  We study the generalized Kompaneets equation (kinetic equation for the photon
distribution function taking into account the Compton scattering by electrons)
using a relativistically covariant formalism. Using the generalized Kompaneets
equation, we derive an analytic expression for the Sunyaev-Zel'dovich effect,
which takes into account up to O(theta**5) terms, where theta=kT/mc**2 is the
relativistic expansion parameter and T is the electron temperature. We
carefully study the applicable region of the obtained analytic expression by
comparing with the result of the direct numerical integration. We conclude that
the present analytic expression can be reliably applied to the calculation of
the Sunyaev-Zel'dovich effect for kT < 15keV, which is the highest electron
temperature in the presently known clusters of galaxies. Therefore, the present
analytic expression can be applied to all known clusters of galaxies.
