# 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. Commented Jul 8, 2019 at 17:55

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


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

• Note that it may be useful to create a 3D version of such "domain coloring," which perhaps should be called "range coloring." Namely, ComplexPlot3D[z^5 - z^4 - z + 1, {z, -4 - 4 I, 4 + 4 I}]. That function was new in Mathematica 12 and was updated in 12.1 and 13.0. Commented Apr 28, 2022 at 19:40
• In case of an older version of Mathematica, one can use ComplexFnPlot, essentially the same 3D function, which is defined in terms of Plot3D in the new book "The Calculus of Complex Functions," by William Johnston. Commented Apr 28, 2022 at 19:43