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Radiative corrections (RC) to the Compton scattering cross section arecalculated in the leading and next-to leading logarithmic approximation to thecase of colliding high energy photon-electron beams.  RC to the double Compton scattering cross section in the same experimentalset-up are calculated in the leading logarithmic approximation.  We consider the case when no pairs are created in the final state. We showthat the differential cross section can be written in the form of the Drell-Yanprocess cross-section.  Numerical values of the $K$-factor and the leading order distribution on thescattered electron energy fraction and scattering angle are presented

Compton and double-compton scattering processes at colliding electron-photon beams