The asymptotic solution of the system of Vlasov and Maxwell equations for a plasma slab with a given current distribution is derived for weak absorption and weak equilibrium density inhomogeneity, but without the usual restriction that the perpendicular wavelengths be larger than the Larmor radii of the thermal particles. The equilibrium magnetic field is homogeneous. No model equations or phenomenological assumptions are introduced, except that the interaction of the components of the electric field parallel and perpendicular to the equilibrium magnetic field (corresponding to the ordinary and extraordinary waves in a homogeneous plasma) has been neglected. The electric field in vacuum is determined by a condition on the analytical form of the Fourier transform of the field that must necessarily also apply to the exact solution. The example presented to illustrate the results considers the emission at the harmonics of the electron gyrofrequency of a population of fast electrons in a plasma with cold ions; the theoretical results are in a very good qualitative agreement with experiment.