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Analytical studies on the Sunyaev-Zeldovich effect in the cluster of galaxies for three Lorentz frames II: single integral formula

We study the Sunyaev-Zeldovich effect for clusters of galaxies. The Boltzmann equations for the cosmic microwave background photon distribution function are studied in three Lorentz frames. We extend the previous work and derive analytic expressions for the integrated photon redistribution functions over the photon frequency. We also derive analytic expressions in the power series expansion approximation. By combining two formulas, we offer a simple and accurate tool to analyse observation data. These formulas are applicable to the non-thermal electron distributions as well as the standard thermal distribution. The Boltzmann equation is reduced to a single integral form of the electron velocity.