Integral Formulas for Slice Cauchy–Riemann Operator and Applications

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.