  Based upon the rate equations for the photon distribution function obtained
in the previous paper, we study the formal solutions in three different
representation forms for the Sunyaev-Zeldovich effect. By expanding the formal
solution in the operator representation in powers of both the derivative
operator and electron velocity, we derive a formal solution that is equivalent
to the Fokker-Planck expansion approximation. We extend the present formalism
to the kinematical Sunyaev-Zeldovich effect. The properties of the frequency
redistribution functions are studied. We find that the kinematical
Sunyaev-Zeldovich effect is described by the redistribution function related to
the electron pressure. We also solve the rate equations numerically. We obtain
the exact numerical solutions, which include the full-order terms in powers of
the optical depth.
