Incomplex analysis,domain coloring or acolor wheel graph is a technique forvisualizingcomplex functions by assigning acolor to each point of thecomplex plane. By assigning points on the complex plane to different colors and brightness, domain coloring allows for a function from the complex plane to itself, whose graph would normally require four spatial dimensions, to be easily represented and understood. This provides insight to the fluidity of complex functions and shows natural geometric extensions ofreal functions.

Agraph of areal function can be drawn in two dimensions because there are two represented variables,${\displaystyle x}$ and${\displaystyle y}$. However,complex numbers are represented by two variables and therefore two dimensions; this means that representing a complex function (more precisely, acomplex-valued function of onecomplex variable${\displaystyle f:\mathbb{C}\to \mathbb{C}}$) requires the visualization of four dimensions. One way to achieve that is with aRiemann surface, but another method is by domain coloring.

The term "domain coloring" was coined by Frank Farris, possibly around 1998.^[1][2] There were many earlier uses of color to visualize complex functions, typically mappingargument (phase) to hue.^[3]Larry Crone used the method in the late 1980s.^[4]Dan Kucerovsky used it in 1990.^[5] The technique of using continuous color to map points from domain to codomain or image plane was used in 1999 by George Abdo and Paul Godfrey^[6] and colored grids were used in graphics byDoug Arnold that he dates to 1997.^[7]

HL plot ofz, as per the simple color function example described in the text (left), and the graph of the complex functionz³ − 1 (right) using the same color function, showing the three zeros as well as the negative real numbers as pink rays starting at the zeros.

Representing a four dimensional complex mapping with only two variables is undesirable, as methods like projections can result in a loss of information. However, it is possible to add variables that keep the four-dimensional process without requiring a visualization of four dimensions. In this case, the two added variables are visual inputs such as color and brightness because they are naturally two variables easily processed and distinguished by the human eye. This assignment is called a "color function". There are many different color functions used. A common practice is to represent thecomplex argument,${\displaystyle \arg z}$, (also known as "phase" or "angle") with ahue following thecolor wheel, and themagnitude by other means, such asbrightness orsaturation.

The following example colors theorigin in black,1 ingreen,−1 inmagenta, and a point at infinity in white:Where H ishue, S issaturation, and L islightness.There are a number of choices for the function${\displaystyle \ell :[0,\mathrm{\infty })\to [0,1)}$.${\displaystyle \ell }$ should bestrictly monotonic andcontinuous.Another desirable property is${\displaystyle \ell (1/r)=1-\ell (r)}$ such thatthe inverse of a function is exactly as light as the original function is dark (and the other way around). Possible choices include

• ${\displaystyle {\ell }_1(r)=\frac{2}{\pi }\arctan (r)}$ and
• ${\displaystyle {\ell }_2(r)=\frac{r^a}{r^a+1}}$ (with some parameter${\displaystyle a>0}$). With${\displaystyle a=2}$, this corresponds to thestereographic projection onto theRiemann sphere.

A widespread choice which does not have this property is the function${\displaystyle {\ell }_3(r)=1-a^{|r|}}$ (with some parameter) which for${\displaystyle a=1/2}$ and${\displaystyle 0\le r\le 1}$ is very close to${\displaystyle {\ell }_1}$.

This approach uses theHSL (hue, saturation, lightness) color model. Saturation is always set at the maximum of 100%. Vivid colors of the rainbow are rotating in a continuous way on the complex unit circle, so the sixthroots of unity (starting with 1) are: green, cyan, blue, magenta, red, and yellow.

Since the HSL color space is not perceptually uniform, one can see streaks of perceived brightness at yellow, cyan, and magenta (even though their absolute values are the same as red, green, and blue) and a halo aroundL =⁠1/2⁠. More modern color spaces, e.g, theLab color space orCIECAM02, correct this, making the images more accurate and less saturated.

Many color graphs have discontinuities, where instead of evenly changing brightness and color, it suddenly changes, even when the function itself is still smooth. This is done for a variety of reasons such as showing more detail or highlighting certain aspects of a function, likelevel sets.

Unlike the argument, which has finite range, the magnitude of a complex number can range from0 to∞. Therefore, in functions that have large ranges of magnitude, changes in magnitude can sometimes be hard to differentiate when a very large change is also pictured in the graph. This can be remedied with a discontinuous color function which shows a repeating brightness pattern for the magnitude based on a given equation. This allows smaller changes to be easily seen as well as larger changes that "discontinuously jump" to a higher magnitude. In the graph on the right, these discontinuities occur in circles around the center, and show a dimming of the graph that can then start becoming brighter again. A similar color function has been used for the graph on top of the article.

Equations that determine the discontinuities may be linear, such as for everyinteger magnitude, exponential equations such as every magnitude${\displaystyle 2^n}$ wheren is an integer, or any other equation.

Discontinuities may be placed where outputs have a certain property to highlight which parts of the graph have that property. For instance, a graph may, instead of showing the color cyan, jump from green to blue. This causes a discontinuity that is easy to spot, and can highlight lines such as where the argument is zero.^[8] Discontinuities may also affect large portions of a graph, such as a graph where the color wheel divides the graph into quadrants. In this way, it is easy to show where each quadrant ends up with relations to others.^[9]

People who experiencecolor blindness may have trouble interpreting such graphs when they are made with standardcolor maps.^[10][11] This issue can be partially fixed by creating alternate versions using color maps that fit within the color space discernible to those with color blindness.^[12] For example, for use by those with totaldeuteranopia, a color map based on blue/grey/yellow may be more readable than the conventional map based on blue/green/red.^[12]

Wikimedia Commons has media related toComplex color plots.

• Samuel Li's function plotter
• High-quality, browser-based interactive complex function plotter by Ricky Reusseur
• Color Graphs of Complex Functions
• Visualizing complex-valued functions in the plane.
• Gallery of Complex Functions
• Complex Mapper by Alessandro Rosa
• John Davis software,S-Lang script for domain coloring
• Open source C and Python domain coloring software
• Enhanced 3D Domain coloring
• Domain Coloring Method on GPU
• Java domain coloring software (In development)
• MATLAB routines[1]
• Python script for GIMP by Michael J. Gruber
• Matplotlib andMayaVi implementation of domain coloring by E. Petrisor[2]
• MATLABroutines with user interface and various color schemes
• MATLABroutines for 3D-visualization of complex functions
• Color wheel method
• Real-Time Zooming Math Engine
• Fractal Zoomer : Software that utilizes domain coloring
• cplot, a domain-coloring package for Python

• Complex analysis
• Numerical function drawing

• Articles with short description
• Short description matches Wikidata
• Commons category link is on Wikidata