Lattice Green's Functions for the Rectangular and the Square Lattices at Arbitrary Points

Abstract

The lattice Green's functions of the rectangular and the square lattices I rect (a;m,n;α,β)≡ 1 π 2 [double integral operator] 0 π cos mx cos ny dx dy a−iε−α cos x−β cos y , I sq (a;m,n)≡I rect (a;m,n;1,1) are considered. The integral I rect(a, m, n; α, β) for a > α + β is evaluated and expressed in terms of the generalized hypergeometric function F 4. Expressions of I sq(a; m, n) for a > 2, a < 2, and a ∼ 2, and I rect(a; m, m; α, β) in terms of p Fp −1 are presented by the method of the analytic continuation using the Mellin‐Barnes type integral. They are useful for the understanding of the nature of the singularity and for numerical calculation. The behaviors of I sq(a; m, n) are shown in figures.