论文标题
所有度和订单的徽标的定义和属性
Definition and properties of logopoles of all degrees and orders
论文作者
论文摘要
徽标是最近提出的针对拉普拉斯方程的解决方案类别，具有与固体球形和固体球形谐波的有趣联系。他们与前者共享相同的有限线奇异性，并将后者作为负顺序的多底数概括。在[Phys。 Rev. Res。 1，033213（2019）]，我们在特殊情况下引入并讨论了这些新功能的属性和应用（Azimuthal Index $ M = 0 $）。这使我们能够关注物理属性，而无需增加数学并发症。在这里，我们将这些概念扩展到一般情况$ M \ neq 0 $。所选的定义是为了保护$ m = 0 $ case的一些最有趣的属性。这就需要将第二种的legendre函数与度量$ -m \ leq n <m $（除了通常的$ n \ geq | m | $），我们表明这些功能也与外部球体谐波有关。我们表明，也可以针对$ n \ le m $定义logopoles，并在尤其是$ n = -m $的徽标上进行讨论，这与均匀极化密度的线段的潜力相对应。
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.
