Coordinate-free exponentials of general multivector (MV) in Cl(p,q) algebras for p+q=3

TL;DR

This paper derives coordinate-free exponential formulas for general multivectors in specific 3D Clifford algebras, simplifying complex expressions and enabling applications in differential equations, signal processing, and robotics.

Contribution

It provides the first closed-form, coordinate-free exponential expressions for general multivectors in Cl(0,3), Cl(3,0), Cl(1,2), and Cl(2,1).

Findings

Formulas simplify to Moivre-type functions after disentanglement.

Results applicable to solving GA differential equations.

Potential applications in signal processing and robotics.

Abstract

Closed form expressions in a coordinate-free form in real Clifford geometric algebras (GAs) Cl(0,3), Cl(3,0)$, Cl(1,2) and Cl(2,1) are found for exponential function when the exponent is the most general multivector (MV). The main difficulty in solving the problem is connected with an entanglement or mixing of vector and bivector components. After disentanglement, the obtained formulas simplify to the well-known Moivre-type trigonometric/hyperbolic function for vector or bivector exponentials. The presented formulas may find wide application in solving GA differential equations, in signal processing, automatic control and robotics.