摘 要：复数域上的函数论作为数学分析的重要分支，在现代物理学中具有不可替代的作用，其研究背景源于经典分析理论在多维空间及量子力学等领域应用的需求。本文旨在探讨复变函数理论及其在物理现象中的深刻意义，通过引入黎曼面、解析延拓等概念，系统阐述了复函数的性质与结构特征。基于柯西-黎曼方程和留数定理，建立了完整的复变函数理论框架，并将其应用于电磁学、流体力学以及量子场论等多个物理领域。研究采用理论推导与数值模拟相结合的方法，对典型物理模型进行了深入分析，揭示了复变函数在描述波动现象、散射过程等方面的独特优势。特别地，本文首次提出了基于复势能函数的非线性动力学模型，该模型能够更精确地刻画微观粒子的行为特性，为理解量子纠缠提供了新的视角。
关键词：复变函数理论；解析延拓；量子场论应用
Abstract：As an important branch of mathematical analysis, function theory in complex number field plays an irreplaceable role in modern physics. Its research background stems from the application requirements of classical analysis theory in multi-dimensional space and quantum mechanics. This paper aims to explore the theory of complex function function and its profound significance in physical phenomena, and introduces the properties and structural characteristics of the Riemn surface and analytic extension. Based on the Cauchy-Riemannian equation and the residue number theorem, it is developed and applied to many physical fields such as electromagnetism, fluid mechanics and quantum field theory. The study combines theoretical derivation and numerical simulation to make a thorough analysis of typical physical models, and reveals the unique advantages of complex change function in describing fluctuation phenomena and scattering processes. In particular, we present for the first time a nonlinear dynamic model based on the complex potential energy function that is able to characterize the behavior of microscopic particles more precisely, providing a new perspective on the understanding of quantum entanglement.
Key words：Theory of complex variable functions; analytic extension; quantum field theory applications
目 录
1 引言 1
2 复数域函数论基础 1
2.1 复数域的代数结构 1
2.2 复变函数的基本概念 2
2.3 解析函数的性质研究 2
3 复数域上的积分理论 3
3.1 复积分的定义与性质 3
3.2 柯西积分定理分析 3
3.3 留数定理及其应用 4
4 物理中的复数域函数 4
4.1 量子力学中的复数表示 4
4.2 电磁学中的复数应用 5
4.3 流体力学中的复势函数 5
5 复数域函数论的发展与展望 6
5.1 现代数学中的复数域拓展 6
5.2 新兴物理领域中的应用前景 6
5.3 复数域函数论未来发展方向 7
6 结论 7
参考文献 8
关键词：复变函数理论；解析延拓；量子场论应用
Abstract：As an important branch of mathematical analysis, function theory in complex number field plays an irreplaceable role in modern physics. Its research background stems from the application requirements of classical analysis theory in multi-dimensional space and quantum mechanics. This paper aims to explore the theory of complex function function and its profound significance in physical phenomena, and introduces the properties and structural characteristics of the Riemn surface and analytic extension. Based on the Cauchy-Riemannian equation and the residue number theorem, it is developed and applied to many physical fields such as electromagnetism, fluid mechanics and quantum field theory. The study combines theoretical derivation and numerical simulation to make a thorough analysis of typical physical models, and reveals the unique advantages of complex change function in describing fluctuation phenomena and scattering processes. In particular, we present for the first time a nonlinear dynamic model based on the complex potential energy function that is able to characterize the behavior of microscopic particles more precisely, providing a new perspective on the understanding of quantum entanglement.
Key words：Theory of complex variable functions; analytic extension; quantum field theory applications
目 录
1 引言 1
2 复数域函数论基础 1
2.1 复数域的代数结构 1
2.2 复变函数的基本概念 2
2.3 解析函数的性质研究 2
3 复数域上的积分理论 3
3.1 复积分的定义与性质 3
3.2 柯西积分定理分析 3
3.3 留数定理及其应用 4
4 物理中的复数域函数 4
4.1 量子力学中的复数表示 4
4.2 电磁学中的复数应用 5
4.3 流体力学中的复势函数 5
5 复数域函数论的发展与展望 6
5.1 现代数学中的复数域拓展 6
5.2 新兴物理领域中的应用前景 6
5.3 复数域函数论未来发展方向 7
6 结论 7
参考文献 8
