Example 2. Into what curve is the unit circle $|z|=1$ mapped by the function $w=z^{2} ?$
Solution. Since $|z|=1$ by condition, then

$$
|w|=|z|^{2}=1 \text {. }
$$

Thus, the image of the circle $|z|=1$ in the $z$-plane is the circle $|w|=1$ in the $w$-plane, traversed twice. This follows from the fact that since $\boldsymbol{w}=z^{2}$, then $\operatorname{Arg} w=2 \operatorname{Arg} z+2 k \pi$, so when the point $z$ traces out the full circle $|z|=1$, its image traces out the circle $|w|=1$ twice. $\quad \square$