Hyperderivatives in Clifford Analysis and Some Applications to the Cliffordian Cauchy-type Integrals

  • M.E. Luna-Elizarrarás
  • M.A. Macías-Cedeño
  • M. Shapiro
Part of the Trends in Mathematics book series (TM)


We introduce the notion of the derivative as the limit of a quotient where the numerator and the denominator represent a kind of the “increments” of a function and of the independent variable respectively. The directional derivative is introduced where a direction means a hyperplane in ℝm+1 for a Clifford algebra Cl o,m. The latter applies for obtaining a formula showing how to exchange the integral sign and the hyperderivative of the Cliffordian Cauchy-type integral as a hyperholomorphic function.


Hyperderivative m-dimensional directional hyperderivative Cliffordian Cauchy-type integrals 

Mathematics Subject Classification (2000)

Primary 30G35 Secondary 32A10 


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  1. [1]
    R. Delanghe, F. Sommen, V. Souček. Clifford algebra and spinor-valued functions. Kluwer Academic Publishers. Mathematics and its Applications, 53, 1992.Google Scholar
  2. [2]
    J.E. Gilbert, M.A.M. Murray. Clifford algebras and Dirac operators in harmonic analysis. Cambridge Studies in Adv. Math., 26, 1991.Google Scholar
  3. [3]
    K. Gürlebeck, H.R. Malonek. A Hypercomplex Derivative of Monogenic Functions inn+1 and its Applications. Complex Variables, 39, 1999, 199–228.zbMATHGoogle Scholar
  4. [4]
    K. Gürlebeck, W. Sprössig. Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley and Sons, 1997.Google Scholar
  5. [5]
    V.V. Kravchenko, M.V. Shapiro. Integral representations for spatial models of mathematical physics. Addison-Wesley-Longman, Pitman Research Notes in Mathematics, 351, 1996.Google Scholar
  6. [6]
    P. Lounesto. Clifford algebras and Spinors. Second edition, London Math. Soc. Lecture Note Series, 286, 2001.Google Scholar
  7. [7]
    H. Malonek. A New Hypercomplex Structure of the Euclidean Spacem+1 and the Concept of Hypercomplex Differentiability. Complex Variables, 14, 1990, 25–33.zbMATHMathSciNetGoogle Scholar
  8. [8]
    H.R. Malonek. Selected Topics in Hypercomplex Function Theory. Clifford Algebras and Potential Theory, S.-L. Eriksson, ed., University of Joensun. Report Series 7, 2004, 111–150.Google Scholar
  9. [9]
    I.M. Mitelman, M. Shapiro. Differentiation of the Martinelli-Bochner Integrals and the Notion of Hyperderivability. Math. Nachr., 172, 1995, 211–238.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    I.R. Porteous. Clifford Algebras and the Classical Groups. Cambridge University Press, Cambridge, 1995.zbMATHGoogle Scholar
  11. [11]
    M.V. Shapiro, N.L. Vasilevski. Quaternionic ψ-Hyperholomorphic Functions, Singular Integral Operators and Boundary Value Problems I. ψ-Hyperholomorphic Func-tion Theory. Complex Variables, 27, 1995, 17–46.zbMATHMathSciNetGoogle Scholar
  12. [12]
    A. Sudbery. Quaternionic Analysis. Math. Proc. Camb. Phil. Soc. 85, 1979, 199–225.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • M.E. Luna-Elizarrarás
    • 1
  • M.A. Macías-Cedeño
    • 1
  • M. Shapiro
    • 1
  1. 1.Departamento MatemáticasE.S.F.M. del I.P.N.México, D.F.México

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