plotting the Cayley Transformation

Question

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.

Answer 1

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]

Answer 2

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.