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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

DMH16
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  • This was a popular question https://mathematica.stackexchange.com/questions/156339/visualizing-the-complex-map-fz-z2 – yarchik Feb 14 '20 at 19:43
  • Your code produces points z on the complex plane such that the absolute value of z+1/z is equal to 1 and so it is a different than the transformation of the uinit circle under mapping z+1/z. This answer provides almost the same what you want Plotting complex numbers as an Argand Diagram. First read and then it becomes clear. – Artes Feb 14 '20 at 19:55

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