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.
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1Introduction 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 个视频 （总计 119 分钟）, 5 个阅读材料, 2 个测验
5 个视频
Algebra and Geometry in the Complex Plane30分钟
Polar Representation of Complex Numbers32分钟
Roots of Complex Numbers14分钟
Topology in the Plane21分钟
5 个阅读材料
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
1 个练习
Module 1 Homework10分钟
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2Complex 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 个视频 （总计 123 分钟）, 5 个阅读材料, 1 个测验
5 个视频
Sequences and Limits of Complex Numbers30分钟
Iteration of Quadratic Polynomials, Julia Sets25分钟
How to Find Julia Sets20分钟
The Mandelbrot Set18分钟
5 个阅读材料
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
1 个练习
Module 2 Homework10分钟
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3Analytic 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 个视频 （总计 135 分钟）, 5 个阅读材料, 2 个测验
5 个视频
The Cauchy-Riemann Equations29分钟
The Complex Exponential Function24分钟
Complex Trigonometric Functions21分钟
First Properties of Analytic Functions25分钟
5 个阅读材料
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
1 个练习
Module 3 Homework10分钟
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4Conformal 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 个视频 （总计 113 分钟）, 5 个阅读材料, 1 个测验
5 个视频
Conformal Mappings26分钟
Möbius transformations, Part 127分钟
Möbius Transformations, Part 217分钟
The Riemann Mapping Theorem15分钟
5 个阅读材料
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
Lecture Slides10分钟
1 个练习
Module 4 Homework10分钟
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创建者 RK•Apr 6th 2018
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!
创建者 GC•Mar 21st 2017
With this wonderful complex analysis course under your belt you will be ready for the joys of Digital Signal Processing, solving Partial Differential Equations and Quantum Mechanics.
讲师
Dr. Petra Bonfert-Taylor
Former Professor of Mathematics at Wesleyan University / Professor of Engineering at Thayer School of Engineering at Dartmouth关于 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.
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