Example 1. Find the zeros of the function $f(z)=e^{z}-1-z$ and determine their order.
Solution.

1. Find the zeros of the function $f(z)$ by solving the equation $e^{z}-1-z=0$. We get $z=0$.
2. Determine the order of the obtained zero $z=0$. For this, we use the Taylor series expansion of the function $f(z)$ in powers of $z$:

$$
e^{z}-1-z=\left(1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\ldots\right)-1-z=\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\ldots
$$

Since in the obtained expansion the coefficients $c_{0}=c_{1}=0$, and $c_{2}=1 / 2 \neq 0$, the point $z=0$ is a zero of the 2nd order of the function $f(z)$.

Answer. $z=0$ is a zero of the 2nd order of the function $f(z)$.