Can you construct a conformal mapping from exterior of upper semi-disk onto exterior of unit disk and fixes infinity?
2026-03-27 10:09:51.1774606191
Conformal mapping from exterior of semi disk onto exterior of unit disk
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Recall that on the Riemann sphere, lines and circles are the same thing.
The exterior of semi-disk is bounded by two "circular" arcs meeting at $-1,1$. Use a fractional linear transformation to send one to $\infty$, the other to $0$. The image of the complement will be bounded by two lines from $0$ to $\infty$; let one of them be the positive half of real axis. A power map will transform this image to upper halfplane. Let's say $a$ is the image of $\infty$ in this halfplane. Then $$z\mapsto \frac{z-\bar a}{z-a}$$ transforms the upper half-plane to the exterior of unit disk, returning $a$ to infinity from whence it came.