What Are Complex Numbers - Essence of Complex Analysis

Overview

Dive into a comprehensive 32-minute video exploring the fundamentals of complex numbers. Learn about Cartesian and polar representations, Euler's formula and identity, and various operations including addition, multiplication, division, and exponentiation. Discover de Moivre's theorem and its applications in finding nth roots of complex numbers. Explore multiple proofs of Euler's formula and gain insights into the argument issue. Grasp the importance of complex numbers in real-world applications such as circuit analysis, Fourier analysis, and Laplace transforms. Suitable for both beginners and those seeking a refresher, this crash course covers topics typically found in introductory complex analysis courses, providing a solid foundation for further study in this powerful mathematical field.

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.