Argument of x to the n Equals n Times The Argument
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Theorem
Let $z$ be a complex number.
Then:
- $\forall n \in \N_{>0}: \map \arg {z^n} = n \map \arg z$
Proof
For $n = 1$
- $\map \arg {z^1} = 1 \cdot \map \arg z$
Assuming the result is true for $n = k$, we have:
| \(\displaystyle \map \arg {z^{k + 1} }\) | \(=\) | \(\displaystyle \map \arg {z z^k}\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \map \arg z + \map \arg {z^k}\) | Argument of Product equals Sum of Arguments | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \map \arg z + k \map \arg z\) | by our induction hypothesis | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \paren {k + 1} \map \arg z\) |
$\blacksquare$
