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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.2012-06-03

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