The Compton reflection of X-rays by low temperature electrons has received much attention in the recent past. It is a useful framework for the explanation of some features of the spectra of AGNs and a basic ingredient for understanding the X-ray background spectrum, together with the cosmological evolution of AGN. Since the basic works of Illarionov et al. (1979) and of Lightman et al. (1981), principally based on a semi-analytical approach to the problem, many authors have developed careful Monte Carlo methods to include the effects of the Klein-Nishina cross-section and to extend the study up to the hard X- and γ-ray regions. However, the semianalytical approach has some useful advantages. Firstly it allows us to obtain directly the reflected spectrum in the energy region and at the inclination angle of interest; in addition it may be versatile enough to link with other methods for the computation of the spectrum in the UV and IR regions, where other physical processes are important, but where the X-rays reprocessed by thermal matter may be an important part of the overall luminosity.

The basic assumption of this approach is the separability of the spatial and energy transport problems and of the absorption due to bound-free transitions.