Holomorphic Cliffordian Function In Clifford Analysis

In this paper,some properties of the holomorphic Cliffordian functions which are defined in R^2m+2and with values in Clifford algebra,A_2m+2(R)are given and boundary behavior of the Cauchy-type integral is studied.As a genelization of regular functions,holomorphic Cliffordian function is a new kind of function class in Clifford analysis.So there is some theoretical and applicable value to study it.The paper is divided into three parts.In the first part,we introduce the construction and operation of Clifford algebra.Then the definitions of regular functions,holomorphic functions etc.and the Cauchy kernel of holomorphic Cliffordian functions are given.This part is the basis of the following parts.In the second part,some properties of holomorphic functions are proved,such as quasicompleteness, sequential compactness and Cauchy-type integral out of ball.So some properties of regular functions can extend to holomorphic functions.In the third part,the Cauchy-type integral formula for the 2m times continuously differential functions are given.Then Cauchy principal value and Plemelj formula of the Cauchy-type integral formula are discussed.On the base of the Cauchy type integral and Plemelj formula,the Privalov theorem are discussed from three cases:two points on the boundary;one point on the boundary and the other point in the region(outside the region);two points in the region(outside the region).Finally,the last part is a conclusion which summarizes the work of the whole paper and puts forward the target for further work.