How to plot this set in the complex plane?

Question

I'd like to plot the set

s=(z - I Im[z] < 0 && -I z + I Re[z] == 
0) || (-1 + Cos[(9 Arg[z])/2] Im[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) + 
 2 Cos[(9 Arg[z])/2] Im[z]^2 Re[z]^2 (Im[z]^2 + Re[z]^2)^(1/4) + 
 Cos[(9 Arg[z])/2] Re[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) < 0 && 
   Im[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] + 
 2 Im[z]^2 Re[z]^2 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] + 
 Re[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] == 0)

Both

ComplexContourPlot[s, {z, -10 - 10*I, 10 + 10*I}]

and

ComplexRegionPlot[(s,{z,-10-10*I,10+10*I}]

produce empty plots. I know s is not empty as

N[s /. z -> (-1)^(4/9)]

True

shows. I think its dimension equals one. The question is inspired by that question.

Addition. Making use of the Domen's comment, I replace inequalities by Booles,

ComplexContourPlot[Boole[z - I Im[z] < 0]*(-I z + I Re[z] == 0) || 
  Boole[-1 + Cos[(9 Arg[z])/2] Im[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) + 
  2 Cos[(9 Arg[z])/2] Im[z]^2 Re[z]^2 (Im[z]^2 + Re[z]^2)^(1/4) + 
  Cos[(9 Arg[z])/2] Re[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) < 
 0]*(Im[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] + 
  2 Im[z]^2 Re[z]^2 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] + 
  Re[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] == 0), {z,  10}]

, but the result is doubtful to me.

Addition 2. The improved approach

ComplexContourPlot[Boole[z - I Im[z] < 0] == 1 && (-I z + I Re[z] == 0) || 
  Boole[-1 + Cos[(9 Arg[z])/2] Im[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) + 
   2 Cos[(9 Arg[z])/2] Im[z]^2 Re[z]^2 (Im[z]^2 + Re[z]^2)^(1/4) +
    Cos[(9 Arg[z])/2] Re[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) < 0] == 
1 && (Im[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] + 
  2 Im[z]^2 Re[z]^2 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] + 
  Re[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] == 0), {z,  10}]

results in empty plot. PlotPoints -> 300 does not help.

Answer 1

• ComplexRegionPlot in Mathematica seems can't handle 2D region and 1D lines simultaneously.

• Here we try to use ComplexContourPlot to plot the 1D lines and use RegionFunction to limit the 2D region.

Clear[plot1, plot2];
plot1 = ComplexContourPlot[-I z + I Re[z] == 0, {z, 10}, 
  RegionFunction -> Function[{z, f}, z - I Im[z] < 0], 
  ContourStyle -> Green]; plot2 = 
 ComplexContourPlot[
  Im[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] + 
    2 Im[z]^2 Re[z]^2 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] + 
    Re[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] == 0, {z, 10},
   RegionFunction -> 
   Function[{z, 
     f}, -1 + Cos[(9 Arg[z])/2] Im[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) + 
      2 Cos[(9 Arg[z])/2] Im[z]^2 Re[z]^2 (Im[z]^2 + Re[z]^2)^(1/4) + 
      Cos[(9 Arg[z])/2] Re[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) < 0], 
  ContourStyle -> Blue];
Show[plot1, plot2]

• The above method can also illustrate by using MeshFunctions in ComplexRegionPlot. Here we only illustrate the second part.

ComplexRegionPlot[(-1 + 
    Cos[(9 Arg[z])/2] Im[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) + 
    2 Cos[(9 Arg[z])/2] Im[z]^2 Re[z]^2 (Im[z]^2 + Re[z]^2)^(1/4) + 
    Cos[(9 Arg[z])/2] Re[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) < 0), {z, 10},
  PlotPoints -> 200, MaxRecursion -> 4, 
 MeshFunctions -> 
  Function[{z}, 
   Im[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] + 
    2 Im[z]^2 Re[z]^2 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2] + 
    Re[z]^4 (Im[z]^2 + Re[z]^2)^(1/4) Sin[(9 Arg[z])/2]], 
 Mesh -> {{0}}, MeshStyle -> Blue, PlotStyle -> Opacity[.1], 
 BoundaryStyle -> Opacity[.1]]