# plotting the Cayley Transformation

I'm new to mathematica and I have to plot the Cayley Transformation, which maps the upper half plane to the unit disk. I want to plot the z-plane and the w-plane. I wrote this:

f[z] := (z - I)/(z + I)
ComplexContourPlot[{Re[f[z]], Im[f[z]]}, {z, -1 - I, 1 + I}]
ComplexContourPlot[{Re[z], Im[z]}, {z, -1 - I, 1 + I}

I still get some parts of the bottom half plane mapped in my w-plane and it plotted the lines of the z-plane that it seems my function maps vertical lines on the blue circles, but it should map the horizontal lines on the blue circles.

Any ideas how I can fix this? Please I need help.

• Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! Commented Mar 9, 2021 at 3:20

Set the MeshFunctions to #3 and #4.

f[z_] := (z - I)/(z + I);
ParametricPlot[ReIm[f[x + I*y]], {x, -4, 4}, {y, -4, 4}, Mesh -> 20,
MeshStyle -> {{Thick, Red}, {Thick, Green}},
MeshFunctions -> {#3 &, #4 &}, PlotRange -> 4, PlotPoints -> 80,
PlotStyle -> None]

• ah! thanks! that was what I was looking for! Is there any chance that I can let a variable go to infinity? like {x, -infinity, infinity}? So that I get closed circles? Commented Mar 9, 2021 at 8:44

You are not plotting the image of the lines Im = =constant and Re == constant, but the contour lines where Re[z]==const and Im[z]==const.

If you want to plot the image of the lines with real or imaginary part constant you may proceed as followes:

First, we define a couple of lines with constant real or imaginary parts. Note, these lines depend on only one single argument.

reConst[im_] = Table[ReIm@f[re + I im], {re, -1, 1, 0.5}];
imConst[re_] = Table[ReIm@f[re + I im], {im, -1, 1, 0.5}];

With these definitions we may now plot the image under the mapping by the function f:

f[z_] := (z - I)/(z + I)
ParametricPlot[{reConst[w], imConst[w]}, {w, -10, 10},
PlotRange -> 4 {{-1, 1}, {-1, 1}}, Frame -> True]

The orange circles are the images for the lines with imaginary part constant (e.g. the real number line corresponds to the unit circle). The blue circles are the images for the lines with real part constant. The circles are not closed because we do not draw from -Infinity to Infinity, but only from -10 to 10.