Hi everyone,

I have the following regionS = { z : 0 < Im(z) < PI } in Cand I'm looking for its image underz --> w = (1+ie^z)/(1-ie^z).

I'm aware of a technique of finding images by looking at what happens to the boundaries of the region being transformed.

In this case, it looks like I have two lines bounding S,L1: Im(z) = 0andL2: Im(z) = PI

The transformation itself doesn't look too simple. I started wondering whether I could break up the transformation into smaller, logical steps for visualization. Such as:

z --> k = e^z

k --> m = ik

m --> n = (1+m)/(1-m)

If I were take the boundaries of S through these simpler transformations in that order, would I get the correct boundaries of the image of S under z --> w?

On a side note, I can visualize the geometric effect of the first two of my transformations but not the third one. What does it do?

As always - thank you!