  Two relativistic distributions which generalizes the Maxwell Boltzman (MB)
distribution are analyzed: the relativistic MB and the Maxwell-J{\"u}ttner (MJ)
distribution. For the two distributions we derived in terms of special
functions the constant of normalization, the average value, the second moment
about the origin, the variance, the mode, the asymptotic behavior, approximate
expressions for the average value as function of the temperature and the
connected inverted expressions for the temperature as function of the average
value. Two astrophysical applications to the synchrotron emission in presence
of the magnetic field and the relativistic electrons are presented.
