Israel Journal of Mathematics

, Volume 171, Issue 1, pp 385–403 | Cite as

Slice monogenic functions

  • Fabrizio Colombo
  • Irene Sabadini
  • Daniele C. Struppa


In this paper we offer a new definition of monogenicity for functions defined on ℝn+1 with values in the Clifford algebra ℝn following an idea inspired by the recent papers [6], [7]. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra ℝn. We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Resarch Notes in Math., 76, Pitman (Advanced Publishing Program), Boston, MA, 1982.MATHGoogle Scholar
  2. [2]
    F. Colombo, I. Sabadini, F. Sommen and D. C. Struppa, Analysis of Dirac Systems and Computational Algebra, Progress in Mathematical Physics, Vol. 39, Birkhäuser, Boston, 2004.Google Scholar
  3. [3]
    F. Colombo, I. Sabadini and D. C. Struppa, A new functional calculus for noncommuting operators, Journal of Functional Analysis 254 (2008), 2255–2274.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    C. G. Cullen, An integral theorem for analytic intrinsic functions on quaternions, Duke Mathematical Journal 32 (1965), 139–148.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    R. Delanghe, F. Sommen and V. Soucek, Clifford Algebra and Spinor-valued Functions, Mathematics and Its Applications 53, Kluwer Academic Publishers Group, Dordrecht 1992.MATHGoogle Scholar
  6. [6]
    G. Gentili and D. C. Struppa, A new approach to Cullen-regular functions of a quaternionic variable, Comptes Rendus Mathématique Académie des Sciences. Paris, 342 (2006), 741–744.MATHMathSciNetGoogle Scholar
  7. [7]
    G. Gentili and D. C. Struppa, A new theory of regular functions of a quaternionic variable, Advances in Mathematics 216 (2007), 279–301.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    G. Gentili and D. C. Struppa, Regular functions on a Clifford Algebra, Complex Variables and Elliptic Equations 53 (2008), 475–483.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    T. Y. Lam, A First Course in Noncommutative Rings, 2nd edition, Graduate Texts in Mathematics, Vol. 131 Springer-Verlag, New York, 2001.MATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2009

Authors and Affiliations

  • Fabrizio Colombo
    • 1
  • Irene Sabadini
    • 1
  • Daniele C. Struppa
    • 2
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.Department of Mathematics and Computer SciencesChapman UniversityOrangeUSA

Personalised recommendations