  In a previous publication we have derived an expression for the full
distorted spectrum arising when the photons of the cosmic background radiation
are absorbed and emitted by an optically thin gas. The expression simply adds
up the effects of the joint probability that an electron absorbs an incoming
photon of frequency}$\bar{\nu}${\small and emits it with a frequency}$\nu
${\small . We here show that such an expression is identical to the Kompaneets
equation describing the thermal Sunyaev-Zel'dovich effect. This clearly shows
that the two pictures, diffusion and absorption-emission are mathematically
equivalent to first order in the Compton parameter. Other consequences of this
result are also discussed.
