ABSTRACT
Clifford analysis started as an attempt to generalize one-variable complex analysis to higher dimensions using Clifford algebras generated from Euclidean space. More recently, deep and unexpected links to classical harmonic analysis, several complex variables, and representation theory have been discovered. In the early stages the subject was developed exclusively in three and four dimensions using the quaternionic division algebra, which is an example of a Clifford algebra. Later it was realized that results obtained in the quaternionic setting, particularly the generalization of Cauchy’s integral formula, did not exclusively rely on the division algebra property of the quaternions, but that it is sufficient for an algebra to contain a vector subspace where all non-zero vectors are invertible in the algebra. In the Clifford algebra setting, this invertibility corresponds to the usual Kelvin inversion of vectors in Euclidean space. This fact is not too surprising, given that Clifford algebras are specifically designed to help describe the geometric properties of quadratic forms on vectors spaces, see for instance, [2], For some time, it has been understood by most people working with Clifford analysis that most results so far obtained in quaternionic analysis more or less automatically extend to all finite dimensions using Clifford algebras.
