# 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:

 $\ds \exp z$ $=$ $\ds e^x \paren {\cos y + i \sin y}$ Definition of Exponential Function $\ds \leadsto \ \$ $\ds y$ $\in$ $\ds \map \arg {\exp z}$ Definition of Polar Form of Complex Number $\ds \leadsto \ \$ $\ds \map \arg {\exp z}$ $=$ $\ds \set {y + 2 k \pi: k \in \Z}$ Definition of Argument of Complex Number $\ds$ $=$ $\ds \set {\Im z + 2 k \pi: k \in \Z}$ Definition of Imaginary Part

$\blacksquare$