I have a set $S$ and a complex function $f$. How can I use mathematica to visualise $f(S)$? Using ComplexPlot doesn't do what I want, the set in the picture doesn't change, only the colouring of it, whereas I want to see what the new set actually looks like. For example, say I have a unit square about the origin and $f(z)=z+1$. Then I would want to see the square shifted one to the right in the picture.

Thanks in advance.

  • $\begingroup$ Please add the code you have tried and the problems you encountered to your question $\endgroup$
    – bbgodfrey
    Sep 1, 2021 at 14:25
  • $\begingroup$ In fact, Mathematica is still sadly deficient in providing visualization of images of sets in the complex plane under complex-valued maps. It ought to have functions that readily allow obtaining as Graphics objects the images of points, lines, circular arcs, other curves, and various other kinds of sets in the complex plane. It can be done! See, for example, some of the images produced by David J. M. Park, Jr.'s "Presentations" application that appear in mathematica-journal.com/2009/11/23/…. $\endgroup$
    – murray
    Sep 1, 2021 at 19:54

1 Answer 1


This has been covered before, but one (of many) ways to do this is:

With[{z = x + I y},
 f[z_] := z + 1;
 ParametricPlot[ReIm[#], {x, 0, 1}, {y, 0, 1},
    PlotRange -> 3,
    PlotStyle -> None
    ] & /@ {z, f[z]}

complex map


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