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The question is to

Find a conformal map from $D$ onto $D\setminus \left[\frac{-1}{2},1\right)$.

I don't know how to start with. The line $\left[\frac{-1}{2},1\right)$ makes it crazy. Anyone can give me some hints?

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    D is the unit disk.2012-11-26

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First map $D$ onto itself with a Möbius. $ L(z)=\frac{ z-\alpha}{ 1 -\bar{\alpha} z}$ where $\alpha=-\frac{1}{2}$. Then follow with $z \to -z$ and finally with $\sqrt{z}$. That will give you the intersection of $D$ with the right half plane. From there it is easy.

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    @ThomasAndrews You are right. I went the wrong direction. One has to invert all the maps. But I think this is probably the best way to proceed.2012-11-26