  The mathematical properties of the new analytic running coupling (NARC) in
QCD are investigated. This running coupling naturally arises under
``analytization'' of the renormalization group equation. One of the crucial
points in our consideration is the relation established between the NARC and
its inverse function. The latter is expressed in terms of the so-called Lambert
$W$ function. This relation enables one to present explicitly the NARC in the
renorminvariant form and to derive the corresponding $\beta$ function. The
asymptotic behavior of this $\beta$ function is examined. The consistent
estimation of the parameter $\Lambda_{QCD}$ is given.
