I am not the best person to reference on this. I am still trying to clarify my understanding

describe synch. from a power law e- population (fig. 6.12):

low-f rise is nu^+5/2 (eqn. 6.54; optically thick therefore source function)

and decay is nu^-(p-1)/2 (eqn 6.22a; optically think therefore emission function)

… but that's not the applicable spectrum:

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?2000MNRAS.313L...1G&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf

The important paper here is Gabriele Ghisellini, Annalisa Celotti and Davide Lazzati MNRAS 2000.

emission is from a quasi-monoenergetic e- population;

F_nu ~ nu^+1/3 up to nu_peak

nu > nu_peak they don't say.

However, with reasonable consideration of cooling, you get a nu^-1/2 in the X-gamma spectrum, because the cooling frequency is now ~ eV.

Here is a figure from a lecture by Tsvi Piran at Caltech http://ned.ipac.caltech.edu/level5/March04/Piran2/Piran8_2.html:

“Figure 22. Synchrotron spectrum of a relativistic shock with a power-law electron distribution. (a) Fast cooling, which is expected at early times (t < t0). The spectrum consists of four segments, identified as A, B, C, D. Self-absorption is important below nua. The frequencies, num, nuc, nua, decrease with time as indicated; the scalings above the arrows correspond to an adiabatic evolution, and the scalings below, in square brackets, to a fully radiative evolution. (b) Slow cooling, which is expected at late times (t > t0). The evolution is always adiabatic. The four segments are identified as E, F, G, H. From [Sari, R., Piran, T., & Narayan, R., 1998, Ap. J. Lett., 497, L17.].”

Zou, Piran, and Sari 09 are a good reference for SSC http://iopscience.iop.org/1538-4357/692/2/L92/pdf/apjl_692_2_92.pdf