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