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The complex text I'm reading asks the following question: find a conformal mapping of the region R between the circles,|z|=2 and |z-1|=1 onto the unit disc.

I know that the upper half plane can be mapped conformally into the unit disc by $e^{i\theta}\frac{z-z_0}{z-\bar z_0}$ where $z_0$ is in the upper half plane. So if I find a conformal mapping from R to the upper half plane, I can compose it with the previous map to get what I want. But I'm not sure how to proceed.

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Hint: $1/(z-2)$ maps $R$ to a vertical strip, which can be mapped to the upper half plane.

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