Division of Complex Numbers in Exponential Form

Theorem

Let $z_1 := r_1 e^{i \theta_1}$ and $z_2 := r_2 e^{i \theta_2}$ be complex numbers expressed in exponential form.

Then:

$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} e^{i \paren {\theta_1 - \theta_2} }$

$\ds \frac {z_1} {z_2}$

$=$

$\ds \frac {r_1 e^{i \theta_1} } {r_2 e^{i \theta_2} }$

$\ds $

$=$

$\ds \frac {r_1 \paren {\cos \theta_1 + i \sin \theta_1 } } {r_2 \paren {\cos \theta_2 + i \sin \theta_2} }$

$\ds $

$=$

$\ds \frac {r_1} {r_2} \paren {\map \cos {\theta_1 - \theta_2} + i \map \sin {\theta_1 - \theta_2} }$

$\ds $

$=$

$\ds \frac {r_1} {r_2} e^{i \paren {\theta_1 - \theta_2} }$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Operations with Complex Numbers in Polar Form: $7.26$
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $19$