Solution
= Solution
For $z=x+iy$, $e^{iz}=e^{-y}e^{ix}$, so $|e^{iz}|=e^{-y}$. Therefore
$$
\boxed{|e^{iz}|\leq1\iff\operatorname{Im}z\geq0,}
$$
the closed <upper half-plane>. Equality holds on the real axis.
= Solution
For $z=x+iy$, $e^{iz}=e^{-y}e^{ix}$, so $|e^{iz}|=e^{-y}$. Therefore
$$
\boxed{|e^{iz}|\leq1\iff\operatorname{Im}z\geq0,}
$$
the closed <upper half-plane>. Equality holds on the real axis.