Decompositions, Integral Formulae And Expansions In Hermitean Clifford Analysis

In this thesis, we mainly study decompositions, integral formulae and ex-pansions in Hermitean Clifford Analysis (simply, HCA). First we introduce the Clifford algebras and spinor spaces, and then the framework of the Classical or-thogonal Clifford analysis (simply, CA). We discuss the properties of the Teodor-escu operator in CA. We introduce a group of fundamental operators include the derivation operators in the framework of HCA, we get a representation of corresponding Lie algebras. By the results of these operators, we obtain a series of decompositions for complex Clifford algebras. We establish the integral for-mulae for spinor valued functions in HCA by using the projection method, which include the Bochner-Martinelli formulae as its special case, this is a promotion of results in several complex analysis in this sense, and we discussed some other cases, we get a variety of different types of integral formulae. Finally on the basis of this, we also get the Sokhotskii-Plemelj formula of Bochner-Martinelli type. Based on the existing theory of Hermitean monogenic polynomials in HCA, at last we get the Taylor expansions for Hermitean monogenic functions by using the spinor projection operator and Gegenbauer polynomials. It consists of seven chapters as follows:In chapter 1, some research backgrounds and histories are introduced.Chapter 2 is about the Clifford algebras and spinor spaces, we mainly in-troduce the real Clifford algebras and complex Clifford algebras, then we discuss the Clifford groups, the Pin groups, the Spin groups and their Lie algebras. Fi-nally we discuss the spinor spaces in Clifford algebras by introducing the complex structures and the Witt basis. By the results of Fock spaces, we obtain that the spinor space is a minimal left ideal in the complex Clifford algebra. At last we investigate the dual spaces of the spinor spaces.In chapter 3, some preliminaries of Classical orthogonal CA are given for this thesis including the theory of monogenic functions, integral formulae, series expansions and so on. And we also discuss the properties of the Teodorescu operator in CA in this chapter.In chapter 4, we mainly introduce the framework of HCA, including Her-mitean vector variables and Dirac operator, action of the unitary group, Her-mitean monogenic polynomials and Fischer decomposition.We study the decompositions of complex Clifford algebras in HCA in chap-ter 5. We introduce a group of fundamental operators include the derivation operators in the framework of HCA, we get a representation of corresponding Lie algebras. By the results of these operators, we obtain a series of decompositions for complex Clifford algebras.In chapter 6, we establish the integral formulae for spinor valued functions in HCA by using the theory of spinor valued functions and spinor projection opera-tor. These formulae including the Bochner-Martinelli formulae as its special case, and we also get a variety of different types of integral formulae by discussing some other cases. We also discuss Sokhotskii-Plemelj formula of Bochner-Martinelli type.In chapter 7, the subject are series expansions in HCA. By the theory of Her-mitean monogenic polynomials, Gegenbauer polynomials and projection meth-ods, we obtain the Taylor expansion of Hermitean monogenic functions.