Definition and properties of logopoles of all degrees and orders

TL;DR

This paper extends the mathematical framework of logopoles, special solutions to Laplace's equation, to include all degrees and orders, revealing new properties and connections to spheroidal harmonics for complex azimuthal cases.

Contribution

It generalizes the definition of logopoles to non-zero azimuthal index m, incorporating Legendre functions of the second kind and exploring their relation to spheroidal harmonics.

Findings

Logopoles are now defined for all degrees and orders, including negative and zero values.

The inclusion of Legendre functions of the second kind enriches the mathematical structure.

Logopoles of degree n=-m relate to potentials of line segments with uniform polarization.

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 with the former and provide a generalization of the latter as multipoles of negative order. In [Phys. Rev. Res. 1, 033213 (2019)], we introduced and discussed the properties and applications of these new functions in the special case of axi-symmetric 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 $m\neq 0$. 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 $-m\leq n<m$ (in addition to the usual $n\geq |m|$) and we show that these are also related to the exterior spheroidal harmonics. We show that Logopoles can also be defined for $n\le m$, and discuss in particular logopoles of degree $n=-m$, which correspond to the potential of line segments of uniform polarization density.