding analytic coupling is reconstructed
via a dispersion relation. We also consider some other approximations to the
QCD beta-function, when the corresponding couplings are solved in terms of the
Lambert function. The Landau gauge gluon propagator has been considered in the
renormalization group invariant analytic approach (IAA). It is shown that there
is a nonperturbative ambiguity in determination of the anomalous dimension
function of the gluon field. Several analytic solutions for the propagator at
the one-loop order are constructed. Properties of the obtained analytical
solutions are discussed.
