The Cauchy Integral Formula for Biregular Function in Octonionic Analysis

Yonghua Guo; Haiyan Wang

doi:doi:10.11648/j.ijtam.20200603.12

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The Cauchy Integral Formula for Biregular Function in Octonionic Analysis

Yonghua Guo, Haiyan Wang

Published in International Journal of Theoretical and Applied Mathematics (Volume 6, Issue 3)

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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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

Yonghua Guo

School of Science, Tianjin University of Technology and Education, Tianjin, China
Haiyan Wang

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

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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}
}

Copy | Download

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  -

Copy | Download