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I have read some similar equations in other places, but it seems like theirs suggestions does not work in my case.

I would like to plot a complex equation in complex plane by the ContourPlot, the equation is ${\rm Re}[f(z)]=0$, where $f(z)=\sqrt{-1-z^2}-\arctan\left(\sqrt{-1-z^2}\right)$. The @Mathematica gives the following picture: It is obvious that there is a missing contour, connecting $\pm i$. Currently I have a dirty solution to this problem, i.e., replacing the original function by $i g(z)=i\sqrt{1+z^2}-i\text{ArcTanh}\left(\sqrt{1+z^2}\right)$. Instead, now I am going to plot ${\rm Im}[g(z)]=0$. It works, there is no missing any longer, see the following picture But in general it can not always modify the functions, such that ${\rm Re}[f(z)]=0$ becomes ${\rm Im}[g(z)]=0$.

My questions are:

  1. What's wrong with ContourPlot in this case?
  2. How can I solve this problem by a universal method?

Thanks in advance!

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    – bbgodfrey
    Commented Oct 30, 2020 at 13:33

1 Answer 1

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The problem appears to be that the missing line segment lies on a branch cut. This issue can be circumvented by moving the contour slightly off the branch cut and excluding none of the plot.

f[z] := Sqrt[-1 - z^2] - ArcTan[Sqrt[-1 - z^2]]
ComplexContourPlot[Re[f[z]] == -10^-4, {z, -3 - 3/2 I, 3 + 3/2 I}, Exclusions -> None]

enter image description here

Response to comment

The more complicated function given in a comment below yields the following plot.

f7[z] := Sqrt[-1 - z^7] - ArcTan[Sqrt[-1 - z^7]]
ComplexContourPlot[Re[f7[z]] == -10^-2, {z, -3 - 3/2 I, 3 + 3/2 I}, 
    Exclusions -> None, PlotPoints -> 1000]

enter image description here

That some tuning of PlotPoints and of the right side of Re[f7[z]] == -10^-2 is necessary should not be surprising. ContourPlot apparently was not designed to display contours overlapping branch cuts, which ordinarily are excluded. Locations of the branch cuts can be displayed by

ComplexContourPlot[Re[f7[z]], {z, -3 - 3/2 I, 3 + 3/2 I}, 
    Contours -> 0, PlotPoints -> 100]

enter image description here

A more colorful version is provided by

ComplexPlot3D[f7[z], {z, -3 - 3/2 I, 3 + 3/2 I}, PlotPoints -> 100, 
    ViewPoint -> Above]

enter image description here

Viewing this 3D plot from different ViewPoints shows that Re[f7[z]] is zero on the seven short branch cuts, but not on the other seven.

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    $\begingroup$ Thanks for your reply. 1, please do not replace ${\rm Im}[g(z)]$ by ${\rm Re}[g(z)]$, it is not right; 2, to check if you method is universal, you could change the power in the square root, say f[z] := Sqrt[-1 - z^7] - ArcTan[Sqrt[-1 - z^7]], I do know the correct result, but you code does not show it. $\endgroup$
    – user142288
    Commented Oct 30, 2020 at 6:49
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    $\begingroup$ @user142288 I have plotted the more complex function, as requested. I do not understand your comment on g[z], which I have not used in my answer. $\endgroup$
    – bbgodfrey
    Commented Oct 30, 2020 at 13:24

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