Example 1. Find the zeros of the function $f(z)=1+\cos z$ and determine their orders.
Solution. Setting $f(z)$ to zero, we get $\cos z=-1$, from which $z_{n}=$ $(2 n+1) \pi(n=0, \pm 1, \pm 2, \ldots)$ - these are the zeros of the given function. Next,

$$
\begin{aligned}
& f^{\prime}[(2 n+1) \pi]=-\sin (2 n+1) \pi=0 \\
& f^{\prime \prime}[(2 n+1) \pi]=-\cos (2 n+1) \pi=1 \neq 0
\end{aligned}
$$

Therefore, the points $z_{n}=(2 n+1) \pi(n=0, \pm 1, \pm 2, \ldots)$ are zeros of the second order of the given function.