Modulus of Exponential of Imaginary Number is One/Corollary

Corollary to Modulus of Exponential of Imaginary Number is One

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

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

\(=\)

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

Definition of $t$ to the Power of $ix$

\(\ds \)

\(=\)

\(\ds 1\)

Modulus of Exponential of Imaginary Number is One, as $x \ln t$ is wholly real for $t > 0$

$\blacksquare$