Regular functions on a Clifford Algebra

The authors offered an alternative definition and theory of regularity for functions of a quaternionic or octonionic variable, inspired by an idea of Cullen. This alternative theory is intriguing because it allows the study of natural power series (and polynomials) with quaternionic or octonionic coefficients, which is excluded when the Fueter approach is followed. In this paper the authors follow the same ideas to offer a new definition of regularity on the Clifford Algebra Cl(0, 3). It turns out that this setting presents a family of new phenomena, which did not appear in the cases of quaternions and octonions. It is nevertheless possible to find a power series expansion for functions regular on an appropriate subset of Cl(0, 3).