Wesleyan University

About this Course

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.

Section

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 videos (Total 119 min), 5 readings, 2 quizzes

Algebra and Geometry in the Complex Plane30m

Polar Representation of Complex Numbers32m

Roots of Complex Numbers14m

Topology in the Plane21m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Module 1 Homework10m

Section

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 videos (Total 123 min), 5 readings, 1 quiz

Sequences and Limits of Complex Numbers30m

Iteration of Quadratic Polynomials, Julia Sets25m

How to Find Julia Sets20m

The Mandelbrot Set18m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Module 2 Homework10m

Section

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 videos (Total 135 min), 5 readings, 2 quizzes

The Cauchy-Riemann Equations29m

The Complex Exponential Function24m

Complex Trigonometric Functions21m

First Properties of Analytic Functions25m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Module 3 Homework10m

Section

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 videos (Total 113 min), 5 readings, 1 quiz

Conformal Mappings26m

Möbius transformations, Part 127m

Möbius Transformations, Part 217m

The Riemann Mapping Theorem15m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Module 4 Homework10m

Section

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 videos (Total 141 min), 5 readings, 2 quizzes

Complex Integration - Examples and First Facts32m

The Fundamental Theorem of Calculus for Analytic Functions19m

Cauchy’s Theorem and Integral Formula32m

Consequences of Cauchy’s Theorem and Integral Formula28m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Module 5 Homework10m

Section

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 videos (Total 114 min), 5 readings, 1 quiz

Power Series25m

The Radius of Convergence of a Power Series28m

The Riemann Zeta Function And The Riemann Hypothesis23m

The Prime Number Theorem15m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Module 6 Homework10m

Section

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 videos (Total 114 min), 6 readings, 1 quiz

Isolated Singularities of Analytic Functions28m

The Residue Theorem17m

Finding Residues13m

Evaluating Integrals via the Residue Theorem9m

Bonus: Evaluating an Improper Integral via the Residue Theorem16m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Lecture Slides10m

Module 7 Homework10m

Section

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. ...

1 quiz

Final Exam40m

4.8

got a tangible career benefit from this course

By 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!

By 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.

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.
...

When will I have access to the lectures and assignments?

Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.

What will I get if I pay for this course?

If you pay for this course, you will have access to all of the features and content you need to earn a Course Certificate. If you complete the course successfully, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. Note that the Course Certificate does not represent official academic credit from the partner institution offering the course.

What is the refund policy?

Is financial aid available?

Yes! Coursera provides financial aid to learners who would like to complete a course but cannot afford the course fee. To apply for aid, select "Learn more and apply" in the Financial Aid section below the "Enroll" button. You'll be prompted to complete a simple application; no other paperwork is required.

More questions? Visit the Learner Help Center

Coursera provides universal access to the world’s best education,
partnering with top universities and organizations to offer courses online.