Periodicity of the Complex Exponential and Its Kernel

In the setting of Complex Analysis: Standing Notation, let $\exp:\mathbb{C}\to\mathbb{C}$ be the complex exponential function, and let $\pi$ be the real number of The Number Pi (i).

(i) Special values.

$$\exp\bigl(\tfrac{\pi}{2}i\bigr)=i,\qquad \exp(\pi i)=-1,\qquad \exp(2\pi i)=1 .$$

(ii) The period. For every $z\in\mathbb{C}$,

$$\exp(z+2\pi i)=\exp z .$$

(iii) The kernel. For $z\in\mathbb{C}$, $\exp z=1$ if and only if $z=2\pi k i$ for some $k\in\mathbb{Z}$.