Argument of Quotient equals Difference of Arguments
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Theorem
Let $z_1$ and $z_2$ be complex numbers.
Then:
- $\map \arg {\dfrac {z_1} {z_2} } = \map \arg {z_1} - \map \arg {z_1} + 2 k \pi$
where:
- $\arg$ denotes the argument of a complex number
- $k$ can be $0$, $1$ or $-1$.
Proof
Let $z_1$ and $z_2$ be expressed in polar form.
- $z_1 = \polar {r_1, \theta_1}$
- $z_2 = \polar {r_2, \theta_2}$
From Division of Complex Numbers in Polar Form:
- $\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} \paren {\map \cos {\theta_1 - \theta_2} + i \, \map \sin {\theta_1 - \theta_2} }$
By the definition of argument:
- $\map \arg {z_1} = \theta_1$
- $\map \arg {z_2} = \theta_2$
- $\map \arg {\dfrac {z_1} {z_2} } = \theta_1 - \theta_2$
There are $3$ possibilities for the size of $\theta_1 + \theta_2$:
- $(1): \quad \theta_1 - \theta_2 > \pi$
Then:
- $-\pi < \theta_1 - \theta_2 - 2 \pi \le \pi$
and we have:
\(\ds \map \cos {\theta_1 - \theta_2}\) | \(=\) | \(\ds \map \cos {\theta_1 - \theta_2 - 2 \pi}\) | Cosine of Angle plus Full Angle | |||||||||||
\(\ds \map \sin {\theta_1 - \theta_2}\) | \(=\) | \(\ds \map \sin {\theta_1 - \theta_2 - 2 \pi}\) | Sine of Angle plus Full Angle |
and so $\theta_1 + \theta_2 - 2 \pi$ is the argument of $\dfrac {z_1} {z_2}$ within its principal range.
- $(2): \quad \theta_1 - \theta_2 \le -\pi$
Then:
- $-\pi < \theta_1 - \theta_2 + 2 \pi \le \pi$
and we have:
\(\ds \map \cos {\theta_1 - \theta_2}\) | \(=\) | \(\ds \map \cos {\theta_1 - \theta_2 + 2 \pi}\) | Cosine of Angle plus Full Angle | |||||||||||
\(\ds \map \sin {\theta_1 - \theta_2}\) | \(=\) | \(\ds \map \sin {\theta_1 - \theta_2 + 2 \pi}\) | Sine of Angle plus Full Angle |
and so $\theta_1 - \theta_2 + 2 \pi$ is within the principal range of $\dfrac {z_1} {z_2}$.
- $(3): \quad -\pi < \theta_1 + \theta_2 \le \pi$
Then $\theta_1 - \theta_2$ is already within the principal range of $\dfrac {z_1} {z_2}$.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Multiplication and Division: $3.7.15$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $27 \ \text{(b)}$