Electostatic edge modes of a dielectric wedge

The calculation of the electrostatic edge modes of a dielectric wedge by Dobrzynski and Maradudin is reconsidered. It is shown that the electric fields are singular and the field energy is infinite near the edge. Rounding the edge by taking the boundary to be a hyperbolic cylinder removes the singularity, but changes the spectrum from continuous and independent of $q$ (where $\phi \propto e^{\mathrm{iqz}}$ to discrete and dependent upon $q$. For the rounded edge these modes possess an orthogonality property convenient for calculating the response of the dielectric to an external charge.