On Constructing Complicated Compositions of Quaternionic Holomorphic Functions

Michael Parfenov^1,

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

Cite this paper:
Michael Parfenov. On Constructing Complicated Compositions of Quaternionic Holomorphic Functions. American Journal of Mathematical Analysis. 2021; 9(1):6-26. doi: 10.12691/ajma-9-1-2

Abstract

The issue of constructing complicated quaternionic holomorphic (-holomorphic) functions in the Cayley-Dickson doubling form is considered. The way of -holomorphic substitutions, allowing us to construct -holomorphic composite functions of any degree of difficulty, is presented. The new –representation form for -holomorphic functions is established as a consequence of the earlier proved commutative behavior of the quaternionic multiplication in the case of -holomorühic functions. The specific polar form of -holomorphic functions with a real-valued modulus and argument similar to complex one is obtained. The -holomorphic generalizations of the logarithmic and inverse trigonometric and hyperbolic functions are implemented. The obtained results reaffirm that any complicated -holomorphic function can be constructed from its complex holomorphic analog. The processing of -holomorphic functions of any degree of difficulty is provided through high-speed programmes in system Wolfram Mathematica^® represented in the Appendix.

Keywords:
quaternionic analysis quaternionic holomorphic functions Cauchy-Riemann's equations compositions of holomorphic functions polar form holomorphic logarithms quaternionic inverse thigonometric and hyperbolic functions

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