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.
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