  We consider the radiative transfer problem in a plane-parallel slab of
thermal electrons in the presence of an overcritical magnetic field. Under such
conditions, the magnetic field behaves as a birefringent medium for the
propagating photons, and the electromagnetic radiation is split into two
polarization modes, ordinary and extraordinary, having different
cross-sections. When the optical depth of the slab is large, the ordinary-mode
photons are strongly Comptonized and the photon field is dominated by an
isotropic component. The radiative transfer problem in strong magnetic fields
presents many mathematical issues and analytical or numerical solutions can be
obtained only under some given approximations. We investigate this problem both
from the analytical and numerical point of view, providing a test of the
previous analytical estimates and extending these results introducing numerical
techniques. We consider the case of low temperature blackbody photons
propagating in a sub-relativistic temperature plasma, which allows us to deal
with a semi Fokker-Planck approximation of the radiative transfer equation. We
report the numerical solution obtained for eigenvalues and eigenfunctions of
the space operator, and the emerging Comptonization spectrum of the
ordinary-mode photons for any eigenvalue of the space equation and for energies
significantly less than the cyclotron energy, which is of the order of MeV for
the intensity of the magnetic field here considered. We derived the specific
intensity of the ordinary photons, under the approximation of large angle and
large optical depth. These assumptions allow the equation to be treated using a
diffusion-like approximation.
