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The classical impulse invariance method in Z-transform theory is found to be incorrect and inaccurate when the impulse response is discontinuous at initial time t = 0 . Such inaccuracy results in higher numerical errors if it is used to develop the update equations for frequency-dependent finite-difference time-domain (FDTD) methods. In this paper, thorough discussions of corrected impulse invariance method in the realm of Z-transform theory for dispersive media are presented. The correction is shown to be necessary for dispersive media which exhibit discontinuity at t = 0 in the time domain susceptibility function. A (corrected) Z-transform table is provided to facilitate the conversion from frequency to Z domain. With the aid of the table, various formulations of frequency-dependent FDTD update equations using both corrected and classical impulse invariance methods are carried out conveniently. Detailed performance measures such as numerical permittivity, leading error term, dispersion relation, normalized phase and attenuation errors as well as memory storage requirements are included along with some extensive numerical comparisons.