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The concept of complex time is employed to derive the time domain representation of the Green's function of a periodic structure for the first time. The spatial domain Green's function of the periodic structure in the frequency domain is separated into singular and nonsingular terms. The nonsingular terms are approximated by using exponential functions, resulting in the well known complex time representation of the time domain Green's function. In order to find a closed form expression for the singular terms, a novel time domain representation of the Sommerfeld-type integrals is derived. The proposed procedure is applied to the computation of the time domain Green's function of a one dimensional periodic structure. Numerical experiments demonstrate the high accuracy of this method.