複素数の積・商

複素数の積

$z_{1} = |z_{1}|(\cos \theta_{1} + i\sin \theta_{1}), z_{2} = |z_{2}|(\cos \theta_{2} + i\sin \theta_{2})$とおく。

\begin{eqnarray*}
z_{1}z_{2} &=& |z_{1}||z_{2}| (\cos \theta_{1} + i\sin \theta_{1})(\cos \theta_{2} + i\sin \theta_{2})\\
&=& |z_{1}||z_{2}| (\cos \theta_{1} \cos \theta_{2} - \sin\theta_{1}\sin\theta_{2} +i\sin\theta_{1}\cos\theta_{2} + i\cos\theta_{1}\sin\theta_{2})\\
&=& |z_{1}||z_{2}| (\cos(\theta_{1} + \theta_{2}) + i\sin(\theta_{1} + \theta_{2}))
\end{eqnarray*}

複素数の商

\begin{eqnarray*}
\frac{z_{1}}{z_{2}} &=& \frac{ |z_1|(\cos\theta_1 + i\sin\theta_1) }{|z_2|(\cos\theta_2+i\sin\theta_2)}\\
&=& \frac{ |z_1|(\cos\theta_1 + i\sin\theta_1)(\cos\theta_2-i\sin\theta_2) }{|z_2|(\cos\theta_2+i\sin\theta_2)(\cos\theta_2-i\sin\theta_2)}\\
&=& \frac{ |z_1|}{|z_2|}(\cos\theta_1 + i\sin\theta_1)(\cos\theta_2-i\sin\theta_2) \\
&=& \frac{|z_1|}{|z_2|} (\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2 + i\sin\theta_1\cos\theta_2 - i\cos\theta_1i\sin\theta_2) \\
&=& \frac{|z_1|}{|z_2|} (\cos(\theta_1-\theta_2) +i\sin(\theta_1-\theta_2)) \end{eqnarray*}