Finding a conformal map from the intersection of two regions to a unit disk [duplicate]

I'm trying to solve a problem which asks me to find a conformal mapping from the intersection of $$\{z\in \mathbb{C}: |z-i|< \sqrt2$$ with $$|z+i|>\sqrt2\}$$ 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 disk and the other region intersect at $$±1$$, but I have no idea how to go about using this to find the result.

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.

marked as duplicate by Martin R, Did, user91500, mrtaurho, Lee David Chung LinJan 3 at 17:06

• @MartinR sorry, I realised I'd made a mistake - the second inequality should actually have been $>\sqrt2$ instead of $<$, and I assumed that was such a fundamental difference it merited a new question. Thanks so much for answering it though - it definitely did help with my understanding regardless. – oxsam Jan 2 at 22:59