Modified Quaternionic Analysis on the Ball

Abstract

A natural generalization of the classical complex analysis to higher dimensions is the theory of monogenic functions (see, e.g. Brackx et al. in Clifford analysis, Pitman, Boston, 1982). Unfortunately the powers of the underlying variable are not in this system. But they are in the modified system introduced by the first author in Leutwiler (Complex Var Theory Appl 20:19–51, 1992). Since this modified system lives in the upper half space {mathbb {R}}_{+}^{3} it is natural to transplant it—with the so-called Cayley mapping—to the unit ball B(0, 1) in {mathbb {R}}^{3}. It is this transplanted system which we are going to investigate in this article. It represents the counterpart to the theory of (H)-solutions studied in Leutwiler (Complex Var Theory Appl 20:19–51, 1992).