FACTORIZATION OF POTENTIAL AND FIELD DISTRIBUTIONS WITHOUT UTILIZING THE ADDITION THEOREM

Abstract

The construction of multipole expansions for potential- and field distribution functions relies on the existence of certain factorization formulae, which are conventionally formulated by the so-called addition theorem. The main objective in this talk is the presentation of a recipe for obtaining factorization formulae without the need for the addition theorem. Factorization, as used here, refers to the multiplicative decomposition of the potential (field) functions into functions, which depend on the co-ordinates of the source- and the observation point individually. The multiplicative decomposition is crucial in all those instances, where we need to, e.g. integrate over a large number of source points, while keeping the observation point constant. Such circumstances routinely arise in large scale engineering computations based on singular surface integrals (boundary element method applications). Simply speaking, the basic idea behind the factorization is, to break down the Euclidean distance between the source- and the observation point, into terms, which depend on the co-ordinates of the source point position vector and the observation point position vector, separately. There are several possibilities to achieve this objective, which will be described in our presentation: (1) Multipole expansions and the underlying addition theorem result in factorized forms with functions being in real space. (2) An alternative approach for obtaining factorized forms consists of employing integral transforms with certain translational shift invariance properties; a prominent example being the Fourier transform, which results in spectral decompositions. It should be pointed out that obtaining factorized forms in spectral domain is based on the diagonalization of the underlying partial differential equations with respect to a distinguished direction in space. (3) In this talk we will diagonalize the governing equations with respect to the radial direction and use field expansions in the real space. It will be shown that our diagonalization allows the factorization of the field distribution functions directly. The presentation can be outlined as follows: we first briefly discuss the addition theorem. Then we show how to diagonalize the Poisson's equation in spherical co-ordinates. Finally, we provide a detailed discussion on how to factorize the field expansion functions. The derivation procedure should be appealing to engineers, who will easily recognize the wide range applicability of the proposed iterative method. It can shown that the proposed recipe is applicable to electrodynamic and elastodynamic problems involving isotropic, anisotropic, or bi-anisotropic materials. Furthermore, it can be shown that whenever a system of PDEs permits diagonalization, our procedure is applicable. In a recent contribution we formulated a conjecture claiming that every physically-realizable system of PDEs allows diagonalization, with respect to a certain distinguished direction in space. Based upon this conjecture it is reasonable to expect, that our formulation, applied to various problems, will result in generalizations of the expansion formulae appearing in the addition theorem. We conclude that our method as a pleasant byproduct, and perhaps ironically, can be instrumental in formulating new addition theorems