Complex Analysis and Operator Theory

, Volume 6, Issue 5, pp 971–985 | Cite as

Cauchy Integral Formulae in Quaternionic Hermitean Clifford Analysis

  • Ricardo Abreu-Blaya
  • Juan Bory-Reyes
  • Fred Brackx
  • Hennie De SchepperEmail author
  • Frank Sommen


The theory of complex Hermitean Clifford analysis was developed recently as a refinement of Euclidean Clifford analysis; it focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two Hermitean Dirac operators constituting a splitting of the traditional Dirac operator. In this function theory, the fundamental integral representation formulae, such as the Borel–Pompeiu and the Clifford–Cauchy formula have been obtained by using a (2 × 2) circulant matrix formulation. In the meantime, the basic setting has been established for so-called quaternionic Hermitean Clifford analysis, a theory centred around the simultaneous null solutions, called q-Hermitean monogenic functions, of four Hermitean Dirac operators in a quaternionic Clifford algebra setting. In this paper we address the problem of establishing a quaternionic Hermitean Clifford–Cauchy integral formula, by following a (4 × 4) circulant matrix approach.


Quaternionic Hermitean Clifford analysis Cauchy integral formula 

Mathematics Subject Classification (2000)



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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Ricardo Abreu-Blaya
    • 1
  • Juan Bory-Reyes
    • 2
  • Fred Brackx
    • 3
  • Hennie De Schepper
    • 3
    Email author
  • Frank Sommen
    • 4
  1. 1.Facultad de Informática y MatemáticaUniversidad de HolguínHolguínCuba
  2. 2.Departamento de MatemáticaUniversidad de OrienteSantiago de CubaCuba
  3. 3.Department of Mathematical Analysis, Faculty of Engineering and ArchitectureGhent UniversityGhentBelgium
  4. 4.Department of Mathematical Analysis, Faculty of SciencesGhent UniversityGhentBelgium

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