Complex Analysis and Operator Theory

, Volume 7, Issue 5, pp 1675–1711

Complex Laplacian and Derivatives of Bicomplex Functions

  • M. E. Luna-Elizarrarás
  • M. Shapiro
  • D. C. Struppa
  • A. Vajiac


In this paper we study in detail the theory of bicomplex holomorphy, in the context of the several ways in which bicomplex numbers can be considered. In particular we will show how the notions of bicomplex derivability and bicomplex holomorphy can be interpreted in these different ways, and the consequences that can be derived.


Bicomplex derivability Bicomplex differentiability Bicomplex holomorphic functions Complex and hyperbolic Laplacians 

Mathematics Subject Classification (2010)

30G35 32A30 32A10 


  1. 1.
    Campos, H.M., Kravchenko, V.V.: Fundamentals of bicomplex pseudoanalytic function theory: Cauchy integral formulas, negative formal powers and Schrödinger equations with complex coefficients. Complex Anal. Oper. Theory 7(3), 634–668 (2013)Google Scholar
  2. 2.
    Charak, K.S., Rochon, D., Sharma, N.: Normal families of bicomplex holomorphic functions. Fractals 17(3), 257–268 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Colombo, F., Sabadini, I., Struppa, D.C., Vajiac, A., Vajiac, M.B.: Singularities of functions of one and several bicomplex variables. Ark. Math. 49(2), 277–294 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Colombo, F., Sabadini, I., Struppa, D.C., Vajiac, A., Vajiac, M.B.: Bicomplex hyperfunctions. Ann. Math. Pura Appl. (2) 190(2), 247–261 (2011)Google Scholar
  5. 5.
    Gal, S.G.: Introduction to Geometric Function Theory of Hypercomplex Variables. Nova Science Publishers, Inc., Hauppauge (2004)Google Scholar
  6. 6.
    Krantz, S.G.: Function Theory of Several Complex Variables. AMS Chelsea Publishing, Providence (1992)MATHGoogle Scholar
  7. 7.
    Luna-Elizarrarás, M.E., Shapiro, M.: A survey on the (hyper-) derivatives in complex, quaternionic and Clifford analysis. Milan J. Math. 79(2), 521–542 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex numbers and their elementary functions. CUBO Math. J. 14(2), 61–80 (2012)CrossRefMATHGoogle Scholar
  9. 9.
    Pogorui, A.A., Rodriguez-Dagnino, R.M.: On the set of zeros of bicomplex polynomials. Complex Var. Elliptic Equ. 51(7), 725–730 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Price, G.B.: An introduction to multicomplex spaces and functions. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 140. Marcel Dekker, Inc., New York (1991)Google Scholar
  11. 11.
    Rochon, D., Shapiro, M.: On algebraic properties of bicomplex and hyperbolic numbers. An. Univ. Oradea Fasc. Math. 11, 71–110 (2004)MathSciNetMATHGoogle Scholar
  12. 12.
    Rochon, D.: On a relation of bicomplex pseudoanalytic function theory to the complexified stationary Schrödinger equation. Complex Var. Elliptic Equ. 53, 501–521 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ryan, J.: Complexified Clifford analysis. Complex Var. Elliptic Equ. 1, 119–149 (1982)CrossRefMATHGoogle Scholar
  14. 14.
    Ryan, J.: \({\mathbb{C}}^2\) extensions of analytic functions defined in the complex plane. Adv. Appl. Clifford Algebras 11, 137–145 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Shapiro, M.: Some remarks on generalizations of the one-dimensional complex analysis: hypercomplex approach. Functional Analytic Methods in Complex Analysis and Applications to Partial Differential Equations, pp. 379–401. World Scientific, Singapore (1995)Google Scholar
  16. 16.
    Struppa, D.C., Vajiac, A., Vajiac, M.B.: Remarks on holomorphicity in three settings: complex, quaternionic, and bicomplex. In: Hypercomplex Analysis and Applications, Trends in Mathematics, pp. 261–274. Springer, Berlin (2011)Google Scholar
  17. 17.
    Vajiac, A., Vajiac, M.: Multicomplex hyperfunctions. Complex Var. Elliptic Equ. 57(7–8), 751–762 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Zorich, V.A.: Mathematical Analysis I. Springer, Berlin (2004)Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • M. E. Luna-Elizarrarás
    • 1
  • M. Shapiro
    • 1
  • D. C. Struppa
    • 2
  • A. Vajiac
    • 2
  1. 1.Escuela Superior de Fisica y MatemáticasInstituto Politécnico NacionalMéxico CityMexico
  2. 2.Schmid College of Science and TechnologyChapman UniversityOrangeUSA

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