In this course we’ll explore complex analysis, complex dynamics, and some applications of these topics. This course provides an introduction to complex analysis, that is the theory of complex functions of a complex variable. We'll start by introducing the complex plane along with the algebra and geometry of complex numbers and make our way via differentiation, integration, complex dynamics and power series representation into territories at the edge of what's known today.
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).
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.
Course Introduction Video