The theory of slice regular functions has been developed rapidly in the past few years, and most properties are given in slices at the early stage. In 2013, Colombo et al. introduced a non-constant coefficients differential operator to describe slice regular functions globally, and this brought the study of slice regular functions in a global sense. In this article, we introduce a slice Cauchy–Riemann operator, which is motivated by the non-constant coefficients differential operator mentioned above. Then, A Borel–Pompeiu formula for this slice Cauchy–Riemann operator is discovered, which leads to a Cauchy integral formula for slice regular functions. A Plemelj integral formula for the slice Cauchy–Riemann operator is introduced, which gives rise to results on slice regular extension at the end.

切片正则函数理论这几年发展很快，早期大部分性质都是在切片中给出的。 2013 年，科伦坡等人。引入非常数系数微分算子来全局描述切片正则函数，从而将切片正则函数的研究带到了全局意义上。在本文中，我们介绍了切片柯西-黎曼算子，它的动机是上述非常系数微分算子。然后，发现了该切片柯西-黎曼算子的Borel-Pompeiu公式，从而导出了切片正则函数的柯西积分公式。引入了切片柯西-黎曼算子的Plemelj积分公式，最终得到了切片正则扩展的结果。