QED Compton scattering in high-energy electron-proton collisions

We here provide exact analytic formulas and therefrom derive a Monte Carlo program without approximations, for QED Compton events in the reaction $\mathrm{ep}\to e\gamma X$ produced in head-on $\mathrm{ep}$ collisions. Those events correspond to final states with an electron and a photon, nearly coplanar, observed at a finite angle within the detector. The process here studied can be used, at high-energy electron-proton colliders, for various purposes: measurement of the luminosity; electromagnetic calibration of detectors; search for an $e^{*}$; and finally, determination of the photon content of the proton.