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Let $z \in \mathbb C$ be an arbitrary complex number. Call it

  • Red, if $\Im(z^7) > 0$
  • Blue, if $\Im(z^7) < 0$
  • Gray, if $\Im(z^7) = 0$.

I want to make a plot in which the red, blue and gray regions of the complex plane are painted in the appropriate color. I have the following helper function:

proc[f_, r_] := ComplexContourPlot[f, {z, 2},
  Contours -> r, ContourShading -> {LightBlue, LightPink}];

The naïve way to use this function is

proc[Im[z^7], {0}]

This does not work as nicely as one would hope. The plot is distorted near the origin due to numerical stability issues in the calculation of $\Im(z^7)$ for very small $z$.

Another attempt would be

r = Range[-Pi, Pi, Pi/7];
proc[Arg[z], r]

This almost works, but not quite. The negative real axis is not painted correctly. This behavior is not unexpected, because Arg[z] is discontinuous at the negative real axis. But it is still annoying.

One rather ugly way to make this work is

p1 = proc[Arg[z], r];
p2 = proc[Arg[-z], {0}];
Show[p2, p1]

Is there anything less egregious than this that will still work?

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    $\begingroup$ Is cranking up PlotPoints not an option? I don't have MM 12, so I can't use ComplexContourPlot, but using ContourPlot[Im[(x + I y)^7], {x, -2, 2}, {y, -2, 2}, PlotPoints -> 100] in MM 11 seems to give a reasonable result. $\endgroup$
    – Michael Seifert
    May 14, 2020 at 18:02
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    $\begingroup$ Building off @Michael's proposal, here's one way to color it: ContourPlot[Im[(x + I y)^7], {x, -2, 2}, {y, -2, 2}, ColorFunction -> (Blend[{Blue, Gray, Red}, LogisticSigmoid[10 #]] &), ColorFunctionScaling -> False, PlotPoints -> 100, PlotRange -> All] $\endgroup$
    – J. M.'s slightly less busy ♦
    May 14, 2020 at 18:11
  • $\begingroup$ @MichaelSeifert That did the trick, thanks! $\endgroup$
    – pyon
    May 14, 2020 at 18:19

1 Answer 1

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If processing time is not a concern, the option PlotPoints can be used to force Mathematica to perform more sampling of the function being plotted.

ComplexContourPlot[f, {z, 2},  Contours -> r, 
 ContourShading -> {LightBlue, LightPink}, PlotPoints->100]

(Please note: I have not tried this code, as I do not have MM 12. If it does not work, please let me know.)

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  • $\begingroup$ It does work, thanks! $\endgroup$
    – pyon
    May 14, 2020 at 21:12

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