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Functions from $\mathbb{C} \rightarrow \mathbb{C}$ are hard to visualize because of their 4-dimensional nature. One nice way of looking at them is by what's called domain coloring. An example from the wiki article is shown below.

When we look at the graph of a real function ($\mathbb{R} \rightarrow \mathbb{R}$), we can get a feel for some of the function's properties: where its zeros are, if it's continuous, if it's differentiable, etc.

My question is what kinds of properties of complex functions can we "see" by looking at a domain coloring? In the example below, we can obviously see where the zeros are and where it blows up. I'm wondering, can we tell if a function is analytic? A rational function? Can we estimate an integral like we could by looking at a real graph?

$f(z) = \frac{(z^2 − 1)(z − 2 − i)^2}{(z^2 + 2 + 2i)}$

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3 Answers 3

IMHO to analyse the image after transformation one must know the image and it's coordinate before transformation. For example 2 different initial images :

cartesian grid before :

polar grid before

and after transformation ( Moebius )

cartesian after :

polar after :

One can use initial image which highlights the futures which are important and see how they change after transformation.


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See also Phase Plots of a Complex Function: A Journey in Illustration and the book "Visualizing Complex Functions" which is an expansion of that paper.

The phase of an analytic function uniquely determines the function, up to a positive constant scaling factor. So mathematically, you can determine the function modulo a constant. Some insights might not be psychologically realistic that are theoretically possible.

You can tell the difference between a zero and a pole by the direction in which the colors swirl around the point. And you can tell the order of the zero or pole by how many times the colors complete a cycle around the point.

If a function is not analytic, you may be able to tell that from the phase plot.

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+1 Any hints on how to tell if a function's analytic? I suspect you'd need to train your eye and brain to recognize colors that are "out of order". –  I. J. Kennedy Mar 8 '13 at 0:13
Can you tell whether a function $\mathbb{R} \to \mathbb{R}$ is real-analytic (or even smooth) by looking at its graph in $\mathbb{R}^2$? Except where there are obvious singularities, I don't think I could... –  Jesse Madnick Mar 8 '13 at 0:33
@JesseMadnick I think I could—try me! –  I. J. Kennedy Nov 22 '13 at 1:22

Hans Lundmark's pages on domain coloring, and his page on Elliptic functions is good places to start. (Picard's great theorems, Lucas's theorem, Weierstrass $\wp$ function)

I don't think we can see that a function is rational -- just moving the roots in the function that you have displayed above by $\pi/100$ in some random direction would be sufficient to make it not a rational function, but the picture would more or less look the same.

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I suspect that there is no assumed restriction to rational functions with algebraic coefficients, in which case moving the zeros by $\pi/100$ wouldn't necessarily make it not a rational function. –  Jonas Meyer Nov 16 '10 at 23:33

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