Spectrum of the Sunyaev-Zel'dovich effect for high electron temperatures

We analyse in detail the spectral shape of the Sunyaev-Zel'dovich effect in the cosmic background radiation, taking into account the relativistic corrections for a hot electron gas. We calculate the displacement of the zero-signal frequency, which is especially informative in a new method for measuring the millimetric temperature of the background radiation; we also present a simple analytical expression, to be used to fit the experimental data in spectral measurements of the effect in very hot cluster gases.