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I am trying to plot the unit circle $\vert z\vert=1$ under the transformation $z\to z+\frac1z$ using mathematica. In general I want to plot a complex function $f(z)$ under the transformation $\phi(z)$.

I tried

With[{z=x+I y}, ContourPlot[Norm[z+1/z]==1, {x,-0.75,0.75},{y,-3,3}]]

Which does not seem to produce the correct output

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