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Expressions are derived for the total energy loss and photon-production spectrum by the processes of Compton scattering, bremsstrahlung, and synchrotron radiation from highly relativistic electrons. For Compton scattering, the general case, the Thomson limit, and the extreme Klein-Nishina limit are considered. Bremsstrahlung is treated for the cases where the electron is scattered by a pure Coulomb field and by an atom. For the latter case the effects of shielding are discussed extensively. The synchrotron spectrum is derived for an electron moving in a circular orbit perpendicular to the magnetic field and also for the general case where the electron's motion is helical. The total photon-production spectrum is derived for each process when there is a power-law distribution of electron energies. The problems of the effects of the three processes on the electron distribution itself are considered. It is shown that if the electron loses a small fraction of its energy in a single occurrence of a process, the electron distribution function satisfies a continuity equation which is a differential equation in energy space. For the more general case where the electron can lose energy in discrete amounts (as in bremsstrahlung and extreme Klein-Nishina Compton losses), the electron distribution function satisfies an integro-differential equation. Some approximate solutions to this equation are derived for certain special cases.
Blumenthal et al. (1970) studied this question.