Definition and properties of logopoles of all degrees and orders.

Abstract

Logopoles are a recently proposed class of solutions to Laplace's equation with intriguing links to both solid spheroidal and solid spherical harmonics. They share the same finite-line singularity as the former and provide a generalization of the latter as multipoles of negative order. In a previous paper [Majic and Le Ru, Phys. Rev. Res. 1, 033213 (2019)2643-156410.1103/PhysRevResearch.1.033213], we introduced and discussed the properties and applications of these new functions in the special case of axisymmetric problems (with azimuthal index m=0). This allowed us to focus on the physical properties without the added mathematical complications. Here we expand these concepts to the general case m≠0. The chosen definitions are motivated to conserve some of the most interesting properties of the m=0 case. This requires the inclusion of Legendre functions of the second kind with degree -m≤n<m (in addition to the usual n≥|m|) and we show that these are also related to the exterior spheroidal harmonics. We show that logopoles can also be defined for n≤m and discuss in particular logopoles of degree n=-m, which correspond to the potential of line segments of uniform polarization density.