Inversion of synchrotron spectra

The problem of synchrotron radiation spectra is treated from the viewpoint of deconvolving the spectrum of ultrarelativistic source electrons from the observed photon spectrum. It is shown that for homogeneous sources the problem amounts to inversion of a Meijer transform with a modified Bessel finction kernel. A precise analytic inversion is only possible in the complex plane but Meijer transform tables are available for a wide range of functions. More convenient inversion formulae prove possible by use of a Laplace transform approximation or by analysing the spectra in terms of their integral moments. The filtering property of the transform is also established showing that the contribution to the synchrotron spectrum of high frequency components in the electron spectrum declines exponentially with their frequency. Thus, as with other Laplace-like transforms, only a few terms in an electron spectrum expansion can be deconvolved for any plausible noise level in the synchrotron spectrum.