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Holomorphy in Multicomplex Spaces

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Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 221))

Abstract

In this paper we extend our research on bicomplex holomorphy and on the existence of bicomplex differential operators to the nested space of multicomplex numbers.S pecifically, we will discuss the different sets of idempotent representations for multicomplex numbers and we will use them to develop a theory of holomorphic and multicomplex analytic functions.

Mathematics Subject Classification (2000). Primary 30G35; Secondary 16E05, 13P10, 35N05.

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References

  1. Baird, P., Wood, J.C., Harmonic morphisms and bicomplex manifolds, J. Geom. Phys., 61, no. 1, (2011), 46–61.

    Article  MathSciNet  MATH  Google Scholar 

  2. Charak, K.S., Rochon, D., Sharma, N., Normal families of bicomplex holomor-phic functions, Fractals, 17, No. 3 (2009), 257–268.

    Article  MathSciNet  MATH  Google Scholar 

  3. Colombo, F., Sabadini, I., Sommen, F., Struppa, D.C., Analysis of Dirac Systems and Computational Algebra, Birkhäuser, Boston, (2004).

    Book  MATH  Google Scholar 

  4. Colombo, F., Sabadini, I., Struppa, D.C., Vajiac, a., Vajiac M.B.,Singularities of functions of one and several bicomplex variables. Arkiv for matematik, (Institut Mittag-Leffler), (2011).

    Google Scholar 

  5. Colombo, F., Sabadini, I., Struppa, D.C., Vajiac, a., Vajiac M.B., Bicomplex hyperfunctions. Ann. Mat. Pura Appl., (2010), 1–15.

    Google Scholar 

  6. Luna-Elizarraras M.E., Shapiro, M., On modules over bicomplex and hyperbolic numbers, in Applied Complex and Quaternionic Approximation, editors: R.K. Kovacheva, J. Lawrynowicz, and S. Marchiafava, Edizioni Nuova Cultura, Roma, 2009, pp. 76–92.

    Google Scholar 

  7. Garant-Pelletier, v., Rochon, D, On a generalized Fatou-Julia theorem in multicomplex spaces, Fractals, 17, no. 2, (2009), 241–255.

    Article  MathSciNet  MATH  Google Scholar 

  8. Garant-Pelletier, Ensembles de Mandelbrot et de Julia remplis classiques, généralisés aux espaces multicomplexes et théorème de Fatou-Julia généralisé, Master Thesis, (2011).

    Google Scholar 

  9. Luna-ElizarrarÁs, M.E., Macas-CedeÑo, M.A., Shapiro, M., On some relations between the derivative and the two-dimensional directional derivatives of a quaternionic function, AIP Conference Proceedings, v. 936, (2007), 758–760.

    Article  Google Scholar 

  10. Luna-ElizarrarÁs, M.E., Macas-CedeÑo, M.A., Shapiro, M., On the derivatives of quaternionic functions along two-dimensional planes, Adv. Appl. Clifford Algebr., 19, (2009), no. 2, 375–390.

    Article  MathSciNet  MATH  Google Scholar 

  11. Price, G.B., An Introduction to Multicomplex Spaces and Functions, Monographs and Textbooks in Pure and Applied Mathematics, 140, Marcel Dekker, Inc., New York, 1991.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Rochon, D., On a relation of bicomplex pseudoanalytic function theory to the complexified stationary Schrödinger equation, Complex Var. Elliptic Equ. 53,no. 6, (2008), 501–521.

    Article  MathSciNet  MATH  Google Scholar 

  14. Rochon, D., A Bloch constant for hyperholomorphic functions, Complex Var. Elliptic Equ. 44, (2001), 85–101.

    Article  MathSciNet  MATH  Google Scholar 

  15. Ryan, J., Complexified Clifford analysis, Complex Variables and Elliptic Equations 1 (1982), 119–149.

    Article  MATH  Google Scholar 

  16. Ryan, J., C2 extensions of analytic functions defined in the complex plane, Adv. in Applied Clifford Algebras 11 (2001), 137–145.

    Article  MATH  Google Scholar 

  17. Segre, C., Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici, Math. Ann. 40 (1892), 413–467.

    Article  MathSciNet  MATH  Google Scholar 

  18. Scorza Dragoni, G., Sulle funzioni olomorfe di una variabile bicomplessa, Reale Accad. d’ltalia, Mem. Classe Sci. Nat. Fis. Mat. 5 (1934), 597–665.

    Google Scholar 

  19. Spampinato, N., Estensione nel campo bicomplesso di due teoremi, del Levi-Civita e del Severi, per le funzioni olomorfe di due variabili bicomplesse I, II, Reale Accad. Naz. Lincei 22 (1935), 38–43, 96–102.

    Google Scholar 

  20. Spampinato, N., Sulla rappresentazione di funzioni di variabile bicomplessa totalmente derivabili, Ann. Mat. Pura Appli. 14 (1936), 305–325.

    Article  MathSciNet  Google Scholar 

  21. 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, 261–274, Birkhäuser, 2010.

    Google Scholar 

  22. Sudbery, A., Quaternionic Analysis, Math.Proc. Camb. Phil. Soc. 85 (1979), 199–225.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Schmid College of Science and Technology, Chapman University, Orange, CA, 92866, USA

    D. C. Struppa, A. Vajiac & M. B. Vajiac

Authors
  1. D. C. Struppa
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  2. A. Vajiac
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  3. M. B. Vajiac
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Corresponding author

Correspondence to D. C. Struppa .

Editor information

Editors and Affiliations

  1. , Abt. Mathematik V, Universität Ulm, Ulm, 89069, Germany

    Wolfgang Arendt

  2. Department of Mathematics, Virginia Polytechnic Institute, McBryde Hall 524, Blacksburg, 24061, Virginia, USA

    Joseph A. Ball

  3. Institut für Mathematik, AG Modellierung, Numerik und, TU Berlin, Straße des 17 Juni 136, Berlin, 10623, Berlin, Germany

    Jussi Behrndt

  4. Fak. II Mathematik & Naturwissenschaften, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Karl-Heinz Förster

  5. Fak. II Mathematik & Naturwissenschaften, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Volker Mehrmann

  6. , Institut für Mathematik, TU Ilmenau, Weimarer Str. 25, Ilmenau, 98693, Thüringen, Germany

    Carsten Trunk

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Struppa, D.C., Vajiac, A., Vajiac, M.B. (2012). Holomorphy in Multicomplex Spaces. In: Arendt, W., Ball, J., Behrndt, J., Förster, KH., Mehrmann, V., Trunk, C. (eds) Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations. Operator Theory: Advances and Applications, vol 221. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0297-0_37

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