Some Integral Representations and Singular Integral over Plane in Clifford Analysis

In this paper, Cauchy type integral and singular integral over hyper-complex plane ∏ are considered. By using a special Möbius transform, an equivalent relation between H ^ μ class functions over ∏ and H μ class functions over the unit sphere is shown. For H ^ μ class functions over ∏ , we prove the existence of Cauchy type integral and singular integral over ∏ . 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 ∏ and unit sphere is built.