Some Properties Of Complex Monogenic Functions In Complex Clifford Analysis

In this paper, complex monogenic functions in complex Cliford analysis are studiedand then its properties and Taylor expansion are obtained. In fact, the complex mono-genic functions are the extension of holomorphic functions in high dimensional space.This paper is made up of three parts:In the frst part, the preliminaries of this paper, basic calculations and funda-mental lemmas are given. Firstly, we introduce the defnition of complex Cliford al-gebra Anand its basic calculations, then give unique decomposition of any elementa=P a+Qaen, a∈Anand basic properties about mappings P and Q, in the end,defne complex monogenic functions. All of these lay a solid foundation for studying thecomplex monogenic functions.In the second part, condition of equivalence and construction theory of complexmonogenic functions are studied. Firstly, we discuss some essential properties of complexmonogenic functions and obtain diferent conditions of equivalence, namely, condition ofequivalence in the view of mappings P and Q; Secondly, we study the construction theoryof complex monogenic functions, then apply the complex harmonic function, decompo-sition of P, Q and some integrals to construct complex monogenic functions, moreover,obtain the relationship between complex monogenic functions and complex harmonicfunctions, such relationship is similar to the tie between analytic functions and harmonicfunctions in classical analysis.In the third part, the relationship between a complex monogenic function f andTaylor expansion is studied. Firstly, we defne the left inner spherical monogenic oforder k, namely, Pk(z), then fnd a basis Vl1,l2,···lkfor Pk(z) and prove that Vl1,l2,···lkisinner spherical monogenic, at last, expand a complex monogenic function f locally intoa so-called Taylor series: In the view of form and content, such expansion is similar to Taylor series in classicalanalysis, in addition, the polynomials Vl1,l2,···lkplay an analogous role as the powers ofcomplex variable z do in the classical Taylor series for holomorphic functions.Lastly, we analyze the related progress and prospect in complex monogenic functionsin conclusion, whose more research will be expected later.The proceeding content develop the theory of complex monogenic functions andenlarge our vision for functions in Cliford analysis in more broad and deep view, allof these lay a solid foundation for studying the properties and applications of complexmonogenic functions.