  Gamma-Ray Burst (GRB) afterglows are well described by synchrotron emission
from relativistic blast waves expanding into an external medium. The blast wave
is believed to amplify the magnetic field and accelerate the electrons into a
power law distribution of energies promptly behind the shock. These electrons
then cool both adiabatically and by emitting synchrotron and inverse Compton
radiation. The resulting spectra is known to consist several power law
segments, which smoothly join at certain break frequencies. Here, we give a
complete description of all possible spectra under those assumptions, and find
that there are 5 possible regimes, depending on the ordering of the break
frequencies. The flux density is calculated by integrating over the
contributions from all the shocked region, using the Blandford McKee solution.
This allows us to calculate more accurate expressions for the value of these
break frequencies, and describe the shape of the spectral breaks around them.
This also provides the shape of breaks in the light curves caused by the
passage of a break frequency through the observed band. These new, more exact,
estimates are different from more simple calculations by up to a factor of
about 70, and describe some new regimes which where previously ignored.
