Example 2. Find the zeros of the function $f(z)=1-e^{z}$ and determine their orders.
Solution. By setting $f(z)$ to zero, we find the zeros $z_{n}=2 n \pi i (n=0, \pm 1, \ldots)$ of the function $f(z)$. Next,

$$
f^{\prime}(2 n \pi i)=-e^{2 n \pi i}=-1 \neq 0 .
$$

Thus, $f(2 n \pi i)=0, f^{\prime}(2 n \pi i) \neq 0$, therefore, the points $z_{n}=2 n \pi i (n=0, \pm 1, \pm 2, \ldots)$ are simple zeros of the function $f(z)=1-e^{z}$.