Example 11. Find a fractional-linear function that maps the point $z_{1}$ to the point $w_{1}=0$, and the point $z_{2}$ to the point $w_{2}=\infty$.
Solution. Let us take an arbitrary point $z_{3}$, different from points $z_{1}$ and $z_{2}$, and assume that it maps to a point $w_{3}$, different from points $w_{1}$ and $w_{2}$. Then, by formula (7) and with the remark in mind, we have

$$
\frac{w-0}{1} \cdot \frac{1}{w_{3}-0}=\frac{z-z_{1}}{z-z_{2}} \cdot \frac{z_{3}-z_{2}}{z_{3}-z_{1}}
$$

from which

$$
w=K \frac{z-z_{1}}{z-z_{2}}
$$

where

$$
K=\frac{z_{3}-z_{2}}{z_{3}-z_{1}} w_{3}
$$

i.e., $K$ is an arbitrary complex number, $K \neq 0$.