Cauchy’s Formula in Clifford Analysis: An Overview

Abstract

The Clifford-Cauchy integral formula has proven to be a corner stone of the monogenic function theory, as is the case for the traditional Cauchy formula in the theory of holomorphic functions in the complex plane. In the recent years, several new branches of Clifford analysis have emerged. Similarly as hermitian Clifford analysis was introduced in Euclidean space \(\mathbb {R}^{2n}\) of even dimension as a refinement of Euclidean Clifford analysis by the introduction of a complex structure on \(\mathbb {R}^{2n}\), quaternionic Clifford analysis arose as a further refinement by the introduction of a so-called hypercomplex structure \(\mathbb {Q}\), i.e. three complex structures (\(\mathbb {I}\), \(\mathbb {J}\), \(\mathbb {K}\)) which submit to the quaternionic multiplication rules, on Euclidean space \(\mathbb {R}^{4p}\), the dimension now being a fourfold. Two, respectively four differential operators are constructed, leading to invariant systems under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are respectively called hermitian and quaternionic monogenic functions. The basics of hermitian monogenicity have been studied in e.g. Brackx et al. (Compl Anal Oper Theory 1(3):341–365, 2007; Complex Var Elliptic Equ 52(10–11):1063–1079, 2007; Appl Clifford Algebras 18(3–4):451–487, 2008). Quaternionic monogenicity has been developed in, amongst others, Pena-Pena (Complex Anal Oper Theory 1:97–113, 2007), Eelbode (Complex Var Elliptic Equ 53(10):975–987, 2008), Damiano et al. (Adv Geom 11:169–189, 2011), and Brackx et al. (Adv Appl Clifford Alg 24(4):955–980, 2014; Ann Glob Anal Geom 46:409–430, 2014). In this contribution, we give an overview of the ways in which a Cauchy integral representation formula has been established within each of these frameworks.