Definition and properties of logopoles of all degrees and orders

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)], 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 not equal 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 >= vertical bar m vertical bar) 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.

Automated summary

Logopoles are a class of solutions to Laplace's equation that exhibit interesting mathematical properties, particularly in axisymmetric problems. They generalize solid spheroidal and solid spherical harmonics, and have special significance for multipoles with negative order. Their definitions conserve key properties of the m = 0 case, involving Legendre functions of the second kind and connections to exterior spheroidal harmonics.