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I want to prove that the region $\mathbb{C}\setminus\{x+i0:-1\le x\le1\}$ is conformally equivalent to an annulus. I am trying to use a linear fractional transformation to map this region onto a disc or something with a hole, but am having trouble finding the appropriate map. Any suggestions?

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That region is conformally equivalent to the punctured unit disc. So one of your radii must be either $0$ or $\infty$ to be equivalent to that. – WimC Jun 3 '12 at 7:22

The function $z\mapsto z+i\sqrt{1-z^2}$ maps your slit to a circle, with branch points at $-1$ and $1$. By choosing appropriate branches you can get it to map your region to all of $\mathbb C$ with a circular hole. Then you're half done.

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