Analytical Study on the Sunyaev-Zeldovich Effect for Clusters of
  Galaxies
  Starting from a covariant formalism of the Sunyaev-Zeldovich effect for the
thermal and non-thermal distributions, we derive the frequency redistribution
function identical to Wright's method assuming the smallness of the photon
energy (in the Thomson limit). We also derive the redistribution function in
the covariant formalism in the Thomson limit. We show that two redistribution
functions are mathematically equivalent in the Thomson limit which is fully
valid for the cosmic microwave background photon energies. We will also extend
the formalism to the kinematical Sunyaev-Zeldovich effect. With the present
formalism we will clarify the situation for the discrepancy existed in the
higher order terms of the kinematical Sunyaev-Zeldovich effect.
