The Lattice Green Function for the Poisson Equation on an Infinite   Square Lattice
  We derive formulas for the matrix elements of the lattice Green function for the discrete Poisson equation on an infinite square lattice. The partial difference equation for the matrix elements is solved by reducing it to a series of first order difference equations, which can then be solved sequentially. These formulas are useful in solving two dimensional Poisson equation problems using the finite difference approximation. 
Stefan Hollos, Richard Hollos