Modified Quaternionic Analysis in ℝ ⁴

Abstract

This paper deals with classes of solutions of the generalized Cauchy-Riemann system

$$ \left\{ {\matrix{ {s({{\partial u} \over {\partial x}} - {{\partial v} \over {\partial y}} - {{\partial w} \over {\partial t}} - {{\partial r} \over {\partial s}}) + 2r = 0,} \cr {{{\partial u} \over {\partial y}} = - {{\partial v} \over {\partial x}},{{\partial u} \over {\partial t}} = - {{\partial w} \over {\partial x}},{{\partial u} \over {\partial s}} = - {{\partial r} \over {\partial x}},} \cr {{{\partial v} \over {\partial t}} = {{\partial w} \over {\partial y}},{{\partial v} \over {\partial s}} = {{\partial r} \over {\partial y}},{{\partial w} \over {\partial s}} = {{\partial r} \over {\partial t}},} \cr } } \right. $$

where f = u + iv + j w + kr is a C ²-function of the quaternionic variable z = x + iy + jt + ks, defined on an open set Ω ℝ⁴. The system (1) has been examined in the more general, n-dimensional, setting in [7] and in the 3-dimensional case in [8] and [9].