Visualizing Complex-Valued Functions

This video goes over a few means of visualizing complex-valued functions/transformations, including domain coloring, modular surfaces and Riemann surfaces. The information provided isn't completely comprehensive and not at all rigorous, but hopefully will give you at least a little bit of insight and interest in the math for which these visuals are involved.

I should mention that the definition of a Riemann surface I gave at 15:09 is a naïve one - what I showed were examples of Riemann surfaces, however they don't have to necessarily be a plot of the imaginary/real component of some function ( https://en.wikipedia.org/wiki/Riemann... ). Also, as pointed out in the comments, the function in the blue box at 3:08 (arctan[Im(z)/Re(z)]) only works within quadrants 1 and 4. In general one would need a function like this: https://en.wikipedia.org/wiki/Atan2

Chapters:
0:00 Intro
0:25 Fundamentals
3:55 2D graphs
8:05 Domain coloring
11:42 3D & 4D plots
20:02 Making your own plots

Blender Add-ons: https://github.com/timvickman/Blender... (Chances are you can code much better than I can - feel free to use and improve upon the code for your own projects)

Matplotlib Graphs: https://gist.github.com/timvickman/e8...

Online Complex Function Plotter: https://www.dynamicmath.xyz/domain-co...

Proof of Euler's Identity: https://www.khanacademy.org/math/ap-c...

I made all the music in a free, open source DAW called LMMS ( https://lmms.io/ ).