A Quaternionic Potential Conception with Applying to 3D Potential Fields

Michael Parfenov^1,

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

Cite this paper:
Michael Parfenov. A Quaternionic Potential Conception with Applying to 3D Potential Fields. American Journal of Mathematical Analysis. 2019; 7(1):1-10. doi: 10.12691/ajma-7-1-1

Abstract

By analogy with complex analysis, any quaternionic holomorphic function, satisfying the earlier presented quaternionic generalization of Cauchy-Riemann’s equations (the left- and right conditions of holomorphy together), is defined as a quaternionic potential. It can be constructed by simple replacing a complex variable as single whole by a quaternionic one in an expression for complex potential. The performed "transfer" of the complex potential conception to the quaternionic area is based on the earlier proved similarity of formulas for quaternionic and complex differentiation of holomorphic functions. A 3D model of a potential field, corresponding its quaternionic potential, can be obtained as a result of calculations made in the quaternionic space and the subsequent final transition to 3D space. The quaternionic generalizations of Laplace's equations, harmonic functions and conditions of antiholomorphy are presented. It is shown that the equations of the quaternionic antiholomorphy unite differential vector operations just as the complex ones. The example of applying to the 3D fluid flow modeling is considered in detail.

Keywords:
quaternionic analysis and holomorphic functions quaternionic Cauchy-Riemann’s equations quaternionic potential quaternionic Laplace's equations harmonic functions 3D models of potential fields

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