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The Cauchy Integral Formula for Biregular Function in Octonionic Analysis

Received: 8 July 2020     Accepted: 21 August 2020     Published: 31 August 2020
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Abstract

In this paper, we mainly study the Cauchy integral formula and mean value theorem for biregular function in octonionic analysis. Octonion is the extension of complex number to non-commutative and non-associative space. Because of the non-associative properties of multiplication, octonion plays an important role in wave equation, Yang-Mills equations, operator theory and so on. In recent years, octonion has become a hot topic for scholars at home and abroad and got many rich results, such as Fourier transform, Bergman kernel, Taylor series and its applications in quantum mechanics. On the basis of two Stokes theorems, we get Cauchy integral formula for biregular function in octonionic analysis by using the methods in dealing with the Cauchy integral formula for biregular function in Clifford analysis and regular function in octonionic analysis. As a direct result we also get the mean value theorem for biregular function in octonionic analysis. This will generalize the corresponding conclusion in complex analysis and Clifford analysis, and lays a solid foundation for the application of octonionic analysis in physics.

Published in International Journal of Theoretical and Applied Mathematics (Volume 6, Issue 3)
DOI 10.11648/j.ijtam.20200603.12
Page(s) 39-45
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Octonion, Biregular Function, Cauchy Integral Formula, Mean Value Theorem

References
[1] Baez, J. C. (2002) The octonions [J]. Bull Amer Math Soc, 39, 145-205.
[2] Baez, J. C. (2005) On quaternions and octonions: Their geometry, arithmetic and symmetry [J]. Bull Amer Math Soc, 42, 229-243.
[3] Mironov, V. L. and Mironov, S. V. (2009) Octonic representation of electromagnetic field equation [J]. J Math Physics, 50, 1-10.
[4] Okubo, S. (1995) Introduction to Octonion and Other Non-Associative Algebras in Physics [M]. Cam-bridge University Press, New York.
[5] Bossard, G. (2012) Octonionic black holes [J]. Journal of High Energy Physics, 5, 1-78.
[6] Cherkis, S. A. (2015) Octonions, monopoles and knots [J]. Lett Math Phys, 105 (5), 641-659.
[7] Borstena, L. and Dahanayakea, D. and Duffa, M. J. and Ebrahima, H. and Rubensa W. (2009) Black holes, qubits and octonions [J]. Physics Reports, 471 (3-4), 113-219.
[8] Wang, H. M. and Wang, W. (2014) On octonionic regular functions and szego projection on the octonionic Heisenberg group [J]. Complex Anal Oper Theory, 8, 1285-1324.
[9] Calin, O. and Chang, D. C. and Markina, I. (2009) Geometric analysis on H-type groups related to division algebras [J]. Math Nachr, 282 (1), 44-68.
[10] Li, X. M. (1998) Octonion analysis, PhD thesis, Peking University, Beijing.
[11] Baez, J. C. and Huerta, J. (2011) The strangest numbers in string theory [J]. Sci Am, 304 (5), 60-65.
[12] Li, X. M. and Zhao, K. and Peng, L. Z. (2002) The Cauchy integral formulas on the octonions [J]. Bull Belg Math Soc Simon Stevin, 9 (1), 47-64.
[13] Peng, L. Z. and Zhao, J. M. (2000) Hardy space and Bergman space on the Octonions [J]. Approximation Theory and Its Applications, 16 (3), 72-84.
[14] Ludkovsky, S. V. (2010) Feynman integration over octonions with application to quantum mechanics [J]. Math Meth Appl Sci, 33 (9), 1148-1173.
[15] Hahn, S. L. and Snopek, K. M. (2011) The unified theory of n-dimensional complex and hypercomplex analytic signals [J]. Bull Pol Ac Tech, 59 (2), 167-181.
[16] Wang, H. Y. and Bian, X. L. (2017) The right inverse of Dirac operator in octonionic space [J]. Journal of Geometry and Physics, 119, 139-145.
[17] Wang, H. Y. and Ren, G. B. (2014) Octonion analysis of several variables [J]. Communications in Mathematics and Statistics, 2, 163-185.
[18] Wang, J. X. and Li, X. M. (2018) The octonionic Bergman kernel for the unit ball [J]. Adv Appl Clifford Algebras, 28, 60.
[19] Jacobson, N. (1985) Basic Algebra I, W H Freeman and Company, New York.
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  • APA Style

    Yonghua Guo, Haiyan Wang. (2020). The Cauchy Integral Formula for Biregular Function in Octonionic Analysis. International Journal of Theoretical and Applied Mathematics, 6(3), 39-45. https://doi.org/10.11648/j.ijtam.20200603.12

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    ACS Style

    Yonghua Guo; Haiyan Wang. The Cauchy Integral Formula for Biregular Function in Octonionic Analysis. Int. J. Theor. Appl. Math. 2020, 6(3), 39-45. doi: 10.11648/j.ijtam.20200603.12

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    AMA Style

    Yonghua Guo, Haiyan Wang. The Cauchy Integral Formula for Biregular Function in Octonionic Analysis. Int J Theor Appl Math. 2020;6(3):39-45. doi: 10.11648/j.ijtam.20200603.12

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  • @article{10.11648/j.ijtam.20200603.12,
      author = {Yonghua Guo and Haiyan Wang},
      title = {The Cauchy Integral Formula for Biregular Function in Octonionic Analysis},
      journal = {International Journal of Theoretical and Applied Mathematics},
      volume = {6},
      number = {3},
      pages = {39-45},
      doi = {10.11648/j.ijtam.20200603.12},
      url = {https://doi.org/10.11648/j.ijtam.20200603.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20200603.12},
      abstract = {In this paper, we mainly study the Cauchy integral formula and mean value theorem for biregular function in octonionic analysis. Octonion is the extension of complex number to non-commutative and non-associative space. Because of the non-associative properties of multiplication, octonion plays an important role in wave equation, Yang-Mills equations, operator theory and so on. In recent years, octonion has become a hot topic for scholars at home and abroad and got many rich results, such as Fourier transform, Bergman kernel, Taylor series and its applications in quantum mechanics. On the basis of two Stokes theorems, we get Cauchy integral formula for biregular function in octonionic analysis by using the methods in dealing with the Cauchy integral formula for biregular function in Clifford analysis and regular function in octonionic analysis. As a direct result we also get the mean value theorem for biregular function in octonionic analysis. This will generalize the corresponding conclusion in complex analysis and Clifford analysis, and lays a solid foundation for the application of octonionic analysis in physics.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - The Cauchy Integral Formula for Biregular Function in Octonionic Analysis
    AU  - Yonghua Guo
    AU  - Haiyan Wang
    Y1  - 2020/08/31
    PY  - 2020
    N1  - https://doi.org/10.11648/j.ijtam.20200603.12
    DO  - 10.11648/j.ijtam.20200603.12
    T2  - International Journal of Theoretical and Applied Mathematics
    JF  - International Journal of Theoretical and Applied Mathematics
    JO  - International Journal of Theoretical and Applied Mathematics
    SP  - 39
    EP  - 45
    PB  - Science Publishing Group
    SN  - 2575-5080
    UR  - https://doi.org/10.11648/j.ijtam.20200603.12
    AB  - In this paper, we mainly study the Cauchy integral formula and mean value theorem for biregular function in octonionic analysis. Octonion is the extension of complex number to non-commutative and non-associative space. Because of the non-associative properties of multiplication, octonion plays an important role in wave equation, Yang-Mills equations, operator theory and so on. In recent years, octonion has become a hot topic for scholars at home and abroad and got many rich results, such as Fourier transform, Bergman kernel, Taylor series and its applications in quantum mechanics. On the basis of two Stokes theorems, we get Cauchy integral formula for biregular function in octonionic analysis by using the methods in dealing with the Cauchy integral formula for biregular function in Clifford analysis and regular function in octonionic analysis. As a direct result we also get the mean value theorem for biregular function in octonionic analysis. This will generalize the corresponding conclusion in complex analysis and Clifford analysis, and lays a solid foundation for the application of octonionic analysis in physics.
    VL  - 6
    IS  - 3
    ER  - 

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Author Information
  • School of Science, Tianjin University of Technology and Education, Tianjin, China

  • School of Science, Tianjin University of Technology and Education, Tianjin, China

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