Analytical Study on the Sunyaev-Zeldovich Effect for Clusters of
  Galaxies. II. comparison of covariant formalisms
  We study a covariant formalism for the Sunyaev-Zeldovich effects developed in
the previous papers by the present authors, and derive analytic expressions for
the redistribution functions in the Thomson approximation. We also explore
another covariant formalism recently developed by Poutanen and Vurm. We show
that the two formalisms are mathematically equivalent in the Thomson
approximation which is fully valid for the cosmic microwave background photon
energies. The present finding will establish a theoretical foundation for the
analysis of the Sunyaev-Zeldovich effects for the clusters of galaxies.
