Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part

Theorem

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

Let $t > 0$ be wholly real.

Let $t^z$ be $t$ to the power of $z$ defined on its principal branch.

Then:

$\left \vert {t^{z}}\right \vert = t^{\operatorname{Re} \left({z}\right) }$.

Proof

\(\displaystyle \left \vert{ t^z }\right \vert\)

\(=\)

\(\displaystyle \left \vert{ t^{\operatorname{Re}\left({z}\right)+ i \operatorname{Im}\left({z}\right)} }\right \vert\)

\(\displaystyle \)

\(=\)

\(\displaystyle \left \vert{ t^{\operatorname{Re}\left({z}\right)} t^{i \operatorname{Im}\left({z}\right)} }\right \vert\)

Sum of Complex Indices of Real Number

\(\displaystyle \)

\(=\)

\(\displaystyle \left \vert{ t^{\operatorname{Re}\left({z}\right)} }\right \vert \left \vert { t^{i \operatorname{Im}\left({z}\right)} }\right \vert\)

Modulus of Product

\(\displaystyle \)

\(=\)

\(\displaystyle \left \vert{ t^{\operatorname{Re}\left({z}\right)} }\right \vert\)

Modulus of Exponential of Imaginary Number is One:Corollary

\(\displaystyle \)

\(=\)

\(\displaystyle t^{\operatorname{Re} \left({z}\right)}\)

Power of Positive Real Number is Positive

$\blacksquare$