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enter image description here from Wegert Visual complex functions.

I am trying to verify a phase portrait by reproducing it with Mathematica. The only problem is that I am misunderstanding the ComplexPlot syntax. The complex function associated is: $f(z)=z^5-z^4-z+1$.$\text{this is shown in figure 3.5}$ The pictures on page 67, figure 3.6 says zooming out at infinity of the phase portrait of the polynomial $f$. To do this can someone confirm you do $f\left(1\over z\right) $ and the portrait of at zero? The author claims the two yellow lines emanating from infinity meet at 1. I thought it looks like -1(arrowed) in the picture someone please explain.

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  • $\begingroup$ Looks like a picture from Visual complex functions of E. Wegert. If this is really the case, you may want to add a reference to the source. $\endgroup$ – Massimo Ortolano Jul 8 at 17:55
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Evidently you're trying to create a "domain coloring" plot for your function, with some aspect of the plot representing the "phase", by which I presume you mean "argument". But there are literally infinitely many such plots, depending on what color scheme you use to represent different values of the argument — and not just the hue, but the intensity/shading.

Mathematica has no such built-in function as ComplexPhasePlot. The appropriate one to try is ComplexPlot. The documentation explains the syntax.

Note that the correct syntax for you function would be:

f[z_]:=z^5-z^4-z+1

So start with:

ComplexPlot[z^5 - z^4 - z + 1, {z, -4 - 4 I, 4 + 4 I}]

Mathematica graphics

Then use such options as you want to achieve your goal.

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