Causality trick and the analytical signal formulation of the time domain periodic Green´s function

The formulation presented in this paper starts with the transient Weyl´s identity to decompose the analytic elemental Green´s function into a series of transient analytic plane waves. Then the periodic Green´s function can be represented as a series of analytic Floquet modes using the Poisson summation formulae. Every single Floquet mode can be expressed as a singular integral, and it can be expressed as a Bessel function by deforming the path of integration. The analytic periodic Green´s function is obtained by summing all the Floquet modes together, and the physically observable signal can be obtained by taking the real part of the corresponding analytic signal