Research On Cauchy Integral Formulas And The Related Problems Of Some Functions In Clifford Analysis

The properties of functions that is defined on the vector space and valued on the Clif-ford algebra are mainly studied in Clifford analysis.Regular functions,hypermonogenic functions,hypergenic functions and harmonic functions are several important functions in Clifford analysis.As an integral representation,Cauchy integral formula is an important tool to study the properties of various functions.Therefore,studying Cauchy integral formulas are of great importance.In real Clifford analysis,the Cauchy integral formula and the related problems have formed a relatively complete theoretical system,but very little research on the related problems in complex Clifford analysis.Based on the above,firstly,we mainly study the Cauchy integral formula and the related problems for left regular functions,right regular functions and biregular functions in the complex Clifford algebra Cl_n+1,0(?),and then we prove the Almansi-type decomposition theorem for right ?-regular functions defined on a star-like domain centered at the origin O with values in the complex Clifford algebra Cl_n+1,0(?).As a corollary,the Almansi-type decomposition theorem for right polyharmonic functions of degree ? is given.This paper is divided into the following three chapters:In the first chapter,we mainly introduce the related definitions and lemmas in the complex Clifford algebra.In the second chapter,we study the Cauchy integral formula for left regular func-tions,right regular functions and biregular functions respectively in the complex Clifford analysis.In the third chapter,we prove the Almansi-type decomposition theorem for right ?-regular functions defined on a star-like domain centered at the origin O with values in the complex Clifford algebraCl_n+1,0(?),and as a corollary,we give the Almansi-type decomposition theorem for right polyharmonic functions of degree ?.