The Polarisability of Cylinder Arrays in Two Dimensions

The technique of solving the two-dimensional Laplace equation by means of conformal transformations is a very useful one. Here we develop this method into one which is applicable to chains and lattices in which the fundamental unit is a cylinder pair. An exact closed form expression for the polarisability with explicit spectral weights is immediate once the appropriate transformation for the given cylinder arrangement is determined. The present formalism provides simple derivations of certain results, among which is a proof of Keller′s theorem. We derive the polarisability formulae for a pair of separate cylinders and then determine the corresponding results for chains and lattices of cylinder pairs.