Spheroidal Domains and Geometric Analysis in Euclidean Space

TL;DR

This paper explores spheroidal domains and geometric analysis in Euclidean space using Clifford's geometric algebra, focusing on spheroidal projections, the Laplace equation, and symmetries, with a unified approach to key analytical tools.

Contribution

It introduces the concept of quasi-monogenic functions and applies geometric algebra to study spheroidal domains and related differential equations.

Findings

Unified treatment of Cauchy-Kovalevska extension and kernel functions.

Introduction of quasi-monogenic functions.

Analysis of symmetries in spheroidal domains.

Abstract

Clifford's geometric algebra has enjoyed phenomenal development over the last 60 years by mathematicians, theoretical physicists, engineers and computer scientists in robotics, artificial intelligence and data analysis, introducing a myriad of different and often confusing notations. The geometric algebra of Euclidean 3-space, the natural generalization of both the well-known Gibbs-Heaviside vector algebra, and Hamilton's quaternions, is used here to study spheroidal domains, spheroidal-graphic projections, the Laplace equation and its Lie algebra of symmetries. The Cauchy-Kovalevska extension and the Cauchy kernel function are treated in a unified way. The concept of a quasi-monogenic family of functions is introduced and studied.