\(S\)-continued fraction method for the investigation of a complex dielectric constant of two-components composite (Q1376325)

As it was proved by Bergman (see the references cited in the bibliography), the effective dielectric constant \(\varepsilon_e\) of a two-phase medium can be expressed by means of a Stieltjes function of the ratio of the dielectric constants of both components. Starting from this representation, the authors derive its \(S\)-continued fraction and establish bounds suitable for theoretical and numerical investigation of \(\varepsilon_e\) in the complex domain. Explicit inequalities are given in the real case. As an example, these results are applied to the computation of the bulk dielectric constants for regular arrays of spheres. Moreover, bounds for the square and hexagonal arrays of cylinders are obtained numerically.