Proposition

Find Mobius transformations satisfying each of the following. Write your answers in standard form, as $\frac{az + b}{cz + d}$.

  1. $1 \mapsto 0, 2 \mapsto 1, 3 \mapsto \infty$.
  2. $1 \mapsto 0, 1 + i \mapsto 1, 2 \mapsto \infty$.
  3. $0 \mapsto i, 1 \mapsto 1, \infty \mapsto -i$.

Solution

\[\begin{align*} f(z) &= \frac{-z + 1}{z - 3}, \\ f(z) &= \frac{(1 - i)z - (1 - i)}{-iz + 2i}, \\ f(z) &= \frac{z + i}{iz + 1}. \end{align*}\]