  Green's functions for Rossby waves in an azimuthal wind are obtained, in
which the stream-function $\psi$ depends on $r$, $\phi$ and $t$, where $r$ is
cylindrical radius and $\phi$ is the azimuthal angle in the $\beta$-plane
relative to the easterly direction, in which the $x$-axis points east and the
$y$-axis points north. The Rossby wave Green's function with no wind is
obtained using Fourier transform methods, and is related to the previously
known Green's function obtained for this case, which has a different but
equivalent form to the Green's function obtained in the present paper. We
emphasize the role of the wave eikonal solution, which plays an important role
in the form of the solution. The corresponding Green's function for a rotating
wind with azimuthal wind velocity ${\bf u}=\Omega r{\bf e}_\phi$
($\Omega=$const.) is also obtained by Fourier methods, in which the advective
rotation operator in position space is transformed to a rotation operator in
${\bf k}$ transform space. The finite Rossby deformation radius is included in
the analysis. The physical characteristics of the Green's functions are
delineated and applications are discussed. In the limit as $\Omega\to 0$, the
rotating wind Green's function reduces to the Rossby wave Green function with
no wind.
