# Verification of ComplexPlot 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.

• 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. – Massimo Ortolano Jul 8 '19 at 17:55

## 1 Answer

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