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_surface ). 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-Complex-function-plots (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/e87f874a06ef882cf86b886f1251a995 Online Complex Function Plotter: https://www.dynamicmath.xyz/domain-coloring/#adv Proof of Euler's Identity: https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-10-14/v/euler-s-formula-and-euler-s-identity I made all the music in a free, open source DAW called LMMS ( https://lmms.io/ ).