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 ( ). 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: Chapters: 0:00 Intro 0:25 Fundamentals 3:55 2D graphs 8:05 Domain coloring 11:42 3D & 4D plots 20:02 Makin