Matrix Associate with Holomorphic Functions Taking
Values in Clifford Algebras

Nguyen Thi Huyen; Dao Viet Cuong

doi:https://doi.org/10.14445/22315373/IJMTT-V69I5P508

International Journal of Mathematics Trends and Technology

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Volume 69 | Issue 5 | Year 2023 | Article Id. IJMTT-V69I5P508 | DOI : https://doi.org/10.14445/22315373/IJMTT-V69I5P508

Matrix Associate with Holomorphic Functions Taking Values in Clifford Algebras

Nguyen Thi Huyen, Dao Viet Cuong

Received	Revised	Accepted	Published
02 Apr 2023	06 May 2023	17 May 2023	31 May 2023

Abstract

In this paper, we introduce the matrix classes 𝐷̃ (1,2, … , 𝑚; 𝑁) and it associates with functions taking values in Clifford algebras. This is an extension of the holomorphic function class from the complex plane to higher dimensional spaces. Some results and examples are presented with low-dimensional spaces.

Keywords

Holomorphic function, Clifford analysis, Reguler function, Cachy-Riemann system

References

[1] V. Iftimie, “Fonction hypercomplexes,” Mathematical Bulletin of the Society of Mathematical Sciences of the Socialist Republic of Romania, vol. 9, no. 57, pp. 279-332, 1965.
[Google Scholar] [Publisher Link]
[2] V. Iftimie, “Operator of the Moisil–Theodorescu Type,” Mathematical Bulletin of the Society of Mathematical Sciences of the Socialist Republic of Romania, vol. 10, no. 58, pp. 271-305, 1966.
[Google Scholar] [Publisher Link]
[3] Richard Delanghe, “On Regular-Analytic Functions with Valeurs in a Clifford Algebra,” Mathematische Annalen, vol. 185, pp. 91 - 111, 1970.
[CrossRef] [Google Scholar] [Publisher Link]
[4] Richard Delanghe, “On the Singularities of Functions with Valeurs in a Clifford Algebra,” Mathematische Annalen, vol. 196, no. 4, pp. 293 - 319, 1972.
[CrossRef] [Publisher Link]
[5] R. Delanghe, and F. Brackx, “Hypercomplex Function Theory and Hilbert Modules with Reproducing Kernels,” Proceedings of the London Mathematical Society, vol. s3-37, no. 3, pp, 545 - 576, 1978.
[CrossRef] [Google Scholar] [Publisher Link]
[6] N. Theodorescu, “The Areolar Derivative and Its Applications to Mathematical Physics,” Thèse, Paris, 1931.
[Google Scholar] [Publisher Link]
[7] N. Theodorescu, “Spatial Derivatives and Extension of Fundamental Differential Operators: Grad, Divet, rot of Field theory,” Mathematical Bulletin of the Society of Mathematical and Physical Sciences of the Romanian People's Republic, vol. 3, no. 51, pp. 363- 383, 1959.
[Google Scholar] [Publisher Link]
[8] N. Theodorescu, La dérivée aréolaire, Ann Roumains Math – Cahier 3, Bucarest, 1936.
[9] A. Angelescu, Sur une fommule de M. D. Pompéiu, Mathématica. Cluj. 1, p. 107, 1929.
[10] Miron Nicolescu, “Complex Functions in the Plane and in Space,” These, Paris, 1928.
[Google Scholar] [Publisher Link]
[11] Gr. C.Moisil, Sur les systèmes d’équations de M. Dirac, du type elliptique, C. R Paris, 191, p. 1292, 1930.
[12] Gr.C. Moisil, “On Monogeneous Quaternions,” Bulletin of Mathematical Sciences, vol. 55, p. 168, 1931.
[13] Gr. C. Moisil, “Holomorphic Functions in Space,” Mathematica, vol. 5, p. 142, 1931.
[14] Grigore Constantin Moisil, “On a Class of Systems of Partial Differential Equations in Physics,” Mathematics – Bucharest, 1931.
[Google Scholar]
[15] Huan Le Dy, “Some Basic Theorems About Holomorphic Vectors,” Sci. Rec., vol. 56, no. 2, p. 53, 1958.
[16] Neldelcu Coroi M., “An Application of the Spatial Derivative to the Equations of Elasticity in Space,” Rev de Math. Pures at appl., vol. 5, pp. 3-4, 1960.
[17] Neldelcu Coroi M., “The Areolar Polynomial of Order,” I. Rev de Math. Pures et appl, vol. 4, no. 4, 1959.
[18] Walter Nef, “The Inessential Singularities of the Regular Functions of a Quaternion Variable,” Commentarii Mathematici Helvetici, vol. 16, pp. 284-304, 1943.
[CrossRef] [Google Scholar] [Publisher Link]
[19] Luigi Sobréro, “Algebra of Hypercomplex Functions and Its Applications to the Mathematical Theory of Elasticity,” Memoirie dell’ Academia d’Italia, 1934.
[Google Scholar]
[20] Rud Von Fueter, “The Singularities of the Unique Regular Functions of a Quaternion Variable,” Commentarii Mathematici Helvetici, vol. 9, pp. 320-334, 1936.
[CrossRef] [Google Scholar] [Publisher Link]
[21] Rud Fueter, “On the Analytic Representation of the Regular Functions of a Quaternion Variable,” Commentarii Mathematici Helvetici, vol. 8, 371-378, 1935/36.
[Publisher Link]
[22] Bernd Goldschmidt, “Regularity Properties of Generalized Analytic Vectors in ℝn ,” Mathematische Nachrichten, vol. 103, no. 1, pp. 245- 254, 1981.
[CrossRef] [Google Scholar] [Publisher Link]
[23] Bernd Goldschmidt, “Existence and Representation of Solutions of a Class of Elliptic Systèmes of Partial Differential Equations of first Order in the Space,” Mathematische Nachrichten, vol. 108, no. 1, pp. 159-166, 1982.
[CrossRef] [Google Scholar] [Publisher Link]
[24] Bernd Goldschmidt, “A Cauchy Integral Formula for a Class of Elliptic Systèmes of Partial Differential Equations of First Order in the Space,” Mathematische Nachrichten, vol. 108, no. 1, pp. 167-178, 1982.
[CrossRef] [Google Scholar] [Publisher Link]
[25] R.P. Gilbert, and G.N. Hile, “Hilbert Function Modules with Reproducing Kernels,” Nonlinear Analysis: Theory, Methods and Applications, vol. 1, no. 2, pp. 135- 150, 1977.
[CrossRef] [Google Scholar] [Publisher Link]
[26] R.P. Gilbert, and G.N. Hile, “Hypercomplex Function Theory in the Sense of L. BERS,” Mathematische Nachrichten, vol. 72, no. 1, pp. 187- 200, 1976.
[CrossRef] [Google Scholar] [Publisher Link]
[27] David Colton, “Bergman Operators for Elliptic Equations in Four Independent Variables,” SIAM Journal, vol. 3, no. 3, 1972.
[CrossRef] [Google Scholar] [Publisher Link]
[28] David Colton, “Integral Operators for Elliptic Equations in Three Independent Variables,” Applicable Analysis, vol. 4, no. 1, pp. 77-95, 1974.
[CrossRef] [Google Scholar] [Publisher Link]
[29] F. Brackx, F. Sommen, and R. Dalanghe, Clifford Analysis, Pitman, Boston, London, Mebourne, 1982.
[30] W. Tutschke, “Classical and Modern Methods of complex Analysis,” Complex Analysis-Methods, Akademie-Verlag Berlin, 1983. [Google Scholar]
[31] Wolfgang Tutschke, “Reduction of the Problem of Linear Conjugation for First Order Nonlinear Elliptic Systems in the Plane to an Analogous Problem for Holomorphic Functions,” Analytic Functions Kozubnik, pp. 446-455, 1980.
[CrossRef] [Google Scholar] [Publisher Link]

Citation :

Nguyen Thi Huyen, Dao Viet Cuong, "Matrix Associate with Holomorphic Functions Taking Values in Clifford Algebras," International Journal of Mathematics Trends and Technology (IJMTT), vol. 69, no. 5, pp. 82-89, 2023. Crossref, https://doi.org/10.14445/22315373/IJMTT-V69I5P508