This repository contains comprehensive lecture notes for a Complex Analysis course, covering topics such as analytic function, Cauchy integral theorem, Laurent series, residue, conformal mapping and harmonic functions in
These notes were created based on a Complex Analysis course at Ocean University of China. The content is designed for a 4-hour weekly course format. Please note that due to its specialized organization, these materials are not recommended for first-time learners of the subject. Also, to make the structure of this notes much better, some part of materials that are not in the course would be put in.
The notes are organized into the following chapters:
- Complex Numbers and Complex Functions - Basic introduction of complex numbers
- Analytic Functions - How to determine an analytic function and its basic properties
- Cauchy Integral - Cauchy integral theorem and its applications
- Taylor Series of Analytic Functions - Taylor serires theory of complex functions
- Laurent Series - Laurent series theory of complex functions and isolated singularity
- Residue and its applications - Residue theorem and calculation of real integral
- Conformal Mappings - Conformal mapping and its geometric understanding
- Harmonic Functions - Introduction of Harmonic functions
| Section Number | Section Name | Status |
|---|---|---|
| 1.1 | Complex Numbers | ❎ |
| 1.2 | Complex Set and Curve | ❎ |
| 1.3 | Complex Functions | ❎ |
| 1.4 | Extended Complex Plane | ❎ |
| 2.1 | Analytic Functions and Cauchy-Riemann Conditions | ❎ |
| 2.2 | Monodrome Functions | ❎ |
| 2.3 | Multivalued Functions | ❎ |
| 3.1 | Complex Integral | ❎ |
| 3.2 | Cauchy Integral Theorem and its Applications | ❎ |
| 3.3 | Cauchy Integral Formula | ❎ |
| 4.1 | Complex Series: Convergence | ❎ |
| 4.2 | Complex Power Seires | ❎ |
| 4.3 | Complex Power Series of Analytic Functions | ❎ |
| 4.4 | Identity Theorem and Maximum Modulus Theorem | ❎ |
| 5.1 | Laurent Series | ❎ |
| 5.2 | Properties of Laurent Series: Isolated Singularity | ❎ |
| 5.3 | Properties of Laurent Series: Infinite Point | ❎ |
| 6.1 | Residue | ❎ |
| 6.2 | Application of Residue | ❎ |
| 6.3 | Argument Principle and Rouche Theorem | ❎ |
| 7.1 | Geometric Properties of Analytic Functions | ❎ |
| 7.2 | Fraction Linear Mapping | ❎ |
| 7.3 | Conformal Mapping | ❎ |
| 7.4 | Identity Theorem of Conformal Mapping | ❎ |
| 8.1 | Mean Value Theorem and Extremum Theorem | ❎ |
| 8.2 | Possion Integral Formula and Dirichlet Problem | ❎ |
Notice that the whole notes is written in Chinese, the content might have some translation mistakes.
Status Symbol Meanings:
| Status | Symbol |
|---|---|
| ❎ | Not started yet |
| ❇️ | Partially finished |
| ✅ | Already finished |
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