On Properties of Holomorphic Functions in Quaternionic Analysis

Michael Parfenov^1,

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

Cite this paper:
Michael Parfenov. On Properties of Holomorphic Functions in Quaternionic Analysis. American Journal of Mathematical Analysis. 2017; 5(1):17-24. doi: 10.12691/ajma-5-1-4

Abstract

We draw the conclusions from the earlier presented quaternionic generalization of Cauchy-Riemann’s equations. The general expressions for constituents of -holomorphic functions as well as the relations between them are deduced. The symmetry properties of constituents of -holomorphic functions and their derivatives of all orders are proved. For full derivatives it is a consequence of uniting the left and right derivatives within the framework of the developed theory. Some -holomorphic generalizations of - holomorphic functions are discussed in detail to demonstrate particularities of constructing H-holomorphic functions. The power functions are considered in detail.

Keywords:
quaternionic holomorphic functions quaternionic analysis quaternionic generalization of Cauchy-Riemann’s equations functions of hypercomplex variables

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