Skip to main content
SpringerLink
Log in

Cauchy Integral Formula on the Distinguished Boundary with Values in Complex Universal Clifford Algebra

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

As an integral representation for holomorphic functions, Cauchy integral formula plays a significant role in the function theory of one complex variable and several complex variables. In this paper, using the idea of several complex analysis we construct the Cauchy kernel in universal Clifford analysis, which has generalized complex differential forms with universal Clifford basic vectors. We establish Cauchy–Pompeiu formula and Cauchy integral formula on the distinguished boundary with values in universal Clifford algebra. This work is the basis for studying the Cauchy-type integral and its boundary value problem in complex universal Clifford analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brackx, F., Delanghe, R., Sommen, F.C.: Clifford analysis, Research Notes in Mathematics, vol. 76. Pitman (Advanced Publishing Program), Boston (1982)

    MATH  Google Scholar 

  2. Delanghe, R., Sommen, F.C., Soucek, V.: Clifford Algebra and Spinor-Valued Functions. Kluwer, Dordrecht (1992)

    Book  Google Scholar 

  3. Du, J.Y., Xu, N., Zhang, Z.X.: Boundary behavior of Cauchy-type integrals in clifford analysis. Acta Math. Scientia. 29B(1), 210–224 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Du, J.Y., Zhang, Z.X.: A Cauchy’s integral formula for functions with values in a universal Clifford algebra and its applications. Complex Var. Elliptic Equ. 47(10), 915–928 (2002)

  5. Du, J.Y., Xu, N.: On boundary behavior of the Cauchy type integrals with values in a universal Clifford algebra. Adv. Appl. Clifford Algebras. 21, 49–87 (2011)

    Article  MathSciNet  Google Scholar 

  6. Huang, S., Qiao, Y.Y., Wen, G.C.: Real and Complex Clifford Analysis. Springer, Berlin (2006)

    MATH  Google Scholar 

  7. Heinrich, B., Zhang, Z.X., Du, J.Y.: On Cauchy–Pompeiu formula for functions with values in a universal Clifford algebra. Acta Math. Scientia. 23B(1), 95–103 (2003)

    Article  MathSciNet  Google Scholar 

  8. Iftimie, V.: Functions hypercomplex. Bull. Math. Soc. Sc. R. S. R. 4, 279–332 (1965)

    Google Scholar 

  9. Ku, M., Du, J.Y., Wang, D.S.: Some properties of holomorphic Cliffordian functions in complex Clifford analysis. Acta. Math. Sci. 30(3), 747–768 (2010)

    Article  MathSciNet  Google Scholar 

  10. Ku, M., Du, J.Y., Wang, D.S.: On generalization of Martinelli-Bochner integral formula using Clifford analysis. Adv. Appl. Clifford Algebras. 20, 351–366 (2010)

    Article  MathSciNet  Google Scholar 

  11. Krantz, S.G.: Function Theory of Several Complex Variables. Wadsworth Brooks and Cole Advanced and Software, New York (1992)

    MATH  Google Scholar 

  12. Li, Z.F., Yang, H.J., Qiao, Y.Y., Guo, B.C.: Some properties of T-operator with bihypermonogenic kernel in Clifford analysis. Complex Var. Elliptic. 62(7), 938–956 (2017)

    Article  MathSciNet  Google Scholar 

  13. Li, Z.F., Yang, H.J., Qiao, Y.Y.: A new Cauchy integral formula in the complex Clifford analysis. Adv. Appl. Clifford Algebras. 28(75), 1–12 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Li, S.S., Leng, J.S., Fei, M.G.: Spectrums of functions associated to the fractional Clifford–Fourier transform. Adv. Appl. Clifford Algebras. 30(1), 1–6 (2020)

    Article  MathSciNet  Google Scholar 

  15. Lu, Q.K., Zhou, X.Y.: Introduction to Functions of Multiple Complex Variables. Science Press, Beijing (2018)

    Google Scholar 

  16. Lu, J.K.: Boundary value problem of analytic function. Wuhan University Press, Wuhan (2004)

    Google Scholar 

  17. Ryan, J.: Complexied Clifford analysis. Complex Variables. 1, 119–149 (1982)

    Google Scholar 

  18. Ryan, J.: Singularities and Laurent expansions in complexied Clifford analysis. Appl. Anal. 15, 33–49 (1983)

    Article  Google Scholar 

  19. Shi, H.P., Yang, H.J., Li, Z.F., Qiao, Y.Y.: Fractional Clifford Fourier transform and its application. Adv. Appl. Clifford Algebras. 30(68), 1–17 (2020)

    MathSciNet  MATH  Google Scholar 

  20. Shi, H.P., Yang, H.J., Li, Z.F., Qiao, Y.Y.: Two-dided Fourier transform in Clifford analysis and its application. Adv. Appl. Clifford Algebras. 30(67), 1–23 (2020)

    Google Scholar 

  21. Shi, J.H.: Fundamentals of Variable Function Theory. Higher Education Press, Beijing (1996)

    Google Scholar 

  22. Xu, Z.Y.: Riemann problem for regular functions with values on Clifford analysis. Sci. Bull. 32(23), 476–477 (1987)

    Google Scholar 

  23. Yang, H.J., Qiao, Y.Y., Huang, S.: Some properties of Cauchy-Type singular integrals in Clifford analysis. J. Math. Res. Appl. 32(2), 189–200 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Yang, H.J., Qiao, Y.Y., Xie, Y.H., Wang, L.P.: Cauchy integral formula for k-monogenic function with \(\alpha \)-weight. Adv. Appl. Clifford Algebras. 28(2), 1–11 (2018)

    MathSciNet  Google Scholar 

  25. Youssef, E.H.: Titchmarsh’s Theorem in Clifford Analysis. Adv. Appl. Clifford Algebras. 31(10), 1–15 (2021)

  26. Zhang, Z.X.: The Schwarz type lemma in upper half space in Clifford analysis. Adv. Appl. Clifford Algebras. 28(98), 1–15 (2018)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Science Foundation of China (No. 11871191), the Soft Science Research Project of Innovation Capacity Promotion Program of Hebei Province(No. 21557647D), Hebei University of Science and Technology Dr. Foundation (No. 1181348). And special thanks to Professor Yufeng Wang and Professor Fuli He for this work.

Author information

Authors and Affiliations

  1. School of Economics and Management, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei Province, People’s Republic of China

    Na Xu

  2. College of Science, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei Province, People’s Republic of China

    Zunfeng Li & Heju Yang

Authors
  1. Na Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zunfeng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Heju Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Na Xu.

Additional information

Communicated by Fabrizio Colombo.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, N., Li, Z. & Yang, H. Cauchy Integral Formula on the Distinguished Boundary with Values in Complex Universal Clifford Algebra. Adv. Appl. Clifford Algebras 31, 72 (2021). https://doi.org/10.1007/s00006-021-01175-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-021-01175-y

Keywords

Mathematics Subject Classification

Search

Navigation