Exponential and logarithm of multivector in low dimensional (n=p+q<3) Clifford algebras

TL;DR

This paper derives explicit formulas for the exponential and logarithm of multivectors in low-dimensional real Clifford algebras, extending known results to new algebraic structures and identifying conditions for the logarithm's existence.

Contribution

It provides closed-form expressions for multivector exponential and logarithm in Cl(p,q) algebras with n<3, generalizing previous results to quaternionic and hypercomplex cases.

Findings

Explicit formulas for Cl(1,0) and Cl(0,1) cases.

Extension to 2D quaternionic algebras Cl(0,2), Cl(1,1), Cl(2,0).

Identification of coefficient space sectors where logarithm exists.

Abstract

Closed form expressions for a multivector exponential and logarithm are presented in real Clifford geometric algebras Cl(p,q)when n=p+q=1 (complex and hyperbolic numbers) and n=2 (Hamilton, split and conectorine quaternions). Starting from Cl(0,1) and Cl(1,0) algebras wherein square of a basis vector is either -1 or +1, we have generalized exponential and logarithm formulas to 2D quaternionic algebras, Cl(0,2), Cl(1,1), and Cl(2,0). The sectors in the multivector coefficient space where 2D logarithm exists are found. They are related with a square root of the multivector.