# Apply two complex functions to a region

Posted 2 months ago
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 Let I want to sketch the image of the semidisk under the following composite mapping:That is, translate by i, rotate by pi/4.It is explained in the documentation (Applications -> Mapping Complex Regions) how to apply such mapping, for only one function:Definitions: semidisk[z_] := Abs[z] <= 2 && Im[z] >= 0 f[z_] := z + I; Apply the transformation f[z]: ComplexRegionPlot[ semidisk[ InverseFunction[f][z] ] , {z, 3}, Axes -> True, GridLines -> Automatic] My question is: What is the command to apply the transformation g[f[z]]? I tried this: g[z_] := E^(I \[Pi]/4) z; ComplexRegionPlot[ semidisk[ InverseFunction[f][ InverseFunction[g][z] ] ] , {z, 3}, Axes -> True, GridLines -> Automatic] but it doesn't give the correct result.Any help?
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Posted 2 months ago
 Like this perhaps? f[z_] := z + I g[z_] := z Exp[I Pi/4] c[x_] := g[f[x]] // FullSimplify and p1 = ParametricPlot3D[{x, y, Re[c[x + I y]]}, {x, -3, 3}, {y, -3, 3}, PlotStyle -> {Opacity[.5], Red}] p2 = ParametricPlot3D[{x, y, Im[c[x + I y]]}, {x, -3, 3}, {y, -3, 3}, PlotStyle -> {Opacity[.5], Blue}] Show[p1, p2] 
 @Hans Dolhaine thanks a lot for your reply!I am looking for an argument to be inserted into ComplexRegionPlot, and not to use ParametricPlot3D, but your reply did give me an idea to accomplish my task:I defined first a function called translate: translate[z_] := semidisk[InverseFunction[f][z]] and then defined another one called rotation: rotate[z_] := translate[InverseFunction[g][z]] then plotting the region: ComplexRegionPlot[ rotate[z] , {z, 4}, Axes -> True, GridLines -> Automatic] This way, it is easy to see that one can accomplish the task of plotting a complex region applied by any finite number of functions in this way: semidisk[ InverseFunction[f][ InverseFunction[g][z] ] ] Without defining two additional functions, here translate and rotate, only the functions f and g that were defined in the first instance.
 Hello Ehund,it seems you totally solved your problem. That is good.But another remark. I think it is a very interesting idea to do geometry in R2 with complex numbers. As my Mathematica (Version 7) does not know ComplexRegionPlot I strived to find a workaround and did it like this toZ[{x_, y_}] := x + I y (* point to complex number*) frZ[z_] := {Re[z], Im[z]} (* complexnumber to point *) trZ[z_, a_] := z + a (* translation *) roZ[z_, a_] := z Exp[I a] (* rotation *) Here we have a semicircle RegionPlot[ And[Norm[{x, y} - {0, 0}] < 1, {x, y}[[2]] > 0], {x, -1.5, 1.5}, {y, -1.5, 1.5}] (*seimicircle *) And here is the rotated and translated semicircle tt = -.6 + .5 I; ww = -1.2; RegionPlot[ And[ Norm[frZ[roZ[trZ[toZ[{x, y}], tt], ww]]] < 1, frZ[roZ[trZ[toZ[{x, y}], tt], ww]][[2]] > 0], {x, -2, 2}, {y, -2, 2}]