Clifford分析中几类函数的性质及其相关问题研究
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摘要
在H.Grassmann代数的基础上,W.K.Clifford推广了“四元数”的概念,创建了一种可结合不可交换的代数结构,称之为Clifford代数.实(或复)Clifford分析主要研究定义在实(或复)欧氏空间上取值于实(或复)Clifford代数空间中函数的性质及其相关理论.
设Cln+1,0(R)(或Cln+1,0(C))是由{e0,e1,···,en}生成的2n+1维实(或复)Clifford代数空间,e(?)D=1是其单位元,且elej+ejel=2δlj(l,j=0,1,···,n),其中δlj是Kronecker符号.设Clo,n(C))是由{e1,e2,···,en}生成的2n维复Clifford代数空间,e(?)=1是其单位元,且elej+ejel=-2δlj(l,j=1,2,···,n).
本文首先研究了定义在Rn+1上取值于Cln+1,0(R)中的k-hypergenic函数与Clifford Mobius变换复合后函数的性质以及hypergenic拟-Cauchy型积分的边界性质和对偶的hypergenic函数的Cauchy积分公式；其次,研究了定义在Cn+1上取值于Cln+1,0(C)中的复k-hypergenic函数的几种等价刻画和Cauchy积分定理以及复k-hypergenic函数与复k-hypergenic调和函数的关系；最后,研究了定义在Cn+1上取值于Cl0,n(C)中的复k-超单演函数的等价刻画和Cauchy积分定理以及复k-超单演函数与复k-双曲调和函数的关系.
第一章简要介绍本文的研究背景和研究现状,给出重要的定义与符号,并且列出本文的主要结果.
第二章首先研究了Cln+1,0(R)中的Clifford Mobius变换,得到了与Clifford Mobius变换相关的几个重要定理,并且证明了一个k-hypergenic函数与Clifford Mobius变换的复合可以得到一个加权的k-hypergenic函数；其次,借助于hyper-genic函数的Cauchy积分公式得到了hypergenic拟-Cauchy型积分的Plemelj公式,再利用Plemelj公式证明了hypergenic拟-Cauchy型积分的Privalov定理；最后,给出了对偶的hypergenic函数的Cauchy积分公式,利用其证明了(1—n)-hypergenic函数的Cauchy积分公式,并且讨论了对偶的hypergenic函数的Cauchy积分公式中右端积分的性质.
第三章首先研究了复k-hypergenic函数的几种等价刻画；其次,利用Stokes-Green定理证明了复k-hypergenic函数的Cauchy积分定理,在此基础上给出了复k-hypergenic调和函数的Cauchy积分定理；最后,讨论了复k-hypergenic函数与复k-hypergenic调和函数的关系.
第四章首先研究了复k-超单演函数的一种与Cauchy-Riemann方程类似的等价刻画,虽然复k-超单演函数的乘积未必是复k-超单演函数,但是利用上述定理可以得到与复k-超单演函数的乘积相关的几个重要定理；其次,利用Stokes-Green定理证明了复k-超单演函数的Cauchy积分定理,在此基础上给出了复k-双曲调和函数的Cauchy积分定理；最后,讨论了复k-超单演函数与复k-双曲调和函数的关系.
本文的研究工作进一步丰富和完善了Clifford分析中的函数理论,深化了人们对Clifford分析的认识,在理论上和实际中都有一定的意义.
Based on H.Grassmann external algebra, W.K.Clifford generalized the concept of "quaternion" and established an associative and non-commutative algebra struc-ture named Clifford algebra. Real (or complex) Clifford analysis studies mainly the properties and the related theory of functions, which are defined in a real (or com-plex) Euclidean space and whose values are in an associative and non commutative real (or complex) Clifford algebra space.
Suppose Cln+1,0(R)(or Cln+1,0(C)) is a2n+1dimensional real (or complex) Clifford algebra space generated by{eo,e1,… en} and e?=1is its unit element and eiej+ejei=2δlj (l,j=0,1,…, n), where δlj is the Kronecker sign. Suppose Cl0,n(C) is a2n dimensional complex Clifford algebra space generated by{e1, e2,…, en} and e?=1is its unit element and elej+ejel=-2δlj (l,j=1,2,…,n).
Firstly, this dissertation studies the properties of the composition functions of κ-hypergenic functions and Clifford Mobius transformations, where the κ-hypergenic functions are defined on Rn+1and their values are in Cln+1,0(R), and it also studies the boundary properties of hypergenic quasi-Cauchy integrals and the Cauchy in-tegral formula for dual hypergenic functions; secondly, it discusses several types of equivalent characterizations of complex κ-hypergenic functions, which are defined on Cn+1and whose values are in Cln+1,o(C), and it also explores the Cauchy inte-gral theorem for complex κ-hypergenic functions and discusses the relations between complex κ-hypergenic functions and complex κ-hypergenic harmonic functions; fi-nally, it analyzes some characterizations of complex κ-hypermonogenic functions, which are defined on Cn+1and whose values are in Cl0,n(C), and it also demon-strates the Cauchy integral theorem for complex κ-hypermonogenic functions and discusses the relations between complex κ-hypermonogenic functions and complex κ-hypermonogenic harmonic functions.
Chapter1introduces briefly the research background and status quo of this dissertation and gives important definitions and notations as well as its main results.
Chapter2firstly studies Clifford Mobius transformations in Cln+1,o(R) and obtains some important theorems related to Clifford Mobius transformations and proves that the composition of a κ-hypergenic function with a Clifford Mobius transformation leads to a κ-hypergenic function with weight; next, by virtue of the Cauchy integral formula for hypergenic functions it obtains the Plemelj formula for hypergenic quasi-Cauchy integrals and proves the Privalov theorem for hypergenic quasi-Cauchy integrals taking advantage of the Plemelj formula; finally, it gives the Cauchy integral formula for dual hypergenic functions and proves the Cauchy in-tegral formula for (1-n)-hypergenic functions making use of it and discusses the properties of the right integral of the Cauchy integral formula for dual hypergenic functions.
Chapter3firstly studies some kinds of equivalent characterizations of com-plex κ-hypergenic functions; secondly, making use of the Stokes-Green theorem, it proves the Cauchy integral theorem for complex κ-hypergenic functions and based on this, it gives the Cauchy integral theorem for complex κ-hypergenic harmonic functions; finally, it discusses the relations between complex κ-hypergenic functions and complex κ-hypergenic harmonic functions.
Chapter4firstly studies a kind of equivalent characterization of complex k-hypermonogenic functions, which is similar to the Cauchy-Riemann equations, al-though the products of complex κ-hypermonogenic functions are not sure complex κ-hypermonogenic functions, yet some important theorems related to the products of complex κ-hypermonogenic functions can be obtained, taking advantage of the above theorem; secondly, in accordance with the Stokes-Green theorem, the Cauchy integral theorem for complex κ-hypermonogenic functions is proved and based on this, the Cauchy integral theorem for complex κ-hypermonogenic harmonic functions is presented; finally, it discusses the relations between complex κ-hypermonogenic functions and complex κ-hyperbolic harmonic functions.
In all, this research further enriches and perfects function theory of Clifford analysis, deepens the understanding of Clifford analysis, and naturally it is signifi-cant in both theory and practice.
设Cln+1,0(R)(或Cln+1,0(C))是由{e0,e1,···,en}生成的2n+1维实(或复)Clifford代数空间,e(?)D=1是其单位元,且elej+ejel=2δlj(l,j=0,1,···,n),其中δlj是Kronecker符号.设Clo,n(C))是由{e1,e2,···,en}生成的2n维复Clifford代数空间,e(?)=1是其单位元,且elej+ejel=-2δlj(l,j=1,2,···,n).
本文首先研究了定义在Rn+1上取值于Cln+1,0(R)中的k-hypergenic函数与Clifford Mobius变换复合后函数的性质以及hypergenic拟-Cauchy型积分的边界性质和对偶的hypergenic函数的Cauchy积分公式；其次,研究了定义在Cn+1上取值于Cln+1,0(C)中的复k-hypergenic函数的几种等价刻画和Cauchy积分定理以及复k-hypergenic函数与复k-hypergenic调和函数的关系；最后,研究了定义在Cn+1上取值于Cl0,n(C)中的复k-超单演函数的等价刻画和Cauchy积分定理以及复k-超单演函数与复k-双曲调和函数的关系.
第一章简要介绍本文的研究背景和研究现状,给出重要的定义与符号,并且列出本文的主要结果.
第二章首先研究了Cln+1,0(R)中的Clifford Mobius变换,得到了与Clifford Mobius变换相关的几个重要定理,并且证明了一个k-hypergenic函数与Clifford Mobius变换的复合可以得到一个加权的k-hypergenic函数；其次,借助于hyper-genic函数的Cauchy积分公式得到了hypergenic拟-Cauchy型积分的Plemelj公式,再利用Plemelj公式证明了hypergenic拟-Cauchy型积分的Privalov定理；最后,给出了对偶的hypergenic函数的Cauchy积分公式,利用其证明了(1—n)-hypergenic函数的Cauchy积分公式,并且讨论了对偶的hypergenic函数的Cauchy积分公式中右端积分的性质.
第三章首先研究了复k-hypergenic函数的几种等价刻画；其次,利用Stokes-Green定理证明了复k-hypergenic函数的Cauchy积分定理,在此基础上给出了复k-hypergenic调和函数的Cauchy积分定理；最后,讨论了复k-hypergenic函数与复k-hypergenic调和函数的关系.
第四章首先研究了复k-超单演函数的一种与Cauchy-Riemann方程类似的等价刻画,虽然复k-超单演函数的乘积未必是复k-超单演函数,但是利用上述定理可以得到与复k-超单演函数的乘积相关的几个重要定理；其次,利用Stokes-Green定理证明了复k-超单演函数的Cauchy积分定理,在此基础上给出了复k-双曲调和函数的Cauchy积分定理；最后,讨论了复k-超单演函数与复k-双曲调和函数的关系.
本文的研究工作进一步丰富和完善了Clifford分析中的函数理论,深化了人们对Clifford分析的认识,在理论上和实际中都有一定的意义.
Based on H.Grassmann external algebra, W.K.Clifford generalized the concept of "quaternion" and established an associative and non-commutative algebra struc-ture named Clifford algebra. Real (or complex) Clifford analysis studies mainly the properties and the related theory of functions, which are defined in a real (or com-plex) Euclidean space and whose values are in an associative and non commutative real (or complex) Clifford algebra space.
Suppose Cln+1,0(R)(or Cln+1,0(C)) is a2n+1dimensional real (or complex) Clifford algebra space generated by{eo,e1,… en} and e?=1is its unit element and eiej+ejei=2δlj (l,j=0,1,…, n), where δlj is the Kronecker sign. Suppose Cl0,n(C) is a2n dimensional complex Clifford algebra space generated by{e1, e2,…, en} and e?=1is its unit element and elej+ejel=-2δlj (l,j=1,2,…,n).
Firstly, this dissertation studies the properties of the composition functions of κ-hypergenic functions and Clifford Mobius transformations, where the κ-hypergenic functions are defined on Rn+1and their values are in Cln+1,0(R), and it also studies the boundary properties of hypergenic quasi-Cauchy integrals and the Cauchy in-tegral formula for dual hypergenic functions; secondly, it discusses several types of equivalent characterizations of complex κ-hypergenic functions, which are defined on Cn+1and whose values are in Cln+1,o(C), and it also explores the Cauchy inte-gral theorem for complex κ-hypergenic functions and discusses the relations between complex κ-hypergenic functions and complex κ-hypergenic harmonic functions; fi-nally, it analyzes some characterizations of complex κ-hypermonogenic functions, which are defined on Cn+1and whose values are in Cl0,n(C), and it also demon-strates the Cauchy integral theorem for complex κ-hypermonogenic functions and discusses the relations between complex κ-hypermonogenic functions and complex κ-hypermonogenic harmonic functions.
Chapter1introduces briefly the research background and status quo of this dissertation and gives important definitions and notations as well as its main results.
Chapter2firstly studies Clifford Mobius transformations in Cln+1,o(R) and obtains some important theorems related to Clifford Mobius transformations and proves that the composition of a κ-hypergenic function with a Clifford Mobius transformation leads to a κ-hypergenic function with weight; next, by virtue of the Cauchy integral formula for hypergenic functions it obtains the Plemelj formula for hypergenic quasi-Cauchy integrals and proves the Privalov theorem for hypergenic quasi-Cauchy integrals taking advantage of the Plemelj formula; finally, it gives the Cauchy integral formula for dual hypergenic functions and proves the Cauchy in-tegral formula for (1-n)-hypergenic functions making use of it and discusses the properties of the right integral of the Cauchy integral formula for dual hypergenic functions.
Chapter3firstly studies some kinds of equivalent characterizations of com-plex κ-hypergenic functions; secondly, making use of the Stokes-Green theorem, it proves the Cauchy integral theorem for complex κ-hypergenic functions and based on this, it gives the Cauchy integral theorem for complex κ-hypergenic harmonic functions; finally, it discusses the relations between complex κ-hypergenic functions and complex κ-hypergenic harmonic functions.
Chapter4firstly studies a kind of equivalent characterization of complex k-hypermonogenic functions, which is similar to the Cauchy-Riemann equations, al-though the products of complex κ-hypermonogenic functions are not sure complex κ-hypermonogenic functions, yet some important theorems related to the products of complex κ-hypermonogenic functions can be obtained, taking advantage of the above theorem; secondly, in accordance with the Stokes-Green theorem, the Cauchy integral theorem for complex κ-hypermonogenic functions is proved and based on this, the Cauchy integral theorem for complex κ-hypermonogenic harmonic functions is presented; finally, it discusses the relations between complex κ-hypermonogenic functions and complex κ-hyperbolic harmonic functions.
In all, this research further enriches and perfects function theory of Clifford analysis, deepens the understanding of Clifford analysis, and naturally it is signifi-cant in both theory and practice.
引文
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