The linearized relativistic Vlasov-Maxwell system for a hot inhomogeneous relativistic magnetized electron plasma is studied through particle orbit theory using the techniques of Fourier transform. An analytical integral expression in the (k, omega )-space is obtained for the current density for waves propagating across an externally applied uniform static magnetic field (k is the wavenumber and omega is the wave frequency). After applying inverse Fourier transform, differential equations for the electric field are obtained from the expression for the current density combined with Maxwell's equations. These fully relativistic equations are correct up to second order in rc/L. where rc is the electron gyroradius and L is the gradient length of the plasma inhomogeneity.