Example 10. What set of points in the complex plane $z$ is defined by the condition

$$
\operatorname{Im} z^{2}>2 ?
$$
Solution: Let $z=x+i y$. Then

$$
z^{2}=(x+i y)^{2}=\left(x^{2}-y^{2}\right)+i 2 x y
$$

Therefore, $\operatorname{Im} z^{2}=2 x y$.

By the condition $2 x y>2$, or $x y>1$. This inequality defines the set of points in the first and third quadrants, respectively above and below the hyperbola $x y=1$. in the first and third quadrants, respectively above and below the hyperbola $x y=1$.