  The conventional definition of the running coupling $\alpha_{\bar{MS}}(\mu)$
in quantum chromodynamics is based on a solution to the renormalization group
equations which treats quarks as either completely massless at a
renormalization scale $\mu$ above their thresholds or infinitely massive at a
scale below them. The coupling is thus nonanalytic at these thresholds. In this
paper we present an analytic extension of $\alpha_{\bar{MS}}(\mu)$ which
incorporates the finite-mass quark threshold effects into the running of the
coupling. This is achieved by using a commensurate scale relation to connect
$\alpha_{\bar{MS}}(\mu)$ to the physical $\alpha_V$ scheme at specific scales,
thus naturally including finite quark masses. The analytic-extension inherits
the exact analyticity of the $\alpha_V$ scheme and matches the conventional
$\bar {MS}$ scheme far above and below mass thresholds. Furthermore just as in
$\alpha_V$ scheme, there is no renormalization scale ambiguity, since the
position of the physical mass thresholds is unambiguous.
