Domain Coloring

Visualizing complex functions

By Juan Carlos Ponce Campuzano, 15/July/2018

Introduction

Domain coloring is a method that allows us to represent complex functions by assigning a color to each point of the complex plane. The method involves basically two main ideas:

1. Assign a color to every point in the complex plane.
2. Color the domain of $f$ by painting the location $z$ with the color determined by the value $f(z)$. It is common to use the color wheel because it is easy to match the HUE values with the phase (argument) of a complex number $z$ which is usually defined in the interval $[0,2\pi)$, or $(-\pi, \pi]$.

To implement this method in the computer consider a rectangular region of pixels on the screen. This will be a discretized domain $D_h$ for the function $f$. Every pixel $i$ is identified with a complex number $z_i$ where $f$ is evaluated. Then calculate the phase of the value $f(z)$ and its corresponding color. Finally assign the resulting color to that pixel. This procedure is shown in the animation below. Here you will find a set of tools to visualize and explore complex functions using different color schemes and also a gallery containing a wide range of examples.

If you want to know all the details about the implementation in JavaScript, check the source code available at GitHub. If you find useful these tools or found some issues, please let me know: j.ponce@uq.edu.au

You can also support this project with:   Patreon   or   ❤️ PayPal

∞ Thanks!

Basic tools

Click on image to open tool!

Click on image to open tool!

WebGL

Click on the flower below to enter

$f(z)=\dfrac{1/(iz)^{18}-1/(iz)}{1/(iz)-1}$

References

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21. Wikipedia: Domain Coloring