Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part
Jump to navigation
Jump to search
 It has been suggested that this article or section be renamed:
|
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 |
$\blacksquare$
