Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field
Abstract
:1. Introduction
2. Preliminaries
3. Regularity of a Function with Values in
- (i)
- and are differential functions in Ω,
- (ii)
- in Ω, where .
4. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix B
References
- Conway, J.H.; Smith, D.A. On Quaternions and Octonions: Their Geometry and Symmetry; A. K. Peters Ltd.: Natick, MA, USA, 1982. [Google Scholar]
- Smith, W.D. Quaternions, Octonions, 16-Ons and 2n × 2n-Ons; New Kinds of Numbers; Pensylvania State University: Pennsylvania, PA, USA, 2004; pp. 1–68. [Google Scholar]
- Carmody, K. Circular and hyperbolic quaternions, octonions, and sedenions—Further results. Appl. Math. Comput. 1997, 84, 27–47. [Google Scholar] [CrossRef]
- Imaeda, K.; Imaeda, M. Sedenions: Algebra and analysis applied. Appl. Math. Comput. 2007, 188, 942–947. [Google Scholar]
- Kvplinger, J. Sgnature of gravity in conic sedenions. Appl. Math. Comput. 2007, 188, 942–947. [Google Scholar]
- Kim, J.E.; Ha, S.J.; Shon, K.H. Properties of hyperholomorphic functions on dual sedenion numbers. Honam Math. J. 2014, 36, 921–932. [Google Scholar] [CrossRef]
- Kim, J.E. Characteristics of Regular Functions Defined on a Semicommutative Subalgebra of 4-Dimensional Complex Matrix Algebra. J. Math. 2021, 2021, 9. [Google Scholar] [CrossRef]
- Naser, M. Hyperholomorphic functions. Sib. Math. J. 1971, 12, 959–968. [Google Scholar] [CrossRef]
- Nôno, K. Hyperholomorphic functions of a quaternion variable. Bull. Fukuoka Univ. Ed. 1982, 32, 22–37. [Google Scholar]
- Kim, J.E.; Shon, K.H. Exponential form of biquaternionic variables in Clifford analysis. Comm. Korean Math. Soc. 2017, 32, 85–92. [Google Scholar] [CrossRef]
- Kim, J.E.; Shon, K.H. Regular Functions on Dual Split Quaternions in Clifford Analysis. Filomat 2017, 31, 17–27. [Google Scholar] [CrossRef]
- Lim, S.J.; Shon, K.H. Regular functions with values in a noncommutative algebra using Clifford analysis. Filomat 2016, 30, 1747–1755. [Google Scholar] [CrossRef]
- Lim, S.J.; Shon, K.H. Hyperholomorphic functions and hyper-conjugate harmonic functions of octonion variables. J. Ineq. Appl. 2013, 2013, 77. [Google Scholar] [CrossRef]
- Li, X.; Kai, Z.; Peng, L.Z. Characterization of octonionic analytic functions. Complex Var. 2005, 50, 1031–1040. [Google Scholar] [CrossRef]
- Ludkovski, S.V. Differentiable functions of Cayley—Dickson numbers and line integration. J. Math. Sci. 2007, 141, 1231–1298. [Google Scholar] [CrossRef]
- Kauhanen, J.; Orelma, J. Cauchy—Riemann operators in octonionic analysis. Adv. Appl. Clifford Al. 2018, 28, 14. [Google Scholar] [CrossRef]
- Vinogradov, V.S. An analog of the Cauchy—Riemann system in four-dimensional space. Soviet. Math. Dokl. 1964, 5, 10–13. [Google Scholar]
- Hörmander, L. L2 estimates and existence theorems for the operator. Acta Math. 1965, 113, 89–152. [Google Scholar] [CrossRef]
- Krantz, S.G. Characterizations of various domains of holomorphy via estimates and applications to a problem of Kohn. Ill. J. Math. 1979, 23, 267–285. [Google Scholar] [CrossRef]
- Bhat, M.Y.; Almanjahie, I.M.; Dar, A.H.; Dar, J.G. Wigner-Ville distribution and ambiguity function associated with the quaternion offset linear canonical transform. Demonstr. Math. 2022, 55, 786–797. [Google Scholar] [CrossRef]
- Oubba, H. Generalized quaternion algebras. Rend. Circ. Mat. Palermo 2023, 72, 4239–4250. [Google Scholar] [CrossRef]
- Achak, A.; Akhlidj, A.; Daher, R.; Jaafari, A. On estimates for the quaternion linear canonical transform in the space L2(,). Rend. Circ. Mat. Palermo 2024, 1–14. [Google Scholar]
- Abilov, V.A.; Abilova, F.V.; Kerimov, M.K. Some remarks concerning the Fourier transform in the space L2(). Comput. Math. Math. Phys. 2008, 48, 2146–2153. [Google Scholar] [CrossRef]
× | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Kim, J.-E. Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field. Axioms 2024, 13, 291. https://doi.org/10.3390/axioms13050291
Kim J-E. Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field. Axioms. 2024; 13(5):291. https://doi.org/10.3390/axioms13050291
Chicago/Turabian StyleKim, Ji-Eun. 2024. "Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field" Axioms 13, no. 5: 291. https://doi.org/10.3390/axioms13050291
APA StyleKim, J. -E. (2024). Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field. Axioms, 13(5), 291. https://doi.org/10.3390/axioms13050291