On the notion of a quaternionic holomorphic function

Abstract

A physically more adequate definition of a quaternionic holomorphic (H-holomorphic) function of one quaternionic variable compared to known ones and a quaternionic generalization of Cauchy-Riemann's equations are presented. At that a class of introduced H-holomorphic functions consists of those quaternionic functions whose left and right derivatives become equal after the transition to 3D space. The presented theory demonstrates a complete similarity of the algebraic properties and differentiation rules between the classes of H-holomorphic and ordinary complex holomorphic functions, including the fact that quaternionic multiplication of the H-holomorphic functions behaves as commutative and the fact that each H-holomorphic function can be created from its complex holomorphic analogue by replacing a complex variable by a quaternion one. A fairly large number of detailed examples are given to illustrate the presented theory efficiency.