Domain coloring

The following example colors the origin in black, 1 in red, −1 in cyan, and a point at infinity in white:

${\displaystyle {\begin{cases}H&=\arg z,\\L&=\ell (|z|)\\S&=100\%.\end{cases}}}$

There are a number of choices for the function ${\displaystyle \ell$ :[0,\infty )\to [0,1)} . A desirable property is ${\displaystyle \ell (1/r)=1-\ell (r)}$ such that the 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 ${\displaystyle a>0}$).

A widespread choice which does not have this property is the function ${\displaystyle \ell _{3}(r)=1-a^{|r|}}$ (with some parameter ${\displaystyle 0<a<1}$) which for ${\displaystyle a=1/2}$ and ${\displaystyle 0\leq r\leq 1}$ is very close to ${\displaystyle \ell _{1}}$.

This approach uses the HSL (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 sixth roots of unity (starting with 1) are: red, yellow, green, cyan, blue, and magenta. Magnitude is coded by intensity via a strictly monotonic continuous function.

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 around L = 1/2. Use of the Lab color space corrects this, making the images more accurate, but also makes them more drab/pastel.

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.

A discontinuous color function. In the graph, each discontinuity occurs when ${\displaystyle |z|=2^{n}}$ for integers n.

Unlike the finite range of the argument, the magnitude of a complex number can range from 0 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 every integer magnitude, exponential equations such as every magnitude n where ${\displaystyle 2^{n}}$ 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.[1] 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.[2]