Functions of multivector variables

TL;DR

This paper extends elementary functions to multivectors in Clifford algebra, revealing new relationships between complex numbers, quaternions, and vectors, and providing formulas for key multivector operations across dimensions.

Contribution

It introduces a unified approach to elementary functions over Clifford multivectors, uncovering new algebraic relationships and formulas in 2D and 3D spaces.

Findings

Complex raised to vector powers yields quaternions

A single formula for square root, amplitude, and inverse of multivectors

Clifford algebra in three dimensions offers versatile operations

Abstract

As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems. One key relationship that we discover is that a complex number raised to a vector power produces a quaternion thus combining these systems within a single equation. We also find a single formula that produces the square root, amplitude and inverse of a multivector over one, two and three dimensions. Finally, comparing the functions over different dimension we observe that $ C\ell \left (\Re^3 \right) $ provides a particularly versatile algebraic framework.