  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.
