I'm trying to solve a problem which asks me to find a conformal mapping from $\{z\in \mathbb{C}: |z-i|< \sqrt2$ and $|z+i|<\sqrt2\}$ onto the open unit disk.

I'm really new to these and I'm a bit lost because this doesn't resemble the examples I've seen so far.

Obviously the two disks intersect at $±1$, but I don't know where I should be looking to map them to, and while I know usually you'd try to map the region to something familiar and then compose this with another map to the unit disk from there, but I can't see how to map this to something simpler.

As I said, I haven't really done many examples of conformal maps so I'd really appreciate if you could help walk me through this example.

Thanks in advance!


Start with the Möbius transformation $T(z) = \frac{z-1}{z+1}$. $T(1) = 0$ and $T(-1) = \infty$, therefore the two circles $|z \pm i| = \sqrt 2$ are mapped to lines through the origin. Use the preservation of angles to compute the directions of those lines. Conclude that $T$ maps the intersection of the two disks to a certain sector with opening angle $\frac \pi 2$. Then map this sector to a half-plane, and finally to the unit disk.


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