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We investigated the effect of inverse Compton scattering in mildlyrelativistic static and moving plasmas with low optical depth using Monte Carlosimulations, and calculated the Sunyaev-Zel'dovich effect in the cosmicbackground radiation. Our semi-analytic method is based on a separation ofphoton diffusion in frequency and real space. We use Monte Carlo simulation toderive the intensity and frequency of the scattered photons for a monochromaticincoming radiation. The outgoing spectrum is determined by integrating over thespectrum of the incoming radiation using the intensity to determine the correctweight. This method makes it possible to study the emerging radiation as afunction of frequency and direction. As a first application we have studied theeffects of finite optical depth and gas infall on the Sunyaev-Zel'dovich effect(not possible with the extended Kompaneets equation) and discuss the parameterrange in which the Boltzmann equation and its expansions can be used. For hightemperature clusters ($k_B T_e \gtrsim 15$ keV) relativistic corrections basedon a fifth order expansion of the extended Kompaneets equation seriouslyunderestimate the Sunyaev-Zel'dovich effect at high frequencies. Thecontribution from plasma infall is less important for reasonable velocities. Wegive a convenient analytical expression for the dependence of the cross-overfrequency on temperature, optical depth, and gas infall speed. Optical deptheffects are often more important than relativistic corrections, and should betaken into account for high-precision work, but are smaller than the typicalkinematic effect from cluster radial velocities.Comment: LateX, 30 pages and 11 figures. Accepted for publication in the  Astrophysical Journa

Inverse Compton Scattering in Mildly Relativistic Plasma