Relativistic corrections to the Kompaneets equation

Abstract: We study the Sunyaev-Zeldovich effect for clusters of galaxies. We explore the relativistic corrections to the Kompaneets equation in terms of two different expansion approximation schemes, namely, the Fokker-Planck expansion approximation and delta function expansion approximation. We show that two expansion approximation formalisms are equivalent under the Thomson approximation, which is extremely good approximation for the CMB photon energies. This will clarify the situation for existing theoretical methods to analyse observation data.