Analytical and regularized solutions of the synchrotron integral equation

In suitable conditions on the radiation source the synchrotron process can be described by a Fredholm integral equation whose integral kernel is expressed in terms of a modified Bessel function of the second kind. We present a completely general solution of this equation. We point out that the linear inverse problem of determining the electron distribution function in the source from the knowledge of the emitted photon spectrum at discrete frequencies is strongly ill-conditioned. In order to reduce the numerical instability due to the presence of noise on the datum, we apply Tikhonov regularization method to some simulated spectra. In particular, the formulation of the method in a suitable Sobolev space allows the use of a priori information on the solution to improve the restoration accuracy. The case of a real spectrum is also considered.