Example 10. Find the fractional-linear function that maps the points $z_{1}=1, z_{2}=i, z_{3}=-1$ to the points $w_{1}=-1, w_{2}=0, w_{3}=1$.
Solution. Using formula (7), we have

$$
\frac{w+1}{w-0} \cdot \frac{1-0}{1-(-1)}=\frac{z-1}{z-i} \cdot \frac{-1-i}{-1-1}
$$

from which $w=i \frac{i-z}{i+z}$;

Remark. If one of the points $z_{k}$ or $w_{k}(k=1,2,3)$ is infinitely distant, then in formula (7) all differences containing this point should be replaced by ones.