Week 1:

Core Topics in Complex Numbers

1. Definition of imaginary number, complex number, real and imaginary parts of complex numbers, complex conjugate.
2. Complex arithmetic including addition, subtraction, multiplication and division. Proposition 2.1, 2.2 and 2.3 in lecture notes.
3. Definition of complex plane, modulus (absolute value) and argument (phase).
4. Representation of complex numbers in rectangular and polar coordinates. Transformation between rectangular coordinates and polar coordinates. Proposition 3.2, 3.3, 3.4 and 3.5 in lecture notes.
5. Euler's formula and exponential notation of complex numbers (Proposition 3.7 and 3.8). How to use Euler's formula to compute nth roots (Proposition 4.8).
6. Properties of polynomial equations with real coefficients, including number of roots and Proposition 4.5.

Homework #1 (due by 5pm Tuesday, 01/23/2018)

Practice Problems in Katznelson's notes

• Exercises 1.1, 1.2 (page 3)
• Exercises 2.2, 2.6, 2.7 (page 6)
• Exercises 3.1, 3.6, 3.7 (page 15)
• Exercises 4.2, 4.7 (page 22)