Manifestly Covariant Analysis of the QED Compton Process in $e pto e gamma p$ and $e p to e gamma X$

We calculate the unpolarized QED Compton scattering cross section in a manifestly covariant way. Our approach allows a direct implementation of the specific kinematical cuts imposed in the experiments, {it e. g.} HERA-H1. We compare the 'exact' cross section in terms of the structure functions $F_{1,2} (x_B,Q^2)$, assuming the Callan-Gross relation, with the one obtained using the equivalent photon approximation (EPA) as well as with the experimental results. We find that the agreement with the EPA is better in $x_{gamma}$ bins, where $x_{gamma}$ is the fraction of the longitudinal momentum of the proton carried by the virtual photon, compared to the bins in the leptonic variable $x_l$