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

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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:

$\cmod {t^z} = t^{\map \Re z}$


Proof

\(\displaystyle \cmod {t^z}\) \(=\) \(\displaystyle \cmod {t^{\map \Re z + i \map \Im z} }\)
\(\displaystyle \) \(=\) \(\displaystyle \cmod {t^{\map \Re z} t^{i \map \Im z} }\) Sum of Complex Indices of Real Number
\(\displaystyle \) \(=\) \(\displaystyle \cmod {t^{\map \Re z} } \cmod {t^{i \map \Im z} }\) Complex Modulus of Product of Complex Numbers
\(\displaystyle \) \(=\) \(\displaystyle \cmod {t^{\map \Re z} }\) Modulus of Exponential of Imaginary Number is One:Corollary
\(\displaystyle \) \(=\) \(\displaystyle t^{\map \Re z}\) Power of Positive Real Number is Positive

$\blacksquare$