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 $m\ne 0$. 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 $-m\le n<m$ (in addition to the usual $n\ge \left|m\right|$) 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.

Figure 1

Schematic of the offset spherical and spheroidal coordinate systems.

Figure 2

Intensity plots of a representative selection of low-degree logopoles and PSSHs for orders $m=1,2$. For better visualization, the functions have been transformed by scaling and taking the arcsinh, which is similar to plotting on a log scale but allows for negative values. Red represents positive; white, 0; and blue, negative. The PSSHs have been translated up the $z$ axis for a closer visual comparison. $R=1$. The logopoles and $Q_n^m$ functions were computed via forward recurrence, which has no visible error at these low orders.

Figure 3

A representative sample of logopoles for $n<0$. The arcsinh of the functions is plotted for better visualization. Red represents positive; blue, negative; and white, 0. $R=1$. The functions are singular on the axial line segment $0\le z\le R$.