Conformal map of this region

Hi everyone,

I have the following region S = { z : 0 < Im(z) < PI } in C and I'm looking for its image under z --> 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) = 0 and L2: 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!

let f(z)=(1+iz)/(1-iz)=-(z-i)/(z+i), then w = f o exp(z).
exp maps the strip S to the upper half plane, excluding {0}.
f maps the upper half plane to the unit disk, excluding {1, -1}

To visualize f, try to transform it on to the Riemann sphere using Stereographic projection

I figured out how you got that answer with a little help from the textbook. Thanks a lot!

Instead of starting a new thread, I'll ask another conformal map question here.

I want to find a map from S1 = { Im(z) < a } to S2 = D(b;1)

Could I map S1 onto the lower open half-plane via translation by 'a', then map the lower open half-plane onto the unit disk and then finally map the unit disk onto D(b;1) by translating it by 'b' units? Is my strategy valid? (it doesn't matter if it's efficient)

Thanks again.

that should work