Exact evaluation of the simple cubic lattice Green function for a general lattice point

The simple cubic lattice Green function G(l, m, n; w) = 1/pi(3) integral(0)(pi)integral(0)(pi)integral(0)(pi) cosltheta(1) cosmtheta(2) cos ntheta(3)/w-costheta(1)-costheta(2)-costheta(3) dtheta(1) dtheta(2) dtheta(3) is investigated, where {l. m. n) denotes a set of integers and w = u + iv is a complex variable in the (u, v) plane. In particular, it is shown that the modified Green function (G) over bar (l, m, n; w) = (3/w)(l+m+n) wG (l, m, n; w) can be expressed in the xi-parametric form (G) over bar (l, m, n; w) = R-0 (l, m, n; xi) + R-1 (l, m, n; xi) [2/piK(k)](2) +R-2(l,m,n;xi) [2/piK(k)][2/piE(k)] + R-3 (l, m, n; xi)[2/piE(k)](2) where K(k) and E(k) are complete elliptic integrals of the first and second kind respectively, with a modulus k equivalent to k(xi) = 4xi(3/2)/(1 - xi)(3/2)(1 + 3xi)(1/2). The connection between the parameter xi and the variable w is given by xi equivalent to xi(w) = (1/w) [1+root1-(9/w(2))](-1/2) [1 + root1-(1/w(2))](-1/2) and {R-j (l, m, n; xi) : j = 0, 1, 2, 3} is a set of rational functions of xi. It is found that the complete elliptic integral formulae for the Green functions {(G) over bar (n, n, n; w) : n = 1, 2, 3, 4} and ((G) over bar (2n, n, n; w) : n = 1, 2, 3, 4} can be factorized as a product of two linear forms in K(k) and E(k) whose coefficients are rational functions of the parameter xi. On the basis of these explicit results it is conjectured that this factorization propery is valid for all integer values of n.