Notions of Regularity for Functions of a Split-Quaternionic Variable

Abstract

The utility and beauty of the theory holomorphic functions of a complex variable leads one to wonder whether analogous function theories exist for other (presumably higher dimensional) algebras. Over the last several decades it has been shown that much of Complex analysis extends to a similar theory for the family of Clifford Algebras $C\ell _{0,n}$. However, there has yet to be a complete description for the general theory over the family of Clifford algebras $C\ell _{p,q}$ (for $p\ne 0$). In this work, we describe two different approaches from the literature for finding a theory of “holomorphic” functions of a split-quaternionic variable (which is the Clifford algebra $C\ell _{1,1}$). We show that one approach yields a relatively trivial theory, while the other gives a richer one. In the second instance, we describe a simple subclass of “holomorphic” functions and give two examples of an analogue of the Cauchy-Kowalewski extension in this context.