The high requirements for undulators in modern coherent radiation devices and the growing demand for sources with specified radiation characteristics have revived interest in the study of the theoretical aspects of UR. The developed mathematical apparatus of extended and generalized forms of special functions made it possible to accurately describe the effect of the finite size of the electron beam, its divergence, energy spread, the effect of non-periodic components of the magnetic field on the intensity, spectrum, and shape of spectral lines of the UR. In particular, the influence of the length of the undulator on all characteristics of the URI was calculated analytically, taking into account the losses, and the possibility of compensating the divergence of the electron beam with a suitably configured constant magnetic field was shown.

At present, sources of undulator radiation find practical application in research in vacuum ultraviolet radiation and in the harder regions of the spectrum, where there is a shortage of intense tunable sources of electromagnetic radiation. There are already interesting results of their application in research on X-ray fluorescence elemental analysis, nuclear spectroscopy, medical diagnostics, and spectroscopy. In the near future, it is planned to further increase the radiation power of operating lasers and place them on a third-stage FEL with wavelengths in the near infrared range of 5-30 microns [11]. There are always many plans in this interesting and promising field of knowledge.

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