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Cauchy’s Formula in Clifford Analysis: An Overview

  • Fred BrackxEmail author
  • Hennie De Schepper
  • Roman Lávička
  • Vladimir Souček
Chapter
Part of the Trends in Mathematics book series (TM)

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, Peña-Peña (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.

Keywords

Cauchy’s formula Monogenic functions 

Mathematics Subject Classification (2010)

Primary 30G35 

Notes

Acknowledgements

R. Lávička and V. Souček gratefully acknowledge support by the Czech Grant Agency through grant GA CR 17-01171S.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Fred Brackx
    • 1
    Email author
  • Hennie De Schepper
    • 1
  • Roman Lávička
    • 2
  • Vladimir Souček
    • 2
  1. 1.Clifford Research GroupDepartment of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent UniversityGhentBelgium
  2. 2.Charles UniversityFaculty of Mathematics and Physics, Mathematical InstitutePrahaCzech Republic

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