HIERARCHIES OF STIELTJES FUNCTIONS AND THEIR APPLICATION TO THE CALCULATION OF BOUNDS FOR THE DIELECTRIC-CONSTANT OF A 2-COMPONENT COMPOSITE MEDIUM

The bulk effective dielectric constant epsilon(e)(epsilon1, epsilon2) of a two-component composite medium with a fixed microstructure, considered as function of the ratio epsilon1/epsilon2, is a Stieltjes function with special analytical properties. It is shown how other related Stieltjes functions can be constructed by exploiting partial information that is available about the values of epsilon(e)(epsilon1, epsilon2) and about its series expansion. These functions are used to construct continued fraction representations for epsilon(e)(epsilon1, epsilon2) that can provide exact bounds for e, that incorporate that information. Explicit algorithmic constructions are developed for obtaining these bounds in the real as well as in the complex domain.