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Find a conformal mapping from D to $D\backslash [-1/2,1)$, where D is the unit disk. I think that maybe we can map $D\backslash [-1/2,1)$ to upper half plane first and the map upper half plane to $D$ and compute the inverses of the maps. Can we map $D\backslash [-1/2,1)$ to upper half plane? Thank you very much.

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up vote 2 down vote accepted
  1. Map the disk onto right halfplane by $z\mapsto \frac{1-z}{1+z}$
  2. Then onto the slit plane by $z\mapsto z^2$.
  3. Note that the former segment $[-1/2,1]$ is now $[0, 9]$. We should make a cut from $0$ to $9$ in the domain. This means extending the slit to $9$. No problem: $z\mapsto z+9$.
  4. Apply the inverse of 2.
  5. Apply the inverse of 1.
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