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复分析引论

总览授课大纲常见问题解答制作方评分和审阅

主页数学和逻辑

复分析引论

卫斯连大学

关于此课程: 复分析是关于变量为复数的复函数的理论,本课程对该理论进行了初步的介绍。我们将从介绍复平面,复数的代数和几何开始,然后介绍微分,积分,复动力系统,幂级数表示和 Laurent 级数一直到当今研究的前沿。每一模块的课程包含5个视频,视频中还会包括小测试,除此之外还有电子计分的作业。而模块1,3,5中会有同学间的互评。 作业需要花时间思考并练习课上教授的概念。实际上,完成作业将花费相当的时间。我们并不希望你太快地完成作业,而是希望你用纸和笔进行详细地计算。总之,我们希望你能投入6-12个小时在每一个模块上,当然,时间长短取决于你的背景。

此课程适用人群: 这门课程面向任何想要探索数学的一个美丽而且重要的分支的人。(假如有)想要更新微积分知识的意愿会对学习课程有所帮助,但对于学习的热情与兴趣比其他前置知识更重要,


制作方:  卫斯连大学
卫斯连大学

  • Dr. Petra Bonfert-Taylor

    教学方:  Dr. Petra Bonfert-Taylor, Professor of Mathematics

    Mathematics and Computer Science
级别Intermediate
承诺学习时间8周课程,6-12小时/周
语言
English
如何通过通过所有计分作业以完成课程。
用户评分
4.8 星
平均用户评分 4.8查看学生的留言
授课大纲
第 1 周
Introduction to Complex Numbers
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.
5 视频, 5 阅读材料
  1. 视频: History of Complex Numbers
  2. Reading: Lecture Slides
  3. 视频: Algebra and Geometry in the Complex Plane
  4. Reading: Lecture Slides
  5. 视频: Polar Representation of Complex Numbers
  6. Reading: Lecture Slides
  7. 视频: Roots of Complex Numbers
  8. Reading: Lecture Slides
  9. 视频: Topology in the Plane
  10. Reading: Lecture Slides
已评分: Module 1 Homework
已评分: Peer-Graded Assignment #1
第 2 周
Complex Functions and Iteration
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.
5 视频, 5 阅读材料
  1. 视频: Complex Functions
  2. Reading: Lecture Slides
  3. 视频: Sequences and Limits of Complex Numbers
  4. Reading: Lecture Slides
  5. 视频: Iteration of Quadratic Polynomials, Julia Sets
  6. Reading: Lecture Slides
  7. 视频: How to Find Julia Sets
  8. Reading: Lecture Slides
  9. 视频: The Mandelbrot Set
  10. Reading: Lecture Slides
已评分: Module 2 Homework
第 3 周
Analytic Functions
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!
5 视频, 5 阅读材料
  1. 视频: The Complex Derivative
  2. Reading: Lecture Slides
  3. 视频: The Cauchy-Riemann Equations
  4. Reading: Lecture Slides
  5. 视频: The Complex Exponential Function
  6. Reading: Lecture Slides
  7. 视频: Complex Trigonometric Functions
  8. Reading: Lecture Slides
  9. 视频: First Properties of Analytic Functions
  10. Reading: Lecture Slides
已评分: Module 3 Homework
已评分: Peer Graded Assignment #2
第 4 周
Conformal Mappings
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.
5 视频, 5 阅读材料
  1. 视频: Inverse Functions of Analytic Functions
  2. Reading: Lecture Slides
  3. 视频: Conformal Mappings
  4. Reading: Lecture Slides
  5. 视频: Möbius transformations, Part 1
  6. Reading: Lecture Slides
  7. 视频: Möbius Transformations, Part 2
  8. Reading: Lecture Slides
  9. 视频: The Riemann Mapping Theorem
  10. Reading: Lecture Slides
已评分: Module 4 Homework
第 5 周
Complex Integration
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!
5 视频, 5 阅读材料
  1. 视频: Complex Integration
  2. Reading: Lecture Slides
  3. 视频: Complex Integration - Examples and First Facts
  4. Reading: Lecture Slides
  5. 视频: The Fundamental Theorem of Calculus for Analytic Functions
  6. Reading: Lecture Slides
  7. 视频: Cauchy’s Theorem and Integral Formula
  8. Reading: Lecture Slides
  9. 视频: Consequences of Cauchy’s Theorem and Integral Formula
  10. Reading: Lecture Slides
已评分: Module 5 Homework
已评分: Peer-Graded Assignment #3
第 6 周
Power Series
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.
5 视频, 5 阅读材料
  1. 视频: Infinite Series of Complex Numbers
  2. Reading: Lecture Slides
  3. 视频: Power Series
  4. Reading: Lecture Slides
  5. 视频: The Radius of Convergence of a Power Series
  6. Reading: Lecture Slides
  7. 视频: The Riemann Zeta Function And The Riemann Hypothesis
  8. Reading: Lecture Slides
  9. 视频: The Prime Number Theorem
  10. Reading: Lecture Slides
已评分: Module 6 Homework
第 7 周
Laurent Series and the Residue Theorem
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.
6 视频, 6 阅读材料
  1. 视频: Laurent Series
  2. Reading: Lecture Slides
  3. 视频: Isolated Singularities of Analytic Functions
  4. Reading: Lecture Slides
  5. 视频: The Residue Theorem
  6. Reading: Lecture Slides
  7. 视频: Finding Residues
  8. Reading: Lecture Slides
  9. 视频: Evaluating Integrals via the Residue Theorem
  10. Reading: Lecture Slides
  11. 视频: Bonus: Evaluating an Improper Integral via the Residue Theorem
  12. Reading: Lecture Slides
已评分: Module 7 Homework
第 8 周
Final Exam
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.
    已评分: Final Exam

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    制作方
    卫斯连大学
    At Wesleyan, distinguished scholar-teachers work closely with students, taking advantage of fluidity among disciplines to explore the world with a variety of tools. The university seeks to build a diverse, energetic community of students, faculty, and staff who think critically and creatively and who value independence of mind and generosity of spirit.
    评分和审阅
    已评分 4.8,总共 5 个 412 评分
    BHARTI SHARMA

    loved it

    SM

    Very easy to understand, and extremely engaging. The exercises are challenging and fun to do.

    cl

    Excellent to learn about complex analysis without burden of too much proofs; an introduction to Riemann Zeta function.

    Rens Kamphuis

    The lectures were very easy to follow and the exercises fitted these lectures well. This course was not always very rigorous, but a great introduction to complex analysis nevertheless. Thank you!



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