Uniqueness Theorems for Point Source Expansions: DIDACKS V

TL;DR

This paper proves the uniqueness of fields generated by finite point source distributions in various dimensions, using complex analysis techniques to generalize classical results for harmonic functions and point sources.

Contribution

It introduces a new proof approach for uniqueness theorems involving point sources, extendable to dipoles and higher order poles across different dimensions.

Findings

Finite point masses produce unique exterior fields.

Finite point dipoles also produce unique fields.

Higher order poles correspond to unique analytic functions.

Abstract

Finite collections of point masses contained in some bounded domain produce a unique field in the exterior domain, which means that the associated basis functions (often called ``fundamental solutions'') are independent. A new proof of this result is given in this paper that can be generalized to other finite combinations of point source distributions. For example, this paper shows in $\mathbb{R}^3$ that a finite combination of point dipoles produces a unique field. The strategy employed in the paper is to develop results for analytic functions in the complex plane and then carry them over to harmonic functions in the real plane, and from there to harmonic functions in $\mathbb{R}^3$. More results are shown for $\mathbb{R}^2$ and for $\mathbb{C}$ than are shown for more general settings. For example, in the complex plane, the paper shows that a finite combination of higher order poles of any order in the interior of a unit disk always corresponds to a unique analytic function in the exterior of a unit disk.