Analytic properties of the effective dielectric constant of a two-dimensional Rayleigh model

Abstract

Analytic properties of the dimensionless static effective dielectric constant f(p, h) of a two-dimensional Rayleigh model (p is the concentration and h is the ratio of the dielectric constants of components) are considered as a function of the complex variable h. It is shown that the only singularities of the function f(p, h) are first-order poles for real h = h _n < 0 (n = 1, 2, ...) with the condensation point h = −1, which form an infinite discrete (countable) set. The positions of the first ten poles of the function f(p, h) and the residues at these points are calculated and represented graphically versus the concentration. Based on the results obtained, a pole-type approximate formula is proposed that describes the behavior of the function f(p, h) over a wide range of p and complex h.