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The general theoretical aspects of inverse Compton scattering was investigated and an equation for the timeindependent inverse Compton spectrum for photons from a plasma cloud of finite extent was derived. This was done by convolving the Kompaneets equation used for describing the evolution of the photon spectrum from a plasma cloud of infinite extent and the distribution of photons over escape time. By looking for a power law solution to the equation so derived, was revealed that the spectrum produced by inverse Compton scattering depend only on the electron temperature and optical depth.