A multidimensional Z-transform evaluation of the discrete finite difference time domain Green's function

Analytical expressions for frequency and time domain discrete Green's functions are developed below. Rather than discretizing the continuous Green's functions directly, we derive the discrete Green's functions from first principles, i.e., the difference equations. This is done with the purpose of obtaining discrete integral operators that would replicate results obtained via the finite difference time domain (FDTD) method. The main effort is directed at inversions of multidimensional frequency domain expressions in the Z-transform domain. The Green's functions obtained in this way coincide with time domain developments presented earlier both in the form of direct numerical computations and via combinatorics and the Catalan triangle. The Z-transform methodology, resulting from this development, provides an orderly manner for deriving expressions for multidimensional discrete integral operators that can be hybridized with FDTD-based differential operators in a self consistent manner.