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Complex Laplacian and Derivatives of Bicomplex Functions

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Abstract

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.

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References

  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. Charak, K.S., Rochon, D., Sharma, N.: Normal families of bicomplex holomorphic functions. Fractals 17(3), 257–268 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Gal, S.G.: Introduction to Geometric Function Theory of Hypercomplex Variables. Nova Science Publishers, Inc., Hauppauge (2004)

    Google Scholar 

  6. Krantz, S.G.: Function Theory of Several Complex Variables. AMS Chelsea Publishing, Providence (1992)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  11. Rochon, D., Shapiro, M.: On algebraic properties of bicomplex and hyperbolic numbers. An. Univ. Oradea Fasc. Math. 11, 71–110 (2004)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ryan, J.: Complexified Clifford analysis. Complex Var. Elliptic Equ. 1, 119–149 (1982)

    Article  MATH  Google Scholar 

  14. Ryan, J.: \({\mathbb{C}}^2\) extensions of analytic functions defined in the complex plane. Adv. Appl. Clifford Algebras 11, 137–145 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  17. Vajiac, A., Vajiac, M.: Multicomplex hyperfunctions. Complex Var. Elliptic Equ. 57(7–8), 751–762 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zorich, V.A.: Mathematical Analysis I. Springer, Berlin (2004)

    Google Scholar 

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Authors and Affiliations

  1. Escuela Superior de Fisica y Matemáticas, Instituto Politécnico Nacional, 07338 , México City, DF, Mexico

    M. E. Luna-Elizarrarás & M. Shapiro

  2. Schmid College of Science and Technology, Chapman University, One University Drive, Orange, CA, 92866, USA

    D. C. Struppa & A. Vajiac

Authors
  1. M. E. Luna-Elizarrarás
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  2. M. Shapiro
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  3. D. C. Struppa
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  4. A. Vajiac
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Corresponding author

Correspondence to D. C. Struppa.

Additional information

Communicated by Daniel Aron Alpay.

M. E. Luna-Elizarrarás and M. Shapiro have been partially supported by CONACYT projects as well by Instituto Politécnico Nacional in the framework of COFAA and SIP programs. All the four authors are grateful to Chapman University for the support offered in preparing the final stages of this article.

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Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C. et al. Complex Laplacian and Derivatives of Bicomplex Functions. Complex Anal. Oper. Theory 7, 1675–1711 (2013). https://doi.org/10.1007/s11785-013-0284-8

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  • DOI: https://doi.org/10.1007/s11785-013-0284-8

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