Analytical studies on the Sunyaev Zeldovich effect in the cluster of
  galaxies for three Lorentz frames
  We study the Sunyaev-Zeldovich effect for clusters of galaxies. The Boltzmann
equations for the CMB photon distribution function are studied in three Lorentz
frames. We clarify the relations of the SZ effects among the different Lorentz
frames. We derive analytic expressions for the photon redistribution functions.
These formulas are applicable to the nonthermal electron distributions as well
as the standard thermal distribution. We show that the Fokker-Planck expansion
of the Boltzmann equation can be expanded by the power series of the diffusion
operator of the original Kompaneets equation.
