Spectral function for a conducting sheet containing circular inclusions

The conductivity (or dielectric behavior) of a binary composite system is conveniently expressed in terms of a spectral function, which is determined by the geometry of the composite. In this paper we examine the case of circular inclusions in a conducting sheet and keep terms up to second order in the inclusion concentration f. The two-inclusion problem can be solved exactly using multiple images, and we use this solution to construct the spectral function. We show that the spectral function is a truncated Lorentzian that can be calculated in a simple closed form. Both the weight and the width of the spectral function are linear in f.