Apply two complex functions to a region

Questions
,
1

Let

4274e1.png

I want to sketch the image of the semidisk

png.png

under the following composite mapping:

png2.png

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?

2

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]
3

@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]

region.png
region.png1080×1078 8.17 KB

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.

4

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}]