  We obtain an exact, closed-form expression for the time-dependent Green's
function solution to the Kompaneets equation. The result, which is expressed as
the integral of a product of two Whittaker functions, describes the evolution
in energy space of a photon distribution that is initially monoenergetic.
Effects of spatial transport within a homogeneous scattering cloud are also
included within the formalism. The Kompaneets equation that we solve includes
both the recoil and energy diffusion terms, and therefore our solution for the
Green's function approaches the Wien spectrum at large times. We show that the
Green's function can be used to generate all of the previously known
steady-state and time-dependent solutions to the Kompaneets equation. The new
solution allows the direct determination of the spectrum, without the need to
numerically solve the partial differential equation. Based upon the Green's
function, we obtain a new time-dependent solution for the photon distribution
resulting from the reprocessing of an optically thin bremsstrahlung initial
spectrum with a low-energy cutoff. The new bremsstrahlung solution possesses a
finite photon number density, and therefore it displays proper equilibration to
a Wien spectrum at large times. The relevance of our results for the
interpretation of emission from variable X-ray sources is discussed, with
particular attention to the production of hard X-ray time lags, and the Compton
broadening of narrow features such as iron lines.
