  We revisit the description provided by Ph. Biane of the spectral measure of
the free unitary Brownian motion. We actually construct for any $t \in (0,4)$ a
Jordan curve $\gamma_t$ around the origin, not intersecting the semi-axis
$[1,\infty[$ and whose image under some meromorphic function $h_t$ lies in the
circle. Our construction is naturally suggested by a residue-type integral
representation of the moments and $h_t$ is up to a M\"obius transformation the
main ingredient used in the original proof. Once we did, the spectral measure
is described as the push-forward of a complex measure under a local
diffeomorphism yielding its absolute-continuity and its support. Our approach
has the merit to be an easy yet technical exercise from real analysis.
