Regular functions on a Clifford algebra

Abstract

In [Gentili, G. and Stoppato, C., 2007, Zeros of regular functions and polynomials of a quaternionic variable, n. 1 (Dipartimento di Matematica “U. Dini”, Università di Firenze), Preprint; Gentili, G. and Struppa, D.C., 2006, A new approach to Cullen-regular functions of a quaternionic variable. C. R. Acad. Sci. Paris, Ser. I, 342, 741–744; Gentili, G. and Struppa, D.C., 2007, A new theory of regular functions of a quaternionic variable. Adv. Math., 216, 279–301.] the authors offered an alternative definition and theory of regularity for functions of a quaternionic or octonionic variable, inspired by an idea of Cullen [Cullen, C.G., 1965, An integral theorem for analytic intrinsic functions on quaternions. Duke Mathematics Journal, 32, 139–148]. 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 article 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).

Keywords:

• Functions of hypercomplex variables

AMS Subject Classification::

• 30G35

Acknowledgements

The authors acknowledge the support of Chapman University during the preparation of this article. This work has been partially supported by G.N.S.A.G.A. of INdAM and by MIUR.