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 both solid spheroidal and spherical harmonics. They share the same finite-line singularity as former provide generalization latter multipoles negative order. In previous paper [Majic Le Ru, Phys. Rev. Res. 1, 033213 (2019)], we introduced discussed properties applications these new functions in special case axisymmetric problems (with azimuthal index $m=0$). This allowed us focus on physical without added mathematical complications. Here expand concepts general $m\ensuremath{\ne}0$. The chosen definitions motivated conserve some most interesting $m=0$ case. requires inclusion Legendre second kind degree $\ensuremath{-}m\ensuremath{\le}n<m$ (in addition usual $n\ensuremath{\ge}|m|$) show that also related exterior We logopoles can be defined for $n\ensuremath{\le}m$ discuss particular $n=\ensuremath{-}m$, which correspond potential line segments uniform polarization density.