I am very new to complex analysis and am having some trouble with finding a Möbius map that will take two unit discs to half-planes.
I don't have enough reputation to post images, but here is a link showing the initial setup:
http://s22.postimg.org/8sr5z3jlt/Skjermbilde_2013_07_24_kl_10_34_21_AM.png
I would like to map the red and green circles to half-planes such that the space between them becomes an infinite strip. Here is the what I would ultimately like to happen:
s18.postimg.org/kr82wowe1/Skjermbilde_2013_07_24_kl_10_41_24_AM.png
Following the instructions that I found here:
math.fullerton.edu/mathews/c2003/mobiustranformationmod.html
I learned that it is possible to construct a mapping by specifying the images of three points in the $z$-space under a Möbius mapping $S(z)$. I naively hoped that by specifying,
$S(i) = i, S(1 + 2i) = 1 + i, S(3i) = \infty$
I could force the boundary of the red circle to the upper half-plane. Using equation 10-21 on the website linked above, I derived that $S(z) = 2 / (z - 3i)$ This is indeed what happens to the red circle, as intended, but the green circle didn't go where I wanted it to (as expected). Here is an image showing this problem:
http://s15.postimg.org/4scrhqy7f/Skjermbilde_2013_07_24_kl_10_46_29_AM.png
So I'm hoping that someone here can help guide me to a better mapping that will take both circles to half-planes, which is what I really need.
Thank you.
This will not work. Since you want two disjoint circles to turn into lines, two points (one from each circle) must map to infinity.