Half-cylinders and branch-cut response functions

We consider the polarizability of half cylinders (explicitly, circular cylinders cut in half along a diameter), and the permittivity or permeability of dilute arrays of half cylinders, extending the work of of M. Pitkonen. We show that Pitkonen's results for the polarizability of half cylinders may be greatly simplified using a property of the dilogarithm function, and we comment on the correct choice of branches of the resulting simplified expression for complex cylinder permittivities. We discuss the relationship between response functions for systems having discrete resonances, as compared with those having branch-cut spectra, and highlight some of the advantages the latter may have in applications such as energy harvesting and design of metamaterials.