Analytic cliffordian functions

TL;DR

This paper explores the theory of Cliffordian functions in higher dimensions, highlighting differences based on the parity of the dimension and establishing foundational functions like powers of identity.

Contribution

It introduces the structure of holomorphic Cliffordian functions across different dimensions, emphasizing the role of algebra parity and foundational power functions.

Findings

Cliffordian functions depend on the parity of the dimension n.

For odd n, functions are defined via the Cauchy-Riemann operator.

Powers of identity serve as fundamental building blocks in all dimensions.

Abstract

In classical function theory, a function is holomorphic if and only if it is complex analytic. For higher dimensional spaces it is natural to work in the context of Clifford algebras. The structures of these algebras depend on the parity of the dimension n of the underlying vector space. The theory of holomorphic Cliffordian functions reflects this dependence. In the case of odd n the space of functions is defined by an operator (the Cauchy-Riemann equation) but not in the case of even $n$. For all dimensions the powers of identity (z^n, x^n) are the foundation of function theory.