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Time-domain impulse transmission functions of a dispersive half space due to an obliquely incident pulsed plane wave of either TE or TM polarization are presented as inverse Fourier transforms. The frequency domain transmission functions are approximated by using Prony's method as finite series of exponential decaying functions. The inverse Fourier transform of these exponential decaying functions are evaluated analytically in closed forms. Time-domain response due to a specific pulse shape is obtained by convoluting the impulse transmission functions with the required pulse shape.