Arkiv för Matematik

  • Ark. Mat.
  • Volume 49, Number 2 (2011), 277-294.

Singularities of functions of one and several bicomplex variables

Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, Adrian Vajiac, and Mihaela B. Vajiac

Full-text: Open access

Abstract

In this paper we study the singularities of holomorphic functions of bicomplex variables introduced by G. B. Price (An Introduction to Multicomplex Spaces and Functions, Dekker, New York, 1991). In particular, we use computational algebra techniques to show that even in the case of one bicomplex variable, there cannot be compact singularities. The same techniques allow us to prove a duality theorem for such functions.

Article information

Source
Ark. Mat. Volume 49, Number 2 (2011), 277-294.

Dates
Received: 16 June 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907143

Digital Object Identifier
doi:10.1007/s11512-010-0126-0

Zentralblatt MATH identifier
1253.30060

Rights
2010 © Institut Mittag-Leffler

Citation

Colombo, Fabrizio; Sabadini, Irene; Struppa, Daniele C.; Vajiac, Adrian; Vajiac, Mihaela B. Singularities of functions of one and several bicomplex variables. Ark. Mat. 49 (2011), no. 2, 277--294. doi:10.1007/s11512-010-0126-0. https://projecteuclid.org/euclid.afm/1485907143


Export citation

References

  • Adams, W. W., Berenstein, C. A., Loustaunau, P., Sabadini, I. and Struppa, D. C., Regular functions of several quaternionic variables and the Cauchy–Fueter complex, J. Geom. Anal. 9 (1999), 1–15.
  • Charak, K. S., Rochon, D. and Sharma, N., Normal families of bicomplex holomorphic functions, Preprint, 2008.
  • CoCoATeam, CoCoA, A system for doing computations in commutative algebra, 2005.
  • Colombo, F., Damiano, A., Sabadini, I. and Struppa, D. C., A surjectivity theorem for differential operators on spaces of regular functions, Complex Var. Theory Appl. 50 (2005), 389–400.
  • Colombo, F., Sabadini, I., Sommen, F. and Struppa, D. C., Analysis of Dirac Systems and Computational Algebra, Birkhäuser, Boston, 2004.
  • Komatsu, H., Relative cohomology of sheaves of solutions of differential equations, in Lecture Notes in Mathematics 287, pp. 192–261, Springer, Berlin, 1973.
  • Palamodov, V. P., Linear Differential Operators with Constant Coefficients, Springer, New York, 1970.
  • Price, G. B., An Introduction to Multicomplex Spaces and Functions, Monographs and Textbooks in Pure and Applied Mathematics 140, Dekker, New York, 1991.
  • Rochon, D., On a relation of bicomplex pseudoanalytic function theory to the complexified stationary Schrödinger equation, Complex Var. Elliptic Equ. 53 (2008), 501–521.
  • Rochon, D. and Shapiro, M., On algebraic properties of bicomplex and hyperbolic numbers, An. Univ. Oradea Fasc. Mat. 11 (2004), 71–110.
  • Ryan, J., Complexified Clifford analysis, Complex Var. Theory Appl. 1 (1982), 119–149.
  • Ryan, J., $\mathcal{C}^{2}$ extensions of analytic functions defined in the complex plane, Adv. Appl. Clifford Algebr. 11 (2001), 137–145.
  • Scorza Dragoni, G., Sulle funzioni olomorfe di una variabile bicomplessa, Reale Accad. d’Italia, Mem. Classe Sci. Nat. Fis. Mat. 5 (1934), 597–665.
  • Segre, C., Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici, Math. Ann. 40 (1892), 413–467.
  • Spampinato, N., Estensione nel campo bicomplesso di due teoremi, del Levi-Civita e del Severi, per le funzioni olomorfe di due variabili bicomplesse I, Reale Accad. Naz. Lincei 22 (1935), 38–43.
  • Spampinato, N., Estensione nel campo bicomplesso di due teoremi, del Levi-Civita e del Severi, per le funzioni olomorfe di due variabili bicomplesse II, Reale Accad. Naz. Lincei 22 (1935), 96–102.
  • Spampinato, N., Sulla rappresentazione di funzioni di variabile bicomplessa totalmente derivabili, Ann. Mat. Pura Appl. 14 (1936), 305–325.
  • Sudbery, A., Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), 199–225.