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.
Conformal map from a disk onto a disk with a slit
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complex-analysis
conformal-geometry
1 Answers
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- Map the disk onto right halfplane by $z\mapsto \frac{1-z}{1+z}$
- Then onto the slit plane by $z\mapsto z^2$.
- 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: $z\mapsto z+9$.
- Apply the inverse of 2.
- Apply the inverse of 1.
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0I would like to ask why in the first step, it is $\frac{1-z}{1+z}$ but not $\frac{1+z}{1-z}$. – 2018-09-18