Annali di Matematica Pura ed Applicata

, Volume 190, Issue 2, pp 247–261 | Cite as

Bicomplex hyperfunctions

  • F. Colombo
  • I. Sabadini
  • D. C. Struppa
  • A. Vajiac
  • M. Vajiac


In this paper, we consider bicomplex holomorphic functions of several variables in \({{\mathbb B}{\mathbb C}^n}\) .We use the sheaf of these functions to define and study hyperfunctions as their relative 3n-cohomology classes. We show that such hyperfunctions are supported by the Euclidean space \({{\mathbb R}^n}\) within the bicomplex space \({{\mathbb B}{\mathbb C}^n}\), and we construct an abstract Dolbeault complex that provides a fine resolution for the sheaves of bicomplex holomorphic functions. As a corollary, we show how that the bicomplex hyperfunctions can be represented as classes of differential forms of degree 3n − 1.


Bicomplex numbers PDE systems Syzygy Resolutions Hyperfunctions Duality Dolbeault complex 

Mathematics Subject Classification (2000)

16E05 35C15 13P10 35N05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams W.W., Berenstein C.A., Loustaunau P., Sabadini I., Struppa D.C.: Regular functions of several quaternionic variables and the Cauchy–Fueter complex. J. Geom. Anal. 9(1), 1–15 (1999)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Charak K.S., Rochon D., Sharma N.: Normal families of bicomplex holomorphic functions. Fractals: Complex Geom., Patterns, Scaling Nat. Soc. 17, 257–268 (2009)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Colombo, F., Sabadini, I., Struppa, D.C., Vajiac, A., Vajiac, M.: Singularities of functions of one and several bicomplex variables, to appear in Ark. MatematikGoogle Scholar
  4. 4.
    Colombo F., Damiano A., Sabadini I., Struppa D.C.: Quaternionic hyperfunctions on five-dimensional varieties in \({\mathbb{H}^2}\) . J. Geom. Anal. 17(3), 435–454 (2007)MATHMathSciNetGoogle Scholar
  5. 5.
    Colombo F., Sabadini I., Sommen F., Struppa D.C.: Analysis of Dirac Systems and Computational Algebra. Birkhäuser, Boston (2004)CrossRefMATHGoogle Scholar
  6. 6.
    Kato G., Struppa D.C.: Fundamentals of Algebraic Microlocal Analysis. Marcel Dekker, New York (1999)MATHGoogle Scholar
  7. 7.
    Price G.B.: An Introduction to Multicomplex Spaces and Functions. Marcel Dekker, New York (1991)MATHGoogle Scholar
  8. 8.
    Rochon D.: On a relation of bicomplex pseudoanalytic function theory to the complexified stationary Schrödinger equation. Complex Variables Elliptic Equ. 53, 501–521 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Rochon, D., Shapiro, M.: On algebraic properties of bicomplex and hyperbolilc numbers. Anal. Univ. Oradea, Fasc. Math. 11 (2004)Google Scholar
  10. 10.
    Ryan J.: Complexified Clifford analysis. Complex Variables Elliptic Equ. 1, 119–149 (1982)CrossRefMATHGoogle Scholar
  11. 11.
    Ryan J.: C 2 extentions of analytic functions in the complex plane. Adv. Appl. Clifford Algebras 11, 137–145 (2001)CrossRefMathSciNetGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2010

Authors and Affiliations

  • F. Colombo
    • 1
  • I. Sabadini
    • 1
  • D. C. Struppa
    • 2
  • A. Vajiac
    • 2
  • M. Vajiac
    • 2
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.Schmid College of Science, One University DriveChapman UniversityOrangeUSA

Personalised recommendations