Calculation of the Lattice Green's Function for the bcc, fcc, and Rectangular Lattices

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Abstract
Formulas are provided which are convenient for the evaluation of the lattice Green's functions for the bcc, fcc, and rectangular lattices, at an arbitrary complex variable. The formulas involve the complete elliptic integral of the first kind with complex modulus; the integral has been found to be evaluated efficiently by the method of the arithmetic‐geometric mean, generalized for the case with complex modulus. The expansions of the lattice Green's functions around the singular points are given for the bcc and fcc lattices. These lattice Green's functions diverge at a variable. The singular points responsible for the divergences are found to form one‐dimensional lines.