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enter image description here "half sliced unit disk"

Can somebody tell me how to map this conformally to the upper half plane? I think the symmetry principle should be applied here but stuck on that for hours. Pardon my hasty sketch, this is a unit disk cut at every 45 degree.

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It is easier to map onto the unit disk, because it has the same sort of symmetry as this domain. (Then of course you can map it onto halfplane.) So, the goal is to somehow "retract" these spikes into the boundary of the disk.

Take one of these eight pieces, say the sector $0<\theta<\pi/4$. This means we make additional cuts along $re^{i\theta }$ with $0\le r<1/2$, $\theta=0,\pi/4$. Mark them with dashed lines. The sector can be mapped on upper halfdisk by $z\mapsto z^4$, and then onto lower halfplane by $z\mapsto \left(z+\frac{1}{z}\right)$ (Joukowski map). In the process, the dashed lines turned into $(-\infty, a)$ and $(a,\infty)$ where $a$ is some positive number that you can find yourself.

Apply $z\mapsto 2z/a$, so that the dashed line turn to $(-\infty,-2)$ and $(2,\infty)$. Then apply the inverse of the Joukowki map, thus mapping the lower halfplane back to upper half-disk. The dashed lines become radii $(-1,0)$ and $(0,1)$. Finally, apply $z\mapsto z^{1/4}$ to return to the sector of the opening angle $\pi/4$. The only thing we achieved so far was to map all solid lines to the circular boundary, leaving the radii dashed. Finally, use reflection across dashed lines, in effect erasing them. This means that, without any changes in the formula, you now have a map of the entire domain onto the unit disk.

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Did you look at the Cayley Transform,

$$f(z) = \frac{z- i}{z + i}$$

which maps (conformally) the upper half-plane onto the unit disk?

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