The Discrete Green’s Function

Abstract

We first discuss discrete holomorphic functions on quad-graphs and their relation to discrete harmonic functions on planar graphs. Then, the special weights in the discrete Cauchy-Riemann (and discrete Laplace) equations are considered, coming from quasicrystalline rhombic realizations of quad-graphs. We relate these special weights to the 3D consistency (integrability) of the discrete Cauchy-Riemann equations, allowing us to extend discrete holomorphic functions to a multidimensional lattice. Discrete exponential functions are introduced and are shown to form a basis in the space of discrete holomorphic functions growing not faster than exponentially. The discrete logarithm is constructed and characterized in various ways, including an isomonodromic property. Its real part is nothing but the discrete Green’s function.