So far I have thought about first having my transformation, $T$, map $i$ to $\infty$ so that I get two parallel lines. But then I am not sure where to proceed from there.
2026-03-27 23:35:54.1774654554
Mobius transforms which map the region ¨ (C \ D(1, 1)) ∩ D(0, 2) into the strip {|Im z| < 1}.
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You should map $2$ to $\infty$: $z\to\dfrac{1}{z-2}$. This will transform the region into a strip with sides perpendicular to the real axis. Next translate so that $0$ is in the middle of the strip, dilate so that the width of the strip is $2$ and rotate $\pi/2$ radians.