首先,给出了R3中平面和球面方程的超复形式,接着提出了R3中关于平面和球面对称点的概念,并给出了关于平面和球面对称点所满足的等价方程.我们考虑了超复空间Cl3中的一些特殊的Möbius变换,并给出了其一些性质, 比如:保持球面或平面不变性, 保持关于平面和球面对称性不变性,保持交比不变性等. 文中给出了正则函数和Möbius变换的关系. 其次,证明了R3中球内正则函数的推广的Cauchy定理和Cauchy积分公式. 借助于上述正则函数的Cauchy积分公式和其对称点的积分表示, 给出了正则函数的Poisson积分表示.最后,在Möbius变换的性质基础上,给出了Möbius变换下曲面积分的变量替换公式.
Abstract
In this paper, the hyper-complex forms of equations for the plane and the sphere in R3 are given. Then, the concept of symmetric points with respect to the plane and the sphere in R3 is introduced, equivalent equations for symmetric points respect to the plane and the sphere are given. Some special Möbius transformations in hyper-complex space Cl3 are considered. Some properties of the mentioned Möbius transformation are given. For example, the Möbius transformation maps the plane and the sphere in R3 one-to-one onto the plane or the sphere, the Möbius transformation preserves the symmetric points with respect to the plane and the sphere in Cl3, the cross ratio of four distinct points is also invariant under the Möbius transformation. A relation between the monogenic function and Möbius transformation is shown. Secondly, a generalized Cauchy theorem and generalized Cauchy integral formula over the sphere for monogenic functions are proved. By combining the Cauchy integral formula for monogenic functions with the integral representation at the symmetric point with respect to the sphere, the Poisson integral representation for monogenic function is obtained. Finally, based on the properties of Möbius transformations, integral formulas for change of variables under Möbius transformations are constructed.
关键词
Clifford代数 /
Möbius 变换 /
Poisson积分表示
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Key words
Clifford algebra /
Möbius transformation /
Poisson integral representation
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参考文献
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脚注
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基金
DAAD王宽诚教育基金;国家自然科学基金青年基金资助项目(11001206)
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