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I would be able to plot the image of a curve (say the circle with radius $1$ centered at $1+i$) under a complex polynomial (say $p(z)=z^2$) using Mathematica. I know how to find the level curves of $p$, but going the other direction I cannot figure out.

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p[z_] = z^2 + z;
c[t_] = Cos[t] + Sin[t] I;
ParametricPlot[{Re[p[c[t]]], Im[p[c[t]]]}, {t, 0, 2 Pi}]

enter image description here

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  • $\begingroup$ This returned an empty graph for me. $\endgroup$ – Trevor J Richards May 8 '15 at 17:31
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    $\begingroup$ Did you get any message? What version are you running? $\endgroup$ – Julián Aguirre May 8 '15 at 20:20
  • $\begingroup$ For OP's particular example, ParametricPlot[Through[{Re, Im}[(1 + I + Exp[I t])^2], {t, 0, 2 π}]. For newer versions, apparently the function ReIm[] can be used instead. $\endgroup$ – J. M.'s technical difficulties♦ May 17 '15 at 4:15
  • $\begingroup$ Sorry for taking so long to accept the answer! $\endgroup$ – Trevor J Richards Mar 18 '18 at 2:39

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