Multipole Translation Theory for the Three-Dimensional Laplace and Helmholtz Equations

The mathematical theory of multipole translation operators for the three-dimensional Laplace and Helmholtz equations is summarized and extended and an elementary proof of the inner-to-inner translation theorem is proved.

Abstract

The mathematical theory of multipole translation operators for the three-dimensional Laplace and Helmholtz equations is summarized and extended. New results for the Laplace equation include an elementary proof of the inner-to-inner translation theorem, from which follows the definition of a far-field signature function analogous to that of the Helmholtz equation. The theory for the Helmholtz equation is developed in terms of a new convolutional form of the translation operator, which is connected to Rokhlin’s diagonal form by means of Wigner 3-j symbols.