eng Science and Education Publishing American Journal of Mathematical Analysis 2333-8431 2022-01-04 9 1 6 26 10.12691/ajma-9-1-2 AJMA2021912 article Michael Parfenov parfenov.48@bk.ru 1 Bashkortostan Branch of Russian Academy of Engineering, Ufa, Russia 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 ¨Crepresentation 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. http://pubs.sciepub.com/ajma/9/1/2/ajma-9-1-2.pdf quaternionic analysis quaternionic holomorphic functions Cauchy-Riemann's equations compositions of holomorphic functions polar form holomorphic logarithms quaternionic inverse thigonometric and hyperbolic functions