Calculus Revisited: Complex Variables, Differential Equations, and Linear Algebra

Part I: Complex Variables

There is nothing especially real about the “real numbers.” More specifically, the real numbers are simply numbers whose squares are non-negative. The name real number probably comes from the fact that there is a 1-to-1 correspondence between the real numbers and points on the x-axis. However the points on the x-axis make up a rather small portion of the points in the entire xy-plane. In that context, the complex numbers extend the number system from representing points on the x-axis into a larger system that represents points in the entire xy-plane. In this part of the course we discuss the arithmetic of complex numbers and why they are so important.

Lecture 1: The Complex Numbers

Lecture 2: Functions of a Complex Variable

Lecture 3: Conformal Mappings

Lecture 4: Sequences and Series

Lecture 5: Integrating Complex Functions