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# Analysis of a Complex Kind

7 Reviews 110 students interested

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## Overview

Complex analysis is the study of functions that live in the complex plane, i.e. functions that have complex arguments and complex outputs. In order to study the behavior of such functions we’ll need to first understand the basic objects involved, namely the complex numbers. We’ll begin with some history: When and why were complex numbers invented? Was it the need for a solution of the equation x^2 = -1 that brought the field of complex analysis into being, or were there other reasons? Once we’ve answered these questions we’ll devote some time to learn about basic properties of complex numbers that will make it possible for us to use them in more advanced settings later on. We will learn how to do basic algebra with these numbers, how they behave in limiting processes, etc. These facts enable us to begin the study of complex functions, and at this point we can already understand the basics about the construction of the Mandelbrot set and Julia sets (if you have never heard of these that’s quite alright, but do look at http://en.wikipedia.org/wiki/Mandelbrot_set for example to see some beautiful pictures).

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. Don’t worry! We’ll help you remember facts from calculus in case you have forgotten.  After this exploration we will be ready to meet the main players: analytic functions. These are functions that possess complex derivatives in lots of places, a fact which endows these functions with some of the most beautiful properties mathematics has to offer. We’ll explore these properties!

Who would want to differentiate without being able to undo it? Clearly we’ll have to learn about integration as well. But we are in the complex plane, so what are the objects we’ll integrate over? Curves! We’ll study these as well, and we’ll tie everything together via Cauchy’s beautiful and all encompassing integral theorem and formula.

Throughout this course we'll tell you about some of the major theorems in the field (even if we won't be able to go into depth about them) as well as some outstanding conjectures.

## Syllabus

Week One: Introduction to complex numbers, their geometry and algebra, working with complex numbers.

Week Two: The Mandelbrot set, Julia sets, a famous outstanding conjecture, history of complex numbers, sequences of complex numbers and convergence, complex functions.

Week Three: Complex differentiation and the Cauchy-Riemann equations.

Week Four: Conformal mappings, Möbius transformations and the Riemann mapping theorem.

Week Five: Complex integration, Cauchy-Goursat theorem, Cauchy integral formula, Liouville's Theorem, maximum principle, fundamental theorem of algebra.

Week Six: Power series representation of analytic functions, singularities, the Riemann zeta function, Riemann hypothesis, relation to prime numbers.

#### Taught by

Petra Bonfert-Taylor

## Reviews for Coursera's Analysis of a Complex Kind 4.7 Based on 7 reviews

• 5 stars 71%
• 4 stars 29%
• 3 star 0%
• 2 star 0%
• 1 star 0%

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• 1
Anonymous
5.0 5 years ago
completed this course.
It was a very informative class. Besides the analytics of complex numbers, we learned to communicate our reasoning and to plot graphs in the complex plane, which was rated by some of our classmates (peer grading). We also were shown how to plot Julia sets, extremely nice pictures.

Dr. Bonfert-Taylor explained all topics in an understandable and friendly way.
1 person found
Gaetano P
5.0 3 years ago
by completed this course.
Really nice course. Materials, peer graded assignments and homeworks were well proposed.

Prof. Bonfert-Taylor carried out the lessons in clear and friendly way. Excellent.

Anonymous
5.0 5 years ago
completed this course.
Excellent!

I got to learn all about the Mandelbrot set, and so much more!
Anonymous
5.0 5 years ago
completed this course.
Really enjoyed it, well presented and excellent material.
Bhuvan N
4.0 4 years ago
by completed this course, spending 3 hours a week on it and found the course difficulty to be medium.
Francisco J
5.0 3 years ago
completed this course.
Nicolas B
4.0 4 years ago
by completed this course.