Revisiting the discrete planar Laplacian: exact results for the lattice Green function and continuum limit

The paper deals with the discrete Laplacian on a uniform infinite square lattice. definition of its fundamental solution or lattice Green function (LGF) is clarified as Fourier coefficients certain generalized periodic g. Such functional must be regularized and gives LGF up to constant equal $${<}g{>}$$ , mean value For $${<}g{>}=0$$ may expressed in an exact analytic form terms hypergeometric gamma functions. continuum limit finally studied requiring appropriate renormalization order obtain logarithmic Coulomb potential.