Analyticity of the effective dielectric constant of two-phase media

Consider an isotropic composite of two isotropic phases. When the microgeometry is fixed it has a complex effective dielectric constant ε⋆(ε1, ε2), which is a function of the complex dielectric constants ε1 and ε2 of the phases that depend on the frequency ω of the applied field. As a prelude to the proof given in the next section, we will now present a strong argument that shows why ε⋆(ε1, ε2) should have some rather special analytic properties. The argument is based on the premise that analyticity properties of the dielectric constant as a function of the frequency ω should extend to composite materials.

The properties of the function ε1(ω) [or ε2(ω)] are well-known and are discussed, for example, by Jackson (1975); see also section 11.1 on page 222. The function ε1(ω) is analytic in the upper half ω-plane, Im(ω) > 0. When Re(ω) = 0 the function takes real values of ε1(ω) ≥ 1, which decrease and approach 1 as │ω│ → ∞. Positive imaginary values of ε1(ω) occur when ω has a positive real part and negative imaginary values of ε1(ω) when ω has a negative real part. As ω ranges over the upper half-plane ε1(ω) can in principle range anywhere in the cut complex plane, where the cut extends along the real axis from −∞ to 1.