Essentially Adequate Concept of Holomorphic Functions in Quaternionic Analysis

Abstract

The so-called essentially adequate concept of quaternionic holomorphic ( -holomorphic) functions defined as functions, whose quaternionic derivatives are independent of 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.