Research On The Properties Of Several Types Of Functions And The Related Problems In Clifford Analysis

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.