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

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.