Analysis of a Complex Kind

Course Description

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 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

Analysis of a Complex Kind
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Video Lectures & Study Materials

Visit the official course website for more study materials:

# Lecture Play Lecture
I. Introduction to Complex Numbers
1 History of Complex Numbers Play Video
2 Algebra and Geometry in the Complex Plane Play Video
3 Polar Representation of Complex Numbers Play Video
4 Roots of Complex Numbers Play Video
5 Topology in the Complex Plane Play Video
II. Complex Functions and iteration
6 Complex Functions Play Video
7 Sequences and Limits of Complex Numbers Play Video
8 Iteration of Quadratic Polynomials, Julia Sets Play Video
9 How to Find Julia Sets Play Video
10 The Mandelbrot Set Play Video
III. Analytic Functions
11 The Complex Derivative Play Video
12 The Cauchy-Riemann Equations Play Video
13 The Complex Exponential Function Play Video
14 Complex Trigonometric Functions Play Video
15 First Properties of Analytic Functions Play Video
16 Inverse Functions of Analytic Functions Play Video
IV. Conformal Mappings
17 Conformal Mappings Play Video
18 Möbius Transformatios, Part I Play Video
19 Möbius Transformatios, Part II Play Video
20 The Riemann Mapping Theorem Play Video
V. Complex Integration
21 Complex Integration Play Video
22 Complex Integration: Examples and First Facts Play Video
23 The Fundamental Theorem of Calculus for Analytic Functions Play Video
24 Cauchy's Theorem and Integral Formula Play Video
25 Consequences of Cauchy's Theorem and Integral Formula Play Video
VI. Complex Infinite Series
26 Infinite Series of Complex Numbers Play Video
27 Power Series Play Video
28 The Radius of Convergence of a Power Series Play Video
29 The Riemann Zeta Function and the Riemann Hypothesis Play Video
30 The Prime Number Theorem Play Video
VII. Laurent Series and the Residue Theorem
31 Laurent Series Play Video
32 Isolated Singularities of Analytic Functions Play Video
33 The Residue Theorem Play Video
34 Finding Residues Play Video
35 Evaluating Integrals via the Residue Theorem Play Video
36 Evaluating an Improper Integral via the Residue Theorem Play Video


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