The thermal and kinematic Sunyaev-Zel'dovich effects revisited

This paper shows that a simple convolution integral expression based on the mean value of the isotropic frequency distribution corresponding to photon scattering off electrons leads to useful analytical expressions describing the thermal Sunyaev-Zel'dovich effect. The approach, to first order in the Compton parameter is able to reproduce the Kompaneets equation describing the effect. Second order effects in the parameter $z=\frac{kT_{e}}{mc^{2}}$ induce a slight increase in the crossover frequency.