Discrete scattering theory: Green’s function for a square lattice

It is well known that, under certain circumstances, discrete plane waves can propagate through lattices. Waves can also be generated by oscillating one point in the lattice: the corresponding solution of the governing partial difference equations is the discrete Green’s function, gmn. The far-field behaviour of gmn is obtained using three methods: textbook derivations are corrected and a formula for gnn as a Legendre function is derived. The low-frequency behaviour of gmn is also obtained using Mellin transforms. These results are useful in the development of a discrete scattering theory.