  Various aspects of relativistic calculations of the Sunyaev-Zeldovich effect
are explored and clarified. We first formally show that the main previous
approaches to the calculation of the relativistically generalized thermal
component of the effect are equivalent. Our detailed description of the full
effect results in a somewhat improved formulation. Analytic approximations to
the exact calculation of the change of the photon occupation number in the
scattering, $\Delta n$, are extended to powers of the gas temperature and
cluster velocity that are higher than in similar published treatments. For the
purely thermal and purely kinematic components, we obtain identical terms up to
the highest common orders in temperature and cluster velocity, and to second
order in the Thomson optical depth, as reported in previous treatments, but we
get slightly different expressions for the terms that depend on both the gas
temperature and cluster velocity. We also obtain an accurate expression for the
crossover frequency.
