# Conformal mapping into a unit disc

$T$ is the upper half of the unit disc $U$. What is the conformal mapping $f$ of $T$ onto $U$ that transforms $\{-1,0,1\}$ to $\{-1,-i,1\}$?

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First map $\{-1, 0, 1\}$ to $\{0, 1, \infty\}$ by the transform $$-\frac{z+1}{z-1}.$$ This maps $T$ to the first quadrant of $\mathbb{C}$. By squaring $$\left(-\frac{z+1}{z-1}\right)^2$$ we get the whole upper half plane. Using Cayley transform we get the unit circle $U$: $$\frac{(z+1)^2 - i(z-1)^2}{(z+1)^2 + i(z-1)^2}.$$ By direct computation we check that $\{-1,0,1\}$ is actually sent to $\{-1,-i,1\}$, so the orientation is correct. (Otherwise we could rotate the result by multiplying by $e^{i\theta}$ for some $\theta$.)