Partial Primitives for Clifford Algebra-Valued Functions

Abstract

In this paper we generalize the concept of primitivation of monogenic functions taking values in a Clifford algebra, which is on its own a generalization to higher dimension of the primitivation problem for holomorphic functions in the complex plane. This problem can be stated as follows: given a monogenic function \(f(x_0, \underline{x})\) on \({\mathbb{R}}^{m+1}\) , i.e. a solution for the generalized Cauchy-Riemann operator D on \({\mathbb{R}}^{m+1}\) , construct a monogenic function \(g(x_0, \underline{x})\) such that \(\overline{D}g = f\). In view of the fact that, for monogenic functions g, this can be written as \(\partial_{x_0}\)g = f, a straightforward generalization consists in replacing the scalar generator \(\partial_{x_0}\) of translations in the x0-direction by a generator of another transformation group. In this paper we consider translations in more dimensions.