  Compton scattering plays an important role in various astrophysical objects
such as accreting black holes and neutron stars, pulsars, and relativistic
jets, clusters of galaxies as well as the early Universe. In most of the
calculations it is assumed that the electrons have isotropic angular
distribution in some frame. However, there are situations where the anisotropy
may be significant due to the bulk motions, or anisotropic cooling by
synchrotron radiation, or anisotropic source of seed soft photons. We develop
here an analytical theory of Compton scattering by anisotropic distribution of
electrons that can simplify significantly the calculations. Assuming that the
electron angular distribution can be represented by a second order polynomial
over cosine of some angle (dipole and quadrupole anisotropy), we integrate the
exact Klein-Nishina cross-section over the angles. Exact analytical and
approximate formulae valid for any photon and electron energies are derived for
the redistribution functions describing Compton scattering of photons with
arbitrary angular distribution by anisotropic electrons. The analytical
expressions for the corresponding photon scattering cross-section on such
electrons as well as the mean energy of scattered photons, its dispersion and
radiation pressure force are also derived. We applied the developed formalism
to the accurate calculations of the thermal and kinematic Sunyaev-Zeldovich
effects for arbitrary electron distributions.
