四元数分析中光滑曲面上的 Poincaré-Bertrand 公式

数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 534-553.

四元数分析中光滑曲面上的 Poincaré-Bertrand 公式

周宇杰^1,^*(),罗纬宇²(),汪玉峰³(),张忠祥³()

¹澳门大学科技学院澳门 999078
²广西师范大学数学与统计学院广西桂林 541004
³武汉大学数学与统计学院武汉 430079

收稿日期:2024-02-16 修回日期:2024-09-24 出版日期:2025-04-26 发布日期:2025-04-09
通讯作者: 周宇杰 E-mail:651397389@qq.com;luoweiyu18@163.com;wh_yfwang@163.com;zhxzhang.math@whu.edu.cn
作者简介:罗纬宇，E-mail: luoweiyu18@163.com;|汪玉峰，E-mail:wh_yfwang@163.com;|张忠祥，E-mail:zhxzhang.math@whu.edu.cn
基金资助:
国家自然科学基金(11223344)

Poincaré-Bertrand Formula on Smooth Surfaces in Quaternion Analysis

Zhou Yujie^1,^*(),Luo Weiyu²(),Wang Yufeng³(),Zhang Zhongxiang³()

¹Faculty of Science and Technology,University of Macau, Macau 999078
²School of Mathematics and Statistics, Guangxi Normal University, Guangxi Guilin 541004
³School of Mathematics and Statistics, Wuhan University, Wuhan 430079

Received:2024-02-16 Revised:2024-09-24 Online:2025-04-26 Published:2025-04-09
Contact: Yujie Zhou E-mail:651397389@qq.com;luoweiyu18@163.com;wh_yfwang@163.com;zhxzhang.math@whu.edu.cn
Supported by:
NSFC(11223344)

摘要/Abstract

摘要：

四元数代数是一种满足结合律但不满足交换律的代数结构, 对于研究高维空间中的方程和算子具有重要的理论意义和应用价值. 通过先证明四元数分析中的含参变量的 Privalov 定理, 再证明非主值积分的换序公式, 最后采用数学分析方法两边同时取极限证明了光滑曲面上的 Poincaré-Bertrand 公式.

关键词: 四元数分析, Cauchy 型奇异积分, Poincaré-Bertrand 公式

Abstract:

Quaternion algebra is an algebraic structure that satisfies the associative law but not the commutative law. It has important theoretical significance and application value for studying equations and operators in high-dimensional spaces. By first proving the Privalov theorem with parameter variables in quaternion analysis, then proving the commutation formula of non principal value products, and finally using mathematical analysis methods to take limits on both sides, the Poincaré-Bertrand formula on smooth surfaces is proved.

Key words: quaternion analysis, Cauchy type singular integral, Poincaré-Bertrand formula

中图分类号:

O175.23

周宇杰,罗纬宇,汪玉峰,张忠祥. 四元数分析中光滑曲面上的 Poincaré-Bertrand 公式[J]. 数学物理学报, 2025, 45(2): 534-553.

Zhou Yujie,Luo Weiyu,Wang Yufeng,Zhang Zhongxiang. Poincaré-Bertrand Formula on Smooth Surfaces in Quaternion Analysis[J]. Acta mathematica scientia,Series A, 2025, 45(2): 534-553.

参考文献

[1]	杨丕文. 四元数分析与偏微分方程. 北京: 科学出版社, 2009
[1]	Yang P W. Quaternion Analysis and Partial Differential Equations. Beijing: Science Press, 2009
[2]	Brack F, Delanghe R, Sommen F. Clifford Analysis. Boston: Pitman Books Limits, 1982
[3]	路见可. 解析函数边值问题. 武汉: 武汉大学出版社, 2004
[3]	Lu J K. Analytical Function Boundary Value Problems. Wuhan: Wuhan University, 2004
[4]	Kandmanov R M. Bochner-Martinelli integrals and their applications. Sciences Siberia Press, 1992, 100: 209-213
[5]	黄沙. Clifford 分析中奇异积分方程的 Poincaré-Bertrand 公式. 数学学报, 1998, 41(1) : 119-126
[5]	Huang S. Poincaré-Bertrand transformation formulae of the singular integrals in Clifford analysis. Acta Mathematica Sinica, 1998, 41(1): 119-126
[6]	罗纬宇. Clifford 分析中正则函数边值问题及奇异积分方程. 武汉: 武汉大学出版社, 2014
[6]	Luo W Y. The regularization function boundary value problem and singular integral equation in Clifford analysis. Wuhan: Wuhan University, 2014
[7]	Huang S, Qiao Y Y, Wen G C. Real and Complex Clifford Analysis. New York: Springer, 2006
[8]	Xu Q H, Ma T, Yin Z. Functional Analysis. Beijing: Higher Education Press, 2017
[9]	Gilbert R P, Buchanan J L. First order elliptic systems, a function theoretic approach. Mathematics in Science Engineering, 1983, 163: Art 281
[10]	Rankin R A. Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.). The Mathematical Gazette, 1968, 52(382): 412
[11]	Luo W Y, Du J. Generalized Cauchy theorem in Clifford analysis and boundary value problems for regular functions. Advances in Applied Clifford Algebras, 2017, 27: 2531-2583
[12]	Zhang Z X, Du J Y. On certain Riemann boundary value problems and singular integral equations in Clifford analysis. Chinese Journal of Contemporary Mathematics, 2001, 22(3): 237-244
[13]	Du J Y, Zhang Z X. A Cauchy's integral formula for functions with values in a universal Clifford algebral and its applications. Complex Variables, 2002, 47(10): 915-928
[14]	Du J Y, Xu N, Zhang Z X. Boundary behavior of Cauchy-type integrals in Clifford analysis. Acta Math Sci, 2009, 29(1): 210-224