Question

Find zeros of f(z)=1-exp(z) and define their order.

Answer 1

$$f(z) = 1 - e^{z}$$ $$f(z) = 0$$ $$e^{z} = 1$$ $$z_{n} = 2n\pi i, \, n \in \mathbb{Z}$$ $$f'(2n\pi i) = -e^{2n\pi i} = -1 \neq 0$$ $$\text{So:}$$ $$f(2n\pi i) = 0, \, f'(z) < 0$$

So all $z_n$ are simple zeros of $f(z)$.