  We propose a new approximation for the cyclo-synchrotron emissivity of a
single electron. In the second part of this work, we discuss a simple
application for our approximation, and investigate the heating of electrons
through the self-absorption process. Finally, we investigate the self-absorbed
part of the spectrum produced by a power-law population of electrons. In
comparison to earlier approximations, our formula provides a few significant
advantages. Integration of the emissivity over the whole frequency range,
starting from the proper minimal emitting frequency, gives the correct cooling
rate for any energy particle. Further, the spectrum of the emission is well
approximated over the whole frequency range, even for relatively low particle
energies (beta << 0.1), where most of the power is emitted in the first
harmonic. In order to test our continuous approximation, we compare it with a
recently derived approximation of the first ten harmonics. Finally, our formula
connects relatively smooth to the synchrotron emission at beta=0.9. We show
that the self-absorption is a very efficient heating mechanism for low energy
particles, independent of the shape of the particle distribution responsible
for the self-absorbed synchrotron emission. We find that the energy gains for
low energy particles are always higher than energy losses by cyclo-synchrotron
emission. We show also that the spectral index of the self-absorbed part of the
spectrum at very low frequencies differs significantly from the well known
standard relation I(nu) ~ nu^(5/2).
