无界域上全纯Cliffordian函数的若干性质及其Cauchy型积分的边值特性

Some properties of holomorphic Cliffordian functions and boundary value properties of the Cauchy type integral in unbounded domain

首先介绍了定义于Euclidean空间,取值于实Clifford代数的全纯Cliffordian函数.然后利用正则函数的性质讨论了全纯Cliffordian函数的若干性质和空间性质.主要借用第一类拟置换得出全纯Cliffordian函数的等价条件,研究了它与正则函数之间的关系.接下来以Cauchy积分公式和Plemelj公式为基础,得出了开拓定理.最后定义了无界域上的Cauchy型积分,并证得它在Cauchy主值意义下收敛.同时利用一些重要的积分估值得出了无界域上的Plemelj公式.

In thefirst place,the left and right holomorphic Cliffordian functions are defined in Euclidean space and with values in real Clifford algebra.Then some spacial properties of holomorphic Cliffordian functions are discussed by way of the properties of regular functions.With the help of thefirst class of Quasi-Permutation,the equal conditions are proved from the angle of regular functions,which build the relations between regular functions and holomorphic Cliffordian functions.And then,the extension theorem is discussed based on the Cauchy type integral formula and the Plemelj formula in the bounded domain using some small techniques.Finally,the Cauchy type integral is defined on unbounded domains,and it is proved be convergent under the meaning of Cauchy principal value.And the Plemelj formula is discussed by way of some significant integral estimation and some methods above.