In this article the theory of two-index Bessel functions is presented. Their generating function, series expansion, and integral representations are discussed. Their usefulness in physical problems is also discussed in the context of analysis of radiation emitted by relativistic electrons in two-frequency undulators. Finally, the theoretical analysis proving addition and multiplication theorems for two-index Bessel functions are completed and their modified forms are introduced.

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We present a simple and inexpensive computer code for free electron laser (FEL) operation that includes high gain and inhomogeneous broadening effects. The code is based on analytical formulae, obtained within the context of the theory of FEL approximants, requires negligible CPU time, and may run on small computers.