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

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$