Generalized Cauchy Theorem in Clifford Analysis and Boundary Value Problems for Regular Functions
Generalized Cauchy Theorem in Clifford Analysis and Boundary Value Problems for Regular Functions
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DOI：
10.1007/s00006-017-0790-2
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发表时间：
2017-05
影响因子：
1.5
通讯作者：
Weiyu Luo;Jinyuan Du
中科院分区：
文献类型：
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作者：
Weiyu Luo;Jinyuan Du
本文建立了Clifford分析中关于仿球面的广义柯西定理和关于强仿球面的广义柯西积分公式。作为应用，分别得到了闭光滑曲面和具有弯曲尖端的柱面上的广义柯西定理和广义柯西积分公式。这直接导致了Painlevé定理和Clifford分析中边值差的Sochocki-Plemelj公式的推广。然后，利用这些结果讨论了Clifford分析中正则函数的Riemann跳跃边值问题和Dirichlet边值问题。文中还对一些奇异积分方程组进行了求解，并根据解出的边值问题得到了柯西主值的反演公式。
In this paper, we establish the generalized Cauchy theorems on the para-sphere and the generalized Cauchy integral formulae on the strong para-sphere in Clifford analysis. As applications, the generalized Cauchy theorems and the generalized Cauchy integral formulae on the closed smooth surface and the cylindroid with crooked tips are respectively obtained. And these directly result in the Painlevé theorem and the generalization of the Sochocki–Plemelj formula for the difference of boundary values in Clifford analysis. Then, by using these results the Riemann jump boundary value problems and Dirichlet boundary value problems for regular functions in Clifford analysis are discussed. Some singular integral equations are also solved and the inversion formula for Cauchy principal value is obtained by the results based on these boundary value problems solved.