  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.
