Modulus of Exponential of Imaginary Number is One

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Theorem

Let $\cmod z$ denote the modulus of a complex number $z$.

Let $e^z$ be the complex exponential of $z$.

Let $x$ be wholly real.


Then:

$\cmod {e^{i x} } = 1$


Corollary

Let $t > 0$ be wholly real.

Let $t^{i x}$ be $t$ to the power of $i x$ defined on its principal branch.


Then:

$\cmod {t^{ix} } = 1$


Proof

\(\displaystyle e^{i x}\) \(=\) \(\displaystyle \cos x + i \sin x\) Euler's Formula
\(\displaystyle \leadsto \ \ \) \(\displaystyle \cmod {e^{i x} }\) \(=\) \(\displaystyle \cmod {\cos x + i \sin x}\)
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {\paren {\map \Re {\cos x + i \sin x} }^2 + \paren {\map \Im {\cos x + i \sin x} }^2}\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {\cos^2 x + \sin^2 x}\) as $x$ is wholly real
\(\displaystyle \) \(=\) \(\displaystyle 1\) Sum of Squares of Sine and Cosine

$\blacksquare$