Modulus and Argument of Complex Exponential

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Theorem

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

Let $\left[{a \,.\,.\, a + 2 \pi}\right)$ be a half open interval of length $2 \pi$.

Let $r \in \left[{0 \,.\,.\, +\infty}\right)$ and $\theta \in \left[{a \,.\,.\, a + 2 \pi}\right)$.


Then $r = \left\vert{z}\right\vert$ and $\theta = \arg \left({z}\right)$ if and only if $z = re^{i \theta}$.


Here, $\left\vert{z}\right\vert$ denotes the modulus of $z$, $\arg \left({z}\right)$ denotes the argument of $z$, and $x \mapsto e^x$ is the complex exponential function.

If $z = 0$ or $r = 0$, then $\theta$ may be any number in $\left[{a \,.\,.\, a + 2 \pi}\right)$.


Proof

Necessary condition

Suppose that $r = \left\vert{z}\right\vert$.

If $z = 0$, we have $z = 0e^{i \theta} = re^{i \theta}$.

Suppose $z \ne 0$ and $\theta = \arg \left({z}\right)$.

By definition of argument, the following two equations hold:

$(1): \quad \dfrac{\operatorname{Re} \left({z}\right) }{r} = \cos \theta$
$(2): \quad \dfrac{\operatorname{Im} \left({z}\right) }{r} = \sin \theta$

where $\operatorname{Re} \left({z}\right)$ denotes the real part of $z$, and $\operatorname{Im} \left({z}\right)$ denotes the imaginary part of $z$.

Then:

\(\displaystyle z\) \(=\) \(\displaystyle \operatorname{Re} \left({z}\right) + i \operatorname{Im} \left({z}\right)\) Definition of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle r \cos \theta + i r \sin \theta\) from $(1)$ and $(2)$
\(\displaystyle \) \(=\) \(\displaystyle r \left({\cos \theta + i \sin \theta}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle re^{i \theta}\) Euler's Formula

$\Box$


Sufficient condition

Let $z = re^{i \theta}$.

From the equations above, we find:

$\operatorname{Re} \left({re^{i \theta} }\right) = r \cos \theta$
$\operatorname{Im} \left({re^{i \theta} }\right) = r \sin \theta$

Then:

\(\displaystyle \left\vert{re^{i \theta} }\right\vert\) \(=\) \(\displaystyle \sqrt{\left({r \cos \theta}\right)^2 + \left({r \sin \theta}\right)^2}\) Definition of Modulus of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle r \sqrt{\cos^2 \theta + \sin^2 \theta}\) as $r \ge 0$
\(\displaystyle \) \(=\) \(\displaystyle r\) Sum of Squares of Sine and Cosine

If $r \ne 0$, we find $\arg \left({re^{i \theta} }\right)$ by solving the two equations by definition of argument:

$(1): \quad \dfrac{r \cos \theta}{r} = \cos \left({\arg \left({re^{i \theta} }\right) }\right)$
$(2): \quad \dfrac{r \sin \theta}{r} = \sin \left({\arg \left({re^{i \theta} }\right) }\right)$

We find:

$\arg \left({re^{i \theta} }\right) = \theta$

$\blacksquare$


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