Cut-offs and time dependent acceleration at relativistic shocks

In these proceedings we extend the eigenfunction method used to calculate the power-law spectrum of particles at a relativistic shock front to consider the time dependent acceleration and the effect of synchrotron losses on the acceleration process. Appropriate matching conditions are shown for the two problems and we show initial results illustrating the expected presence of a cut-off in the case of synchrotron losses. This is fundamentally important for astrophysical sources of high energy cosmic rays such as GRBs and AGNs.