Basic Tutorial

Installation, loading, Makie and Plots.jl

DomainColoring.jl provides plotting routines for complex functions. These require either the Makie or Plots.jl plotting libraries. In this tutorial we will use the former.

Makie supports multiple backends, one of which has to be loaded to display the resulting plot. There are two main options:

• GLMakie for interactive plots, and
• CairoMakie for publication quality plots.

In this tutorial we will use CairoMakie to provide output for the documentation, but whilst following along you might want to use GLMakie instead.

To install DomainColoring.jl and either of these packages, enter

]add DomainColoring GLMakie

or

]add DomainColoring CairoMakie

into the Julia REPL. (To return to the Julia REPL after this, simply press backspace.)

After installation your session should in general start with either

using GLMakie, DomainColoring

for interactive work, or for publication graphics with

using CairoMakie, DomainColoring

Plotting our first few phase plots

Julia supports the passing of functions as arguments, even better, it supports the creation of so called 'anonymous' functions. We can for instance write the function that maps an argument $z$ to $2z + 1$ as

z -> 2z + 1

Let us now see how the phase of this function behaves in the complex plane. First, if you haven't already, we need to load DomainColoring.jl and an appropriate backend (our suggestion is GLMakie if you're experimenting from the Julia REPL):

using CairoMakie, DomainColoring

Then a simple phase plot can be made using

domaincolor(z -> 2z + 1)

As expected we see a zero of multiplicity one at $-0.5$, furthermore we see that domaincolor defaults to unit axis limits in each direction.

Something useful to know about phase plots, the order of the colors tells you more about the thing you are seeing:

• red, green and blue (anticlockwise) is a zero; and
• red, blue and green is a pole.

The number of times you go through these colors gives you the multiplicity. A pole of multiplicity two gives for instance:

domaincolor(z -> 1 / z^2)

We've now looked at poles and zeroes, another interesting effect to see on a phase plot are branch cuts. Julia's implementation of the square root has a branch cut on the negative real axis, as we can see on the following figure.

domaincolor(sqrt, [-10, 2, -2, 2])

There are a couple of things of note here. First, Julia allows us to simply pass sqrt, which here is equivalent to z -> sqrt(z). Second, domaincolor accepts axis limits as an optional second argument (for those familiar with Julia: any indexable object will work). Finally, branch cuts give discontinuities in the phase plot (identifying these is greatly helped by the perceptual uniformity of the Phase Wheel used).

We conclude by mentioning that you do not always need to specify all limits explicitly. If you want to take the same limit in all four directions you can simply pass that number. When you pass a vector with only two elements, these will be taken symmetric in the real and imaginary direction respectively. This way we can zoom in on the beauty of the essential singularity of $e^\frac{1}{z}$.

domaincolor(z -> exp(1/z), 0.5)

Plotting the DomainColoring.jl logo

As a final example, let us show off a few more capabilities of the domaincolor function by plotting the DomainColoring.jl logo.

This is a plot of $f(z) = z^3i - 1$ with level curves of the logarithm of the magnitude and an integer grid. You can continue by reading the General Overview to learn more about these and other additional options, and the other provided plotting functions.

domaincolor(z -> im*z^3-1, 2.5, all=true)