On the Green function for the anisotropic simple cubic lattice

The analytical properties of the lattice Green function G(alpha, omega) = 1/pi (3) integral (pi)(0) integral (pi)(0) integral (pi)(0) d theta (1)d theta (2)d theta (3)/omega -cos theta (1)-cos theta (2)-alpha cos theta (3) are investigated, where omega = u + iv is a complex variable in the (u, v) plane and alpha is a real parameter in the interval (0, infinity). In particular, it is shown that the function y(G)(alpha, z) equivalent to omegaG(a, w), where z = 1/omega (2), is a solution of a fourth-order linear differential equation of the type Sigma (4)(j=0) f(j)(alpha, z)D(4-j)y = 0, where f(j)(alpha, z) is a polynomial in the variables alpha and z and D equivalent to d/dz. It is then proved that the solutions of this differential equation can all be expressed in terms of a product of two functions H-1 (alpha, z) and H-2 (alpha, z) which satisfy second-order linear differential equations of the normal type [D-2 + U+(alpha, z)]y = 0, [D-2 + U-(alpha, z)]y = 0, respectively, where U+/- (alpha, z) are complicated algebraic functions of alpha and z. Next Schwarzian transformation theory is used to reduce both these second-order differential equations to the standard Gauss hypergeometric differential equation. From this result it is deduced that omegaG(alpha, omega) = 2/root1-(2-alpha)(2)z + root1-(2+alpha)(2)z [2/pi K(k(+))][2/pi K(k(-))], where k(+/-)(2) equivalent to k(+/-)(2)(alpha, z) = 1/2 - 1/2[root1-(2-alpha)(2)z + root1-(2+alpha)(2)z](-3) x [root1+(2-alpha)rootz root1-(2+alpha)rootz + root1-(2-alpha)rootz root1+(2+alpha)rootz] x {+/- 16z + root1-alpha (2)z [root1+(2-alpha)rootz root1+(2+alpha)rootz + root1-(2-alpha)rootz root1-(2-alpha)rootz](2)} and K(k) denotes the complete elliptic integral of the first kind with a modulus k. This basic formula is valid for all values of omega = u + iv which lie in the (u, v) plane, provided that a cut is made along the real axis from omega = -2 - alpha to omega = 2 + alpha. In the remainder of the paper exact series expansions for G(alpha, omega) are derived which are valid in a sufficiently small neighbourhood of the branch-point singularities at omega = 2 + alpha, omega = alpha, and omega = \2 - alpha \. In all cases it is shown that the real and imaginary parts of the coefficients in the analytic part of these expansions can be expressed in terms of complete elliptic integrals of the first and second kinds, while the coefficients in the singular part of the expansions can be expressed in terms of rational functions of a. The behaviour of G(alpha, omega) in the immediate neighbourhood of omega = 0 is also investigated in a similar manner. Finally, several applications of the results are made in lattice statistics. (C) 2001 Academic Press.