Product of Complex Numbers in Exponential Form

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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:

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


Proof

\(\ds z_1 z_2\) \(=\) \(\ds r_1 e^{i \theta_1} r_2 e^{i \theta_2}\) Definition of Exponential Form of Complex Number
\(\ds \) \(=\) \(\ds r_1 \paren {\cos \theta_1 + i \sin \theta_1 } r_2 \paren {\cos \theta_2 + i \sin \theta_2}\) Euler's Formula
\(\ds \) \(=\) \(\ds r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \map \sin {\theta_1 + \theta_2} }\) Product of Complex Numbers in Polar Form
\(\ds \) \(=\) \(\ds r_1 r_2 e^{i \paren {\theta_1 + \theta_2} }\) Euler's Formula

$\blacksquare$


Sources