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Logopoles are a recently proposed class of solutions to Laplace's equationwith intriguing links to both solid spheroidal and solid spherical harmonics.They share the same finite line singularity with the former and provide ageneralization of the latter as multipoles of negative order. In [Phys. Rev.Res. 1, 033213 (2019)], we introduced and discussed the properties andapplications of these new functions in the special case of axi-symmetricproblems (with azimuthal index $m=0$). This allowed us to focus on the physicalproperties without the added mathematical complications. Here we expand theseconcepts to the general case $m\neq 0$. The chosen definitions are motivated toconserve some of the most interesting properties of the $m=0$ case. Thisrequires 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 arealso related to the exterior spheroidal harmonics. We show that Logopoles canalso be defined for $n\le m$, and discuss in particular logopoles of degree$n=-m$, which correspond to the potential of line segments of uniformpolarization density.

Definition and properties of logopoles of all degrees and orders