Essentially Adequate Concept of Holomorphic Functions in Quaternionic Analysis

Michael Parfenov^1,

¹Bashkortostan Branch of Russian Academy of Engineering, Ufa, Russia

Cite this paper:
Michael Parfenov. Essentially Adequate Concept of Holomorphic Functions in Quaternionic Analysis. American Journal of Mathematical Analysis. 2020; 8(1):14-30. doi: 10.12691/ajma-8-1-3

Abstract

The so-called essentially adequate concept of quaternionic holomorphic ( -holomorphic) functions defined as functions, whose quaternionic derivatives are independent of "the way of their computation", is developed. It is established that -holomorphic functions form one remarkable class of quaternionic functions whose properties are fully similar (essentially adequate) to complex ones: the quaternionic multiplication of these quaternionic functions behaves as commutative, the left quotient equals the right one, the rules for differentiating sums, products, ratios, inverses, and compositions are the same as in complex analysis. One can just verify these properties, constructing -holomorphic functions from their complex holomorphic counterparts by using the presented constructing rule. Several examples, confirming the theory in question, are considered. When using this concept there are no principal restrictions to build a quaternionic analysis similar to complex one. The elementary source flow and elementary vortex flow, allowing us to construct different 3D steady state fluid flows by superposition, are considered. To automate the processing of -holomorphic functions the pack of Mathematica^® Programs is developed, part of which is presented.

Keywords:
quaternionic holomorphic functions quaternionic analysis quaternionic generalization of Cauchy-Riemann’s equations rules for quaternionic differentiation quaternionic potential 3D steady state fluid flows

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	Mathews, J. H., Howell, R. W., Complex Analysis for Mathematics and Engineering, 3rd ed, Jones and Bartlett Publishers, Boston-Toronto-London-Singapore, 1997.
[2]	Sudbery, A., "Quaternionic analysis", Math. Proc. Camb. Phil. Soc., 85 (1979), 199-225.
[3]	Dzagnidze, O., "On the differentiability of quaternion functions", arXiv: math.CV, March 2012. Available: arxiv.org/abs/1203.5619.
[4]	Parfenov, M., "On Properties of Holomorphic Functions in Quaternionic Analysis", American Journal of Mathematical Analysis, Vol. 5, No. 1, pp. 17-24, July 2017.
[5]	Khaled Abdel-Khalek, "Quaternion Analysis", arXiv:hep-th, July 1996. Available: arxiv.org/abs/hep-th/9607152v2.
[6]	Parfenov, M., "Adequate quaternionic generalization of complex differentiability", viXra: Functions and Analysis, Jan 2017. Available: vixra.org/abs/1609.0006
[7]	Parfenov, M., "A Quaternionic Potential Conception with Applying to 3D Potential Fields", American Journal of Mathematical Analysis, Vol. 7, No. 1, p.p. 1-10, April 2019.
[8]	Kantor, I. L., Solodovnikov, A. S.. Hypercomplex numbers. An Elementary Introduction to Algebras. Springer-Verlag, 1989
[9]	Parfenov, M., "The Similarity between Rules for Essentially Adequate Quaternionic and Complex Differentiation", viXra: Functions and Analysis, 2018. Available: vixra.org/abs/1806.0239.
[10]	Wellin, P.R., Gaylord, R. J., Kamin, S.N., An Introduction to Programming with Mathematica, 3rd ed, Cambridge University Press, New York, 2005.
[11]	Lavrentiev M. A., Shabat B. V., Methods of the Theory of Complex Variable, Nauka, Moscow, 1973. (In Russian)
[12]	Podoksenov, M. N., Prokhozhiy, S. A., Analytic Geometry in the space, "VSU named after P. M. Masherov" Publishers, Vitebsk, 2013.