Argument of Complex Conjugate equals Argument of Reciprocal
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Theorem
Let $z \in \C$ be a complex number.
Then:
- $\arg {\overline z} = \arg \dfrac 1 z$
where:
- $\arg$ denotes the argument of a complex number
- $\overline z$ denotes the complex conjugate of $z$.
Proof
Let $z$ be expressed in polar form:
- $z := r \paren {\cos \theta + i \sin \theta}$
Then:
\(\ds \dfrac 1 z\) | \(=\) | \(\ds \dfrac 1 r \paren {\cos \theta - i \sin \theta}\) | Polar Form of Reciprocal of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 r \paren {\map \cos {-\theta} + i \map \sin {-\theta} }\) | Cosine Function is Even, Sine Function is Odd |
The result follows by definition of the argument of a complex number
$\blacksquare$
Also see
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations