Plotting region $f(S)$ for given complex function $f$ and $S \subseteq \mathbb{C}$

Question

I have a function $f: \mathbb{C} \rightarrow \mathbb{C}$ and I want to be able to see the effect of $f$ on any particular region of $\mathbb{C}$ e.g. what happens to the unit disk under this transformation.

I know how to plot particular regions in the complex plane using RegionPlot

e.g.

 RegionPlot[
 Evaluate@ComplexExpand[
   Abs[z]<1 /. z -> x + I y], {x, -2, 2}, {y, -2, 
  2}, AxesLabel -> Automatic]; 

shades the interior of the unit disk in light purple color.

Is it possible to modify this code snippet to see the effect of $f(z)$ on the above unit disk for any given $f$?

Answer 1

To visualize try this!

Manipulate[With[{alpha = val},Show[RegionPlot[
Evaluate@ComplexExpand[Abs[z] < 1 /. z -> x^alpha + I (2 - alpha) y ],
{x, -12,2}, {y,12, 15},AxesLabel -> Automatic, Frame -> False,PlotRange -> All,
PerformanceGoal -> "Quality",
PlotStyle ->Directive[Opacity[.6], Blend[{Red, Blue}, alpha - 1.]], 
PlotLabel ->TraditionalForm[
  Style[x + I y, Red] -> Style[ x^1.9 + I 0.1 y, Blue]]],
Graphics[{Red, Thick, Dashed, Circle[]}]]
], {val, 1, 1.9, .05}]