Lattice Green's Function for the Simple Cubic Lattice in Terms of a Mellin-Barnes Type Integral

Abstract

The lattice Green's function for the simple cubic lattice I(a)= 1 π 3 [triple intergral operator] 0 π dx dy dz a+iε− cos x− cos y− cos z for a > 3 is expressed as a Mellin‐Barnes type integral. The analytic continuation gives simple and useful expressions in series for the numerical calculation of the real part I R(a) and the imaginary part I I(a) of the integral for 0 < a < 1 and 1 < a < 3. The values at a = 1, a = 0, and a=√5 are obtained exactly: I R(1) = (π/2)[Γ(5/8)Γ(7/8)]−2, I I (1)/(−i)=(√2)I R (1), and I I(0) = 3·2−11/3π−4[(Γ(1/3)]6.