Generalized bessel functions and their use in bremsstrahlung and multi-photon processes

The theory of Generalized Bessel Functions is reviewed and their application to various problems in the study of electro-magnetic processes is presented. We consider the cases of emission of bremsstrahlung radiation by ultra-relativistic electrons in linearly polarized undulators, including also exotic configurations, aimed at enhancing the harmonic content of the emitted radiation. The analysis is eventually extended to the generalization of the FEL pendulum equation to treat Free Electron Laser operating with multi-harmonic undulators. The paper aims at picking out those elements supporting the usefulness of the Generalized Bessel Functions in the elaboration of the theory underlying the study of the spectral properties of the bremsstrahlung radiation emitted by relativistic charges, along with the relevant flexibility in accounting for a large variety of apparently uncorrelated phenomenolgies, like multi-photon processes, including non linear Compton scattering.