Polar Form

[!Theorem 7.3.4]

Let $z_1=r_1(cos{\theta }_1+isin{\theta }_1)$ and $z_2=r_2(cos{\theta }_2+isin{\theta }_2)$

Then $z_1{z}_2=r_1{r}_2(cos({\theta }_1+{\theta }_2)+isin({\theta }_1+{\theta }_2))$

[!de Moivre’s Theorem]

If $z=r(cos\theta +isin\theta )\ne 0$, then

$z^n=r^n(cos(n\theta )+isin(n\theta ))$

for any $n\in \mathbb{Z}$

[!Complex Exponential Form]

Let $\theta \in \mathbb{R}$. The expression $e^{i\theta }$ is defined to mean

$e^{i\theta }=cos\theta +isin\theta$

If $z=r(cos\theta +isin\theta )$ is the polar form of $z\in \mathbb{C}$, then $z=r{e}^{i\theta }$ is the complex exponential form of z

[!Euler’s Identity]

$e^{i\pi }=cos\pi +isin\pi =-1+i(0)=-1$

Can be rewritten as

$e^{i\pi }+1=0$