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The complex-envelope finite-difference time-domain (CE-FDTD) method has been developed in order to efficiently simulate electromagnetic structures with band-limited signals. This method has most often been formulated in terms of wave equations, leaving its numerical properties significantly understudied. In this paper, full-wave CE-FDTD formulations, and a comprehensive analysis of the numerical stability and dispersion of this method are presented. It is found that that the maximum time step allowed for stability is strongly dependent upon the ratio of the carrier wavelength to spatial step sizes. When this ratio is larger than a threshold value, the time step is bounded by a Courant-Frederick-Levy (CFL) condition. When this ratio is lower than the threshold value, the time step is no longer bounded by this stability condition. However, in such instances, the associated numerical dispersion errors become unacceptably large (more than 200%). Therefore, in comparison with the conventional FDTD method, the CE-FDTD method is proven to be not advantageous in terms of computation efficiency for an accuracy of a similar level.