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:

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

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


Proof

Let $z = x + iy$.

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

We have:

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

$\blacksquare$


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