Some Integral Representations and Singular Integral over Plane in Clifford Analysis

In this paper, Cauchy type integral and singular integral over hyper-complex plane ({prod}) are considered. By using a special Möbius transform, an equivalent relation between ({widehat{H}^mu}) class functions over ({prod}) and ({H^mu}) class functions over the unit sphere is shown. For ({widehat{H}^mu}) class functions over ({prod}) , we prove the existence of Cauchy type integral and singular integral over ({prod}) . Cauchy integral formulas as well as Poisson integral formulas for monogenic functions in upper-half and lower-half space are given respectively. By using Möbius transform again, the relation between the Cauchy type integrals and the singular integrals over ({prod}) and unit sphere is built.