On the Radial Part of the Cauchy-Riemann Operator

Abstract

Using the Clifford algebra C _n , with generators ${e_{1}}, \ldots ,{e_{n}},e_{i}^{2} = - 1,$ the Cauchy-Riemann operator in ℝⁿ⁺¹ is defined as $\bar{\partial } = \frac{\partial }{{\partial xo}} + \sum\limits_{{i = 1}}^{n} {\frac{\partial }{{\partial {x_{i}}}}{e_{i}}} ,$ where $x = {x_{0}} + \sum\limits_{{i = 1}}^{n} {{x_{i}}{e_{i}}} ,$ is a paravector. We consider Fueter-type paravector functions f = u( x _o, p) + I( x) v( x _o, p), where ${p^{2}} = \sum\limits_{{i = 1}}^{n} {x_{i}^{2}} ,I\left( x \right): = \frac{1}{p} = \sum\limits_{{i = 1}}^{n} {{x_{i}}{e_{i}}} ,$ and u, v are real-valued. The equation $\bar{\partial }f = 0$ splits into two parts. One of them depends only on x_o,p. This leads to a system of partial differential equations which coincides with the system defining hypermonogenic functions. These functions arise for example as solutions of the Dirac equation in the upper half space ℝ ⁿ⁺¹₊ endowed with the Poincaré metric.