# Finding a conformal map from the intersection of two disks to the unit disk.

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.

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.