Modulus of Exponential of Imaginary Number is One

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

\(\ds e^{i x}\)

\(=\)

\(\ds \cos x + i \sin x\)

Euler's Formula

\(\ds \leadsto \ \ \)

\(\ds \cmod {e^{i x} }\)

\(=\)

\(\ds \cmod {\cos x + i \sin x}\)

\(\ds \)

\(=\)

\(\ds \sqrt {\paren {\map \Re {\cos x + i \sin x} }^2 + \paren {\map \Im {\cos x + i \sin x} }^2}\)

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds \sqrt {\cos^2 x + \sin^2 x}\)

as $x$ is wholly real

\(\ds \)

\(=\)

\(\ds 1\)

Sum of Squares of Sine and Cosine

$\blacksquare$