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The theory of spectral formation in thermal X$-$ray sources, where the effects of Comptonization and Klein$-$Nishina corrections are important, is presented. Analytical expressions are obtained for the produced spectrum as a function of such input parameters as the plasma temperature, the optical depth of the plasma cloud and the injected soft photon spectrum. The analytical theory developed here takes into account the dependence of the scattering opacity on the photon energy. It is shown that the plasma temperature as well as the asymptotic rate of photon escape from the plasma cloud determine the shape of the upscattered hard tail in the emergent spectra, even in the case of very small optical depths. The escape distributions of photons are given for any optical depth of the plasma cloud and their asymptotic for very small and large optical depths are examined. The comparison of the new analytical theory with extensive Monte-Carlo calculations are also presented. It is shown that this new generalized approach matches extremely well the Monte-Carlo calculation in very wide ranges of plasma temperature (1$-$500 keV) and plasma cloud optical depths (0.1$-$10). The fits of spectra by the analytical Comptonization model for a large variety of hard X$-$ray sources and determination of the plasma temperature in the region of main energy release in Cyg X$-$1 and the Seyfert galaxy NGC 4151 are discussed.
