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

The lattice Green's functions of the rectangular and the square lattices Irect(a;m,n;α,β)≡1π2double integral operator0πcosmxcosny dx dya−iε−αcosx−βcosy,Isq(a;m,n)≡Irect(a;m,n;1,1)are considered. The integral Irect(a, m, n; α, β) for a α + β is evaluated and expressed in terms of the generalized hypergeometric function F4. Expressions of Isq(a; m, n) for a 2, a 2, and a ∼ 2, and Irect(a; m, m; α, β) in terms of pFp−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 Isq(a; m, n) are shown in figures.