Domain coloring: Visualizing Complex Functions

What is domain coloring? How can we use it to visualize and study properties of complex functions? How can we use it to create abstract mathematical art?

Note: I re-uploaded this video since there was a missing sign at 5:06.

• Multiple online plotting tools in p5js:

• Learn more about Complex Analysis with this interactive textbook:

0:00 Intro
1:53 What is domain coloring?
4:23 Basic examples: phase portraits
5:13 Enhaced phase portraits
6:33 Examples of Enhanced phase portraits
8:42 Mathematical art and other color schemes

REFERENCES

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Music Credits 🎵
"Dreams" by Benjamin Tissot
License code: J6W4UHIC4XTSNBDT

∞ Thanks to my Patreons:
Juan Pablo Xandri Antuña, Ruan Ramon, Miguel Díaz, bleh, Dennis Watson, Doug Kuhlmann, mirror, Newnome Beauton, Adam Parrott, Sophia Wood (Fractal Kitty), pmben, Abei, Edward Huff.

#complexnumbers #p5js #mathart

[douglasstrother]

Frank Farris deserves a medal!
I didn't know who originated this coloring scheme.

[douglasstrother]

"Applied Complex Variables" by John W. Dettman (Dover Publishers) is a great read (The Math Sorcerer has a video on it.): the first part covers the geometry/topology of the complex plane from a Mathematician's perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicist's perspective. I've used Smith Charts (RF/microwave engineering) for years, but learned from Dettman that the "Smith Chart" is an instance of a Möbius Transformation.

For practical reasons, a typical "Math Methods for Physics & Engineering" course introduces the Cauchy-Riemann Conditions, Conformal Mapping, Contour Integrals and applications of the Residue Theorem, but has to omit a lot interesting details.

The Schaum's Outline on "Complex Variables" is a great companion book for more problems/solutions and content.

[tamfang]

I've also seen versions in which zeros are black and poles are white.