Complex Function Range Computation 

Use FunctionRange to compute the range of complex functions.

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X FunctionRange[Sqrt[z], z, w, Complexes]

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Find the image of the unit square through the complex mapping .

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X FunctionRange[{z^3 + z^2 + z, 0 < Re[z] < 1 && 0 < Im[z] < 1}, z, w, Complexes]

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X FunctionRange[{z^3 + z^2 + z, 0 < Re[z] < 1 && 0 < Im[z] < 1}, z, w, Complexes]; RegionPlot[ Evaluate[% /. {Re[w] -> a, Im[w] -> b}], {a, -2.5, 3.5}, {b, -0.5, 5.5}, PlotPoints -> 100]

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