  Compton scattering of polarized radiation in a strong magnetic field is
considered. The recipe for calculation of the scattering matrix elements, the
differential and total cross sections based on quantum electrodynamic (QED)
second order perturbation theory is presented for the case of arbitrary initial
and final Landau level, electron momentum along the field and photon momentum.
Photon polarization and electron spin state are taken into account. The correct
dependence of natural Landau level width on the electron spin state is taken
into account in general case of arbitrary initial photon momentum for the first
time. A number of steps in calculations were simplified analytically making the
presented recipe easy-to-use. The redistribution functions over the photon
energy, momentum and polarization states are presented and discussed. The paper
generalizes already known results and offers a basis for accurate calculation
of radiation transfer in strong $B$-field, for example, in strongly magnetized
neutron stars.
