Adequate quaternionic generalization of complex differentiability

The high efficiency of complex analysis is attributable mainly to the ability to represent adequately the Euclidean physical plane essential properties, which have no counterparts on the real axis. In order to provide the similar ability in higher dimensions of space we introduce the general concept of essentially adequate differentiability, which generalizes the key features of the transition from real to complex differentiability. In view of this concept the known Cauchy-Riemann-Fueter equations can be characterized as inessentially adequate. Based on this concept, in addition to the usual complex definition, the quaternionic derivative has to be independent of the method of qua- ternion division: on the left or on the right. Then we deduce the generalized quaternionic Cauchy- Riemann equations as necessary and sufficient conditions for quaternionic functions to be ℍ-holo- morphic. We prove that each ℍ-holomorphic function can be constructed from the ℂ-holomorphic function of the same kind by replacing a complex variable by a quaternionic in an expression for the ℂ-holomorphic function. It follows that the derivatives of all orders of ℍ- holomorphic functions are also ℍ-holomorphic and can be analogously constructed from the corresponding derivatives of ℂ- holomorphic functions. The examples of Liouvillian elementary functions demonstrate the effi- ciency of the developed theory.