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.


Procedure

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: jcponcemath@gmail.com

Finally, if you find this content useful, please consider supporting my work using the links below.

∞ Thanks!

Basic tools

Click on image to open tool!

Modulus & Phase

Real & Imaginary components

Modulus & Phase

Advanced tools

Click on image to open tool!

JavaScript

Modulus & Phase

Real & Imaginary components

Modulus

Modulus & Phase

Mod/Pha/Re/Im

Modulus & Phase

Plain HSL

HSLuv Modulus & Phase

WebGL

Real & Imaginary components

Modulus & Phase

Click on the flower below to enter

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


References

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  3. Davis, J. E. (2017). Complex Domain Coloring, Available online
  4. Crone, L. (s.f) Color graphs of complex functions.
  5. Farris, F. (1998). Review of Visual Complex Analysis. By Tristan Needham. The American Mathematical Monthly, 105(6), 570-576.
  6. Farris, F. A. (n.d.). Visualizing complex-valued functions in the plane.
  7. Farris, F. A. (2017). Domain Coloring and the Argument Principle. PRIMUS 27:8-9, 827-844, DOI: 10.1080/10511970.2016.1234526
  8. Losada Liste, R. (2014). El color dinámico de GeoGebra. Gaceta De La Real Sociedad Matematica Española, 17, 525–547, Madrid.
  9. Lundmark, H. (2004). Visualizing complex analytic functions using domain coloring.
  10. Needham, T. (1997). Visual Complex Analysis. Oxford University Press, Oxford.
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  13. Ponce Campuzano, J. C. (2021) Domain colouring for visualising and exploring the beauty of complex functions. Australian Mathematical Society Gazette. Vol. 48, No. 4, pp. 162-170.
  14. Ponce Campuzano, J. C. (2019) The use of phase portraits to visualize and investigate isolated singular points of complex functions Journal International Journal of Mathematical Education in Science and Technology. 50(7), pp. 999-1010, DOI: 10.1080/0020739X.2019.1656829
  15. Thaller, B. (1998). Visualization of complex functions, Mathematica J., Vol. 7, issue 2,
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  19. Wegert, E. (2016). Visual exploration of complex functions. In T. Qian and L. R. Rodino (eds.), Mathematical analysis, probability and applications – Plenary Lectures SPMS (Vol. 177. pp. 253–279). Switzerland: Springer International Publishing.
  20. Wegert, E. & Semmler, G. (2011). Phase plots of complex functions: a journey in illustration. Notices American Mathematical Society, 58, 768-780.
  21. Wikipedia: Domain Coloring