Advertisement

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
Article

Abstract

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.

Keywords

Quaternionic Hermitean Clifford analysis Cauchy integral formula 

Mathematics Subject Classification (2000)

30G35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abreu Blaya, R., et al.: Hermitean Cauchy integral decomposition of continuous functions on hypersurfaces. Bound. Value Probl. 2008, article ID 425256 (2008)Google Scholar
  2. 2.
    Abreu Blaya R. et al.: A Hermitean Cauchy formula on a domain with fractal boundary. J. Math. Anal. Appl. 369, 273–282 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Abreu Blaya R., Bory Reyes J., Moreno García T.: Hermitian decomposition of continuous functions on a fractal surface. Bull. Braz. Math. Soc. 40(1), 107–115 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis. In: Research Notes in Mathematics, vol. 76. Pitman, Boston (1982)Google Scholar
  5. 5.
    Brackx F. et al.: Fundaments of Hermitean Clifford analysis. Part I: complex structure. Complex Anal. Oper. Theory 1(3), 341–365 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brackx F. et al.: Fundaments of Hermitean Clifford analysis. Part II: splitting of h-monogenic equations. Complex Var. Elliptic Equ. 52(10–11), 1063–1079 (2007)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brackx F., De Schepper H., Sommen F.: The Hermitian Clifford analysis toolbox. Adv. Appl. Clifford Alg. 18(3–4), 451–487 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brackx F., De Knock B., De Schepper H., Sommen F.: On Cauchy and Martinelli–Bochner integral formulae in Hermitean Clifford analysis. Bull. Braz. Math. Soc. 40(3), 395–416 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Damiano A., Eelbode D., Sabadini I.: Quaternionic Hermitian spinor systems and compatibility conditions. Adv. Geom. 11, 169–189 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Damiano A., Eelbode D., Sabadini I.: Algebraic analysis of Hermitian monogenic functions. C. R. Acad. Sci. Paris Ser. I 346, 139–142 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Delanghe R., Sommen F., Souček V.: Clifford Algebra and Spinor-Valued Functions. Kluwer, Dordrecht (1992)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gürlebeck, K., Habetha, K., Sprössig, W.: Holomorphic functions in the plane and n-dimensional space. Birkhäuser Verlag, Basel (2008). Translated from the 2006 German originalGoogle Scholar
  13. 13.
    Gürlebeck K., Sprössig W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1998)Google Scholar
  14. 14.
    Gilbert J., Murray M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)CrossRefzbMATHGoogle Scholar
  15. 15.
    Krantz S.: Function Theory of Several Complex Variables, 2nd edn. Wadsworth & Brooks/Cole, Pacific Grove (1992)zbMATHGoogle Scholar
  16. 16.
    Kytmanov A.: The Bochner–Martinelli Integral and its Applications. Birkhaüser, Basel-Boston- Berlin (1995)CrossRefzbMATHGoogle Scholar
  17. 17.
    Peña-Peña D., Sabadini I., Sommen F.: Quaternionic Clifford analysis: the Hermitian setting. Complex Anal. Oper. Theory 1, 97–113 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rocha-Chávez, R., Shapiro, M., Sommen, F.: Integral theorems for functions and differential forms in \({{\mathbb{C}}^m}\) . In: Research Notes in Mathematics, vol. 428Google Scholar
  19. 19.
    Ryan J.: Complexified Clifford analysis. Complex Var. Theory Appl. 1(1), 119–149 (1982)CrossRefzbMATHGoogle Scholar
  20. 20.
    Sabadini I., Sommen F.: Hermitian Clifford analysis and resolutions. Math. Methods Appl. Sci. 25(16–18), 1395–1413 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations