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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/complex-analysis
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课程名称：复变分析导论 课程概述：本课程提供复变分析的基础知识，即复变量的复函数理论。我们将从复平面及复数的代数与几何入手，逐步深入到微分、积分、复动力学、幂级数表示及洛朗级数等领域，探索现代数学的前沿知识。每个模块包含五个视频讲座和嵌入式小测验，随后是电子评分的作业。此外，模块 1、3 和 5 还包含同伴评估。作业要求学生深入思考并练习课堂上讨论的概念，学习过程主要在完成作业中进行。整体预计每个模块需要 6-12 小时的学习时间，具体取决于个人背景。 课程大纲： 1. **复数导论**：回顾复数的历史，探讨其诞生的原因。学习复平面的代数与几何，掌握复数的极坐标表示以及复平面的拓扑知识。 2. **复函数与迭代**：研究复平面上的函数，重点了解复函数的光滑性。通过二次多项式探索复动力学以及如何生成美丽的分形图像。 3. **解析函数**：学习复微分及其与柯西-黎曼方程的关系，关注解析函数的美妙性质，并研究复指数和三角函数的复杂性质。 4. **保角映射**：研究解析函数的逆函数，了解保角映射的性质及美比乌斯变换的重要性，并探讨黎曼映射定理。 5. **复积分**：了解复平面上的积分，学习复路径积分及其在整个复分析中的应用，包括柯西积分定理及代数基本定理的证明。 6. **幂级数**：学习解析函数的幂级数表示，研究无穷级数及收敛特性，探讨黎曼ζ函数及其与素数的关系。 7. **洛朗级数与留数定理**：理解洛朗级数在孤立奇点附近的应用，学习重要的定理及其在实际中的运用，掌握如何通过留数定理计算积分。 8. **期末考试**：课程的终结模块，涵盖前七周的所有内容。考试为开放式，设计为两小时内完成，并提供讨论论坛供技术问题反馈。 通过本课程，学习者将全面掌握复变分析的基础知识及其应用，为后续更深入的数学研究奠定坚实基础。
Name:Introduction to Complex Numbers
Description:We’ll begin this module by briefly learning about the history of complex numbers: When and why were they invented? In particular, we’ll look at the rather surprising fact that the original need for complex numbers did not arise from the study of quadratic equations (such as solving the equation z^2+1 = 0), but rather from the study of cubic equations! Next we’ll cover some algebra and geometry in the complex plane to learn how to compute with and visualize complex numbers. To that end we’ll also learn about the polar representation of complex numbers, which will lend itself nicely to finding roots of complex numbers. We’ll finish this module by looking at some topology in the complex plane.
Name:Complex Functions and Iteration
Description:Complex analysis is the study of functions that live in the complex plane, that is, functions that have complex arguments and complex outputs. The main goal of this module is to familiarize ourselves with such functions. Ultimately we’ll want to study their smoothness properties (that is, we’ll want to differentiate complex functions of complex variables), and we therefore need to understand sequences of complex numbers as well as limits in the complex plane. We’ll use quadratic polynomials as an example in the study of complex functions and take an excursion into the beautiful field of complex dynamics by looking at the iterates of certain quadratic polynomials. This allows us to learn about the basics of the construction of Julia sets of quadratic polynomials. You'll learn everything you need to know to create your own beautiful fractal images, if you so desire. We’ll finish this module by defining and looking at the Mandelbrot set and one of the biggest outstanding conjectures in the field of complex dynamics.
Name:Analytic Functions
Description:When studying functions we are often interested in their local behavior, more specifically, in how functions change as their argument changes. This leads us to studying complex differentiation – a more powerful concept than that which we learned in calculus. We’ll begin this module by reviewing some facts from calculus and then learn about complex differentiation and the Cauchy-Riemann equations in order to meet the main players: analytic functions. These are functions that possess complex derivatives in lots of places; a fact, which endows them with some of the most beautiful properties mathematics has to offer. We’ll finish this module with the study of some functions that are complex differentiable, such as the complex exponential function and complex trigonometric functions. These functions agree with their well-known real-valued counterparts on the real axis!
Name:Conformal Mappings
Description:We’ll begin this module by studying inverse functions of analytic functions such as the complex logarithm (inverse of the exponential) and complex roots (inverses of power) functions. In order to possess a (local) inverse, an analytic function needs to have a non-zero derivative, and we’ll discover the powerful fact that at any such place an analytic function preserves angles between curves and is therefore a conformal mapping! We'll spend two lectures talking about very special conformal mappings, namely Möbius transformations; these are some of the most fundamental mappings in geometric analysis. We'll finish this module with the famous and stunning Riemann mapping theorem. This theorem allows us to study arbitrary simply connected sub-regions of the complex plane by transporting geometry and complex analysis from the unit disk to those domains via conformal mappings, the existence of which is guaranteed via the Riemann Mapping Theorem.
Name:Complex Integration
Description:Now that we are familiar with complex differentiation and analytic functions we are ready to tackle integration. But we are in the complex plane, so what are the objects we’ll integrate over? Curves! We’ll begin this module by studying curves (“paths”) and next get acquainted with the complex path integral. Then we’ll learn about Cauchy’s beautiful and all encompassing integral theorem and formula. Next we’ll study some of the powerful consequences of these theorems, such as Liouville’s Theorem, the Maximum Principle and, believe it or not, we’ll be able to prove the Fundamental Theorem of Algebra using Complex Analysis. It's going to be a week filled with many amazing results!
Name:Power Series
Description:In this module we’ll learn about power series representations of analytic functions. We’ll begin by studying infinite series of complex numbers and complex functions as well as their convergence properties. Power series are especially easy to understand, well behaved and easy to work with. We’ll learn that every analytic function can be locally represented as a power series, which makes it possible to approximate analytic functions locally via polynomials. As a special treat, we'll explore the Riemann zeta function, and we’ll make our way into territories at the edge of what is known today such as the Riemann hypothesis and its relation to prime numbers.
Name:Laurent Series and the Residue Theorem
Description:Laurent series are a powerful tool to understand analytic functions near their singularities. Whereas power series with non-negative exponents can be used to represent analytic functions in disks, Laurent series (which can have negative exponents) serve a similar purpose in annuli. We’ll begin this module by introducing Laurent series and their relation to analytic functions and then continue on to the study and classification of isolated singularities of analytic functions. We’ll encounter some powerful and famous theorems such as the Theorem of Casorati-Weierstraß and Picard’s Theorem, both of which serve to better understand the behavior of an analytic function near an essential singularity. Finally we’ll be ready to tackle the Residue Theorem, which has many important applications. We’ll learn how to find residues and evaluate some integrals (even some real integrals on the real line!) via this important theorem.
Name:Final Exam
Description:Congratulations for having completed the seven weeks of this course! This module contains the final exam for the course. The exam is cumulative and covers the topics discussed in Weeks 1-7. The exam has 20 questions and is designed to be a two-hour exam. You have one attempt only, but you do not have to complete the exam within two hours. The discussion forum will stay open during the exam. It is against the honor code to discuss answers to any exam question on the forum. The forum should only be used to discuss questions on other material or to alert staff of technical issues with the exam.
This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. We will start by introducing the complex plane, along with the algebra and geometry of complex numbers, and then we will make our way via differentiation, integration, complex dynamics, power series representation and Laurent series into territories at the edge of what is known today. Each module consists of five video lectures with embedded quizzes, followed by an electronically graded homework assignment. Additionally, modules 1, 3, and 5 also contain a peer assessment. The homework assignments will require time to think through and practice the concepts discussed in the lectures. In fact, a significant amount of your learning will happen while completing the homework assignments. These assignments are not meant to be completed quickly; rather you'll need paper and pen with you to work through the questions. In total, we expect that the course will take 6-12 hours of work per module, depending on your background.
