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On Cauchy Integral Theorem for Quaternionic Slice Regular Functions

  • J. Oscar González CervantesEmail author
Article
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Abstract

The aim of this work is to show that the operator G, which has been introduced in Colombo et al. (Trans Am Math Soc 365:303–318, 2013) and whose kernel kerG coincides with the set of quaternionic slice regular functions, is a member of a family of operators with similar properties, such that all the members possess the respective versions of Stokes and Cauchy–type integral theorems. As direct consequences, these theorems are obtained for slice regular functions.

Keywords

Quaternions Non-constant coefficient differential operator Quaternionic Stokes theorem Quaternionic Cauchy integral theorem Quaternionic slice regular functions 

Mathematics Subject Classification

Primary 30G35 

Notes

References

  1. 1.
    Colombo, F., Gentili, G., Sabadini, I., Struppa, D.: Extension results for slice regular functions of a quaternionic variable. Adv. Math. 222, 1793–1808 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Colombo, F., González-Cervantes, J.O., Sabadini, I.: A non constant coefficients differential operator associated to slice monogenic functions. Trans. Am. Math. Soc. 365, 303–318 (2013)CrossRefzbMATHGoogle Scholar
  3. 3.
    Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative functional calculus. Progress in mathematics. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Gentili, G., Stoppato, C., Struppa, D.C.: Regular functions of a quaternionic variable. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Gürlebeck, K., Sprössig, W.: textitQuaternionic analysis and elliptic boundary value problems. Birkhaüser Verlag, Basel (1990)CrossRefzbMATHGoogle Scholar
  6. 6.
    Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford calculus for physicists and engineers. Wiley, Hoboken (1997)zbMATHGoogle Scholar
  7. 7.
    Hamilton, W.R.: On quaternions, or on a new system of imaginaries in algebra. Philos. Mag. 25, 489–495 (1844)Google Scholar
  8. 8.
    Kravchenko, V.V., Shapiro, M.V.: Helmholtz operator with a quaternionic wave number and associated function theory. Deformations of the mathematical structures, pp. 101–128. Kluwer Academic Publishers, Dordrecht (1993)Google Scholar
  9. 9.
    Kravchenko, V.V., Shapiro, M.V.: Integral representation for spatial models of mathematical physics. Pitman research notes in mathematics. CRC Press, Boca Raton (1996)Google Scholar
  10. 10.
    Shapiro, M., Vasilevski, N.L.: Quaternionic \(\psi \)-monogenic functions, singular operators and boundary value problems. I. \(\psi \)-Hyperholomorphy function theory. Complex Var. Theory Appl. 27, 17–46 (1995)Google Scholar
  11. 11.
    Shapiro, M., Vasilevski, N.L.: Quaternionic \(\psi \)-hyperholomorphic functions, singular operators and boundary value problems II. Algebras of singular integral operators and Riemann type boundary value problems. Complex Var. Theory Appl. 27, 67–96 (1995)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Sudbery, A.: Quaternionic analysis. Math. Proc. Philos. Soc. 85, 199–225 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasE.S.F.M. del I.P.NMéxico CityMéxico

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