  Compton scattering by a locally plane parallel atmosphere is analyzed within
the Kompaneets approximation. In contrast to almost all previous analytic
calculations of inverse Compton scattering, we assume neither homogeneity nor
isotropy of the photon distribution. However, we do adopt an assumed angular
distribution of the photons. For any specified energy dependence of the flux
entering the base of the atmosphere, an analytic expression is obtained for the
flux at any optical depth. Only integrals over the input flux are involved. If
the thickness of the atmosphere is much less than its radial position on an
accretion disk, the contribution from all radii may be summed. The initial
motivation (addressed in a subsequent paper) is the computation of the energy
spectrum of (possibly modulated) photons originating from a specified radial
region of an accretion disk, relative to the total energy spectrum.
