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:

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.