$R^n$空间中的Cauchy积分公式

Cauchy Integral Formulae in $\mathbb{R}^n$

中文关键词:  Dirac算子  Cauchy积分公式.

英文关键词:Dirac operator  Cauchy integral formula.

中文摘要:

      本文中$p(\underline{D})={\underline{D}}^m+b_1{\underline{D}}^{m-1}+\cdots+b_m$为$R^n$空间中的多项式Dirac算子, 其中$\underline{D}=\sum^n_{j=1} e_j\frac{\partial }{\partial x_j}$为标准的Dirac算子, 系数$b_j$等均为复常数. 本文讨论了$p(\underline{D})$的所有分解类型, 得到了相应方程$p(\underline{D})f=0$解$f$的Cauchy积分公式显示表达式.

英文摘要:

      In this note $p(\underline{D})={\underline{D}}^m+b_1{\underline{D}}^{m-1}+\cdots+b_m$ is a polynomial Dirac operator in $\mathbb{R}^n$, where $\underline{D}=\sum^n_{j=1} e_j\frac{\partial }{\partial x_j}$ is a standard Dirac operator in $\mathbb{R}^n$, $b_j$ are the complex constant coefficients. In this note we discuss all decompositions of $p(\underline{D})$ according to its coefficients $b_j$, and obtain the corresponding explicit Cauchy integral formulae of $f$ which are the solution of $p(\underline{D})f=0$.