What Are Complex Numbers - Essence of Complex Analysis

This crash course introduces complex numbers through Cartesian and polar representations, Euler’s formula, arithmetic and transcendental operations, and de Moivre’s theorem for finding nth roots.

Syllabus

Sarcastic and serious introductions.
1.1 Complex plane - Cartesian way.
1.2 Complex plane - Polar way (Intro).
1.3 Arguments about arguments.
1.4 Interconversion .
2.1 Euler's formula - classic proof.
2.2 Euler's formula - 2nd proof.
3.1 Operations - addition/subtraction.
3.2 Operations - multiplication.
3.3 Operations - conjugation.
3.4 Operations - division.
3.5 Operations - exponentiation.
3.6 Operations - logarithm.
3.7 Operations - sine/cosine.
4.1 de Moivre's theorem - intro.
4.2 de Moivre's theorem - nth roots.
4.3 de Moivre's theorem - Euler's formula 3rd proof.
Outro.