Slice monogenic functions of a Clifford variable via the $S$-functional calculus

Abstract:

In this paper we define a new function theory of slice monogenic functions of a Clifford variable using the $S$-functional calculus for Clifford numbers. Previous attempts of such a function theory were obstructed by the fact that Clifford algebras, of sufficiently high order, have zero divisors. The fact that Clifford algebras have zero divisors does not pose any difficulty whatsoever with respect to our approach. The new class of functions introduced in this paper will be called the class of slice monogenic Clifford functions to stress the fact that they are defined on open sets of the Clifford algebra $\mathbb {R}_n$. The methodology can be generalized, for example, to handle the case of noncommuting matrix variables.