Tip singularity of the electromagnetic field at the apex of a material cone

Abstract

The tip Singularity of the electromagnetic field at the apex of a cone is investigated in its most general framework. To this end one considers, without loss of generality, a circularly symmetric cone which separates two simple media having different constitutive parameters, and tries to reveal the asymptotic behaviour of the electromagnetic field created near the apex of the cone by any rotationally symmetric source distribution. To cover various boundary conditions which are extensively used in actual investigations, the cone is supposed to be formed by an infinitely thin material sheet having its own constitutive parameters. The problem of determining the type and order of the singularity in question gives rise to a transcendental algebraic equation involving the Legendre functions of the first kind with complex orders. If the order is a simple root of this equation, then the singularity is always of the algebraic type whereas a multiple root gives rise also to logarithmic singularities.