Argument of Exponential is Imaginary Part plus Multiple of 2 Pi
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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 $e^z$, $\cos z$, $\sin z$