Complex differentiability
Channel: fancy manifold
Date: 2021-12-21

This video is an overview of the notion of complex differentiability. It explains what it means for a function to be holomorphic, harmonic, conformal, and analytic, how these properties are related, and what's the difference between them.

📬 Contact info: fancymanifold@protonmail.com


🎵 Music

https://freemusicarchive.org/music/Kai_Engel/Chapter_Four__Fall

- September by Kai Engel
- Cobweb Morning by Kai Engel
- November by Kai Engel
- random chord progressions by me (Technician)


🕰 Timestamps

00:00 Intro
00:30 Outline
01:05 Prerequisites
01:23 1.1. Real differentiation
03:22 1.2. Complex differentiation
08:33 1.3. Cauchy-Riemann equations
10:52 1.4. Wirtinger derivatives
14:47 1.5. Relation to real derivative
16:35 1.6. Geometric picture
17:50 1.7. Conclusions
18:35 2.1. Curves on C
19:36 2.2. Conformal mappings
22:44 2.3. Complex numbers as matrices
25:51 3.1. Harmonic functions
27:51 3.2. Properties of harmonic functions
29:35 3.3. Mean-value property (harmonic functions)
32:34 3.4. Mean-value property (holomorphic functions)
33:50 3.5. Maximum principle
35:33 4.1. Analytic functions in R
37:44 4.2. Analytic functions in C
39:08 5. Final remarks