[en] The coupling of the extra-ordinary and the quasi-electrostatic modes at the second electron cyclotron resonance for a weakly relativistic plasma is investigated, in order to evaluate the effectiveness of the ECRH mechnism to pre-heat the plasma. When the propagation is nearly perpendicular to the external magnetic field, and the plasma density is sufficiently high we find a strong coupling between these waves. In a critical interval of density the geometrical optics model predicts a 100% mode conversion for waves crossing the resonance layer. It is shown that this model fails within a small boundary layer near the resonance, thus requiring a full-wave analysis in this region. The linearized relativistic Vlasov equation is solved in the inner region, and field equations valid within the boundary layer are derived. The outer region is treated using the standard geometrical optics method. The inner solutions are matched to the outer solutions in the usual manner. We have also derived appropriate energy flux theorems so that the energy associated with the wave and the particles can be distinguished