We investigate the process of synchrotron radiation from thermal electrons
at semirelativistic and relativistic temperatures. We find an analytic expr
ession for the emission coefficient for random magnetic fields with an accu
racy significantly higher than those derived previously. We also present an
alytic approximations to the synchrotron turnover frequency, treat Comptoni
zation of self-absorbed synchrotron radiation, and give simple expressions
for the spectral shape and the emitted power. We also consider modification
s of the above results by bremsstrahlung.
We then study the importance of Comptonization of thermal synchrotron radia
tion in compact X-ray sources. We first consider emission from hot accretio
n flows and active coronae above optically thick accretion discs in black h
ole binaries and active galactic nuclei (AGNs). We find that for plausible
values of the magnetic field strength, this radiative process is negligible
in luminous sources, except for those with hardest X-ray spectra and stell
ar masses. Increasing the black hole mass results in a further reduction of
the maximum Eddington ratio from this process. Then, X-ray spectra of inte
rmediate-luminosity sources, e.g. low-luminosity AGNs, can be explained by
synchrotron Comptonization only if they come from hot accretion flows, and
X-ray spectra of very weak sources are always dominated by bremsstrahlung.
On the other hand, synchrotron Comptonization can account for power-law X-r
ay spectra observed in the low states of sources around weakly magnetized n
eutron stars.