  We present an extension of the Kompaneets equation which allows relativistic
effects to be included to any desired order. Using this, we are able to obtain
simple analytic forms for the spectral changes due to the Sunyaev-Zel'dovich
effect in hot clusters, correct to first and second order in the expansion
parameter theta_e=k_B T_e/m c^2. These analytic forms agree with previous
numerical calculations of the effect based upon the multiple scattering
spectrum for k_B T_e up to 10 keV. Our results confirm previous conclusions
that the result of including relativistic corrections in the Sunyaev-Zel'dovich
effect is a small reduction in the amplitude of the effect over the majority of
the spectrum: specifically we find Delta T/T = -2y(1 - 17/10 theta_e + 123/40
theta_e^2) (correct to second-order) in the Rayleigh-Jeans region, where y is
the usual Comptonization parameter. For a typical cluster temperature of 8 keV,
this amounts to a correction downwards to the value of the Hubble constant
derived using combined X-ray and Rayleigh-Jeans Sunyaev-Zel'dovich information
by about 5 percent.
