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

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.