  A kinetic equation for Compton scattering is given that differs from the
Kompaneets equation in several significant ways. By using an inverse
differential operator this equation allows treatment of problems for which the
radiation field varies rapidly on the scale of the width of the Compton kernel.
This inverse operator method describes, among other effects, the thermal
Doppler broadening of spectral lines and continuum edges, and automatically
incorporates the process of Compton heating/cooling. It is well adapted for
inclusion into a numerical iterative solution of radiative transfer problems.
The equivalent kernel of the new method is shown to be a positive function and
with reasonable accuracy near the intitial frequency, unlike the Kompaneets
kernel, which is singular and not wholly positive. It is shown that iterates of
the inverse operator kernel can be easily calculated numerically, and a simple
summation formula over these iterates is derived that can be efficiently used
to compute Comptonized spectra. It is shown that the new method can be used for
initial value and other problems with no more numerical effort than the
Kompaneets equation, and that it more correctly describes the solution over
times comparable to the mean scattering time.
