I want to map $Im(z)>0$ to $|z-i|<1$ using a Möbius transformation. I have a simple question regarding that. I know I can choose 3 points on the domain boundary and map them to the circle, but it is quite tedious. Can I know which point will be mapped to the center of the circle $|z-i|=1$? I can then use the preservation of symmetry to find the transformation. I saw somewhere that I can say $f(i)=i$ in this case, but I have no idea why it is ensured $i$ will go to the center of the circle and not any other points inside the $Im(z)>0$ domain.
2026-03-29 01:35:37.1774748137
Mapping Im(z)>0 to a shifted circle - Möbius transformation
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