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f[z_] = -i/2 z^(1/2) (z - 3);
Needs["Graphics`Colors`"];
Needs["Graphics`ComplexMap`"];
g = CartesiaMap[f, {-3, 4, 0.5}, {0.00001, 2.00001, 0.2},
   PlotStyle -> {Blue, Red},
   Ticks -> {Range[-4, 2, 2], Range[0, 3, 3]}];
Graphics[PlotRange -> {{-2.5, 2.5}, {0, 1.5}},
AxesLabel -> {"u", "v"}, Ticks -> {Range[-2, 2, 1], Range[0, 2, 1]},
DisplayFunction -> $DisplayFunction]
Print["The image of the upper half plane Im[z]>0 under thte mapping"];
Print["w=f[z]=", f[z]];
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CODE:
f[z_] = -I/2 z^(1/2) (z - 3);
ParametricPlot[Through[{Re, Im}[f[x + I*y]]], {x, -3, 4}, {y, 0, 2},
PlotStyle -> None]

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