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.

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## Introduction to Complex Analysis

## About this Course

### Learner Career Outcomes

## 50%

## 25%

### Skills you will gain

### Learner Career Outcomes

## 50%

## 25%

#### 100% online

#### Flexible deadlines

#### Intermediate Level

#### Approx. 35 hours to complete

#### English

### Offered by

#### Wesleyan University

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.

## Syllabus - What you will learn from this course

**5 hours to complete**

## 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 hours to complete**

**5 videos**

**5 readings**

**1 practice exercise**

**3 hours to complete**

## 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.

**3 hours to complete**

**5 videos**

**5 readings**

**1 practice exercise**

**5 hours to complete**

## 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 hours to complete**

**5 videos**

**5 readings**

**1 practice exercise**

**3 hours to complete**

## 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.

**3 hours to complete**

**5 videos**

**5 readings**

**1 practice exercise**

### Reviews

#### 4.8

##### TOP REVIEWS FROM INTRODUCTION TO COMPLEX ANALYSIS

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!

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.

The prof makes it easy to understand yet fascinating. I enjoyed video checkpoints, quizzes and peer reviewed assignments. This course encourages you to think and discover new things.

it was very challenging course , not so easy to pass the assignments but if you have gone through lectures, it will helps a lot while doing the assignments especially the final quiz

Thank Dr. Bonfert-Taylor so much for delivering this course online at coursera. I've really learned a lot from this course. And I hope I'll take another course with her again.

Fun course and learned lots. Its tough to do maths without doing lots of practice so be prepared to do lots of examples outside of the course in order to master the material.

Love this course! Was VERY challenging, but I learned a tremendous amount. I would recommend this to anyone wanting to further their knowledge in this area of mathematics.

Wow, great introductory course on complex analysis, was hoping for a bit more analysis and a bit more actual proofs, but it's Coursera, can't expect a full-on course here.

Some of the latter lessons should have been put up front so as to better present the intuitions of what you're proofing, but overall found it to be a good course.

Helpful introduction to complex analysis. Sketches the intuition behind fundamental theorems without becoming too difficult for amateur mathematicians to follow.

The course is very well presented. Great experience for those individuals who believe in purity and virtue of the ideas enchained by the mathematical consensus.

Outstanding class -- very accessible introduction for a non-math major, but covered all the topics I felt I missed in my undergraduate complex analysis class.

## Frequently Asked Questions

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 purchase the Certificate?

When you purchase a Certificate you get access to all course materials, including graded assignments. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.

What is the refund policy?

Is financial aid available?

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