# A new class of solutions to Laplace equation: Regularized multipoles of negative orders

## TL;DR

This paper introduces logopoles, a new class of solutions to the Laplace equation involving spherical harmonics of the second kind, which extend multipole methods to negative orders and may enhance physical problem-solving.

## Contribution

It presents logopoles as a novel solution class, establishing their relation to existing harmonics and extending multipole techniques to negative orders.

## Key findings

- Logopoles have finite line singularities similar to spheroidal harmonics.
- They relate to solid spherical harmonics and extend the multipole ladder.
- Potential applications in physical problems using spherical or spheroidal harmonics.

## Abstract

We introduce a new class of solutions to Laplace equation, dubbed logopoles, and use them to derive a new relation between solutions in prolate spheroidal and spherical coordinates. The main novelty is that it involves spherical harmonics of the second kind, which have rarely been considered in physical problems because they are singular on the entire z axis. Logopoles, in contrast, have a finite line singularity like solid spheroidal harmonics, but are also closely related to solid spherical harmonics and can be viewed as an extension of the standard multipole ladder toward the negative multipolar orders. These new solutions may prove a fruitful alternative to either spherical or spheroidal harmonics in physical problems.