Argument of Exponential is Imaginary Part plus Multiple of 2 Pi

Theorem
Let $z \in \C$ be a complex number.

Let $\exp z$ denote the complex exponential of $z$.

Let $\arg z$ denote the argument of $z$.

Then:


 * $\map \arg {\exp z} = \set {\Im z + 2 k \pi: k \in \Z}$

where $\Im z$ denotes the imaginary part of $z$.

Proof
Let $z = x + i y$.

Let $\theta \in \map \arg {\exp z}$.

We have: