PX153 - B2 - polar representation

consider $z = x + i y$
- by analogy, with polar coordinates, write the complex number in terms of its modulus $| z | = r$ , and argument $θ$
$x = r \cos θ, y = r s i n θ$ $z = r \cos θ + i r \sin θ$

euler's formula

z = r e^{i θ} = r \cos θ + i r \sin θ

multiplication:

z_{1} z_{2} = r_{1} r_{2} e^{i (θ_{1} + θ_{2})}

z z^{*} = r e^{i θ} r e^{- i θ} = r^{2}

division:

\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}} e^{i (θ_{1} - θ_{2})}

important corollaries

$\cos θ = \frac{1}{2} (e^{i θ} + e^{- i θ})$
$e^{i θ} + e^{- i θ} = (\cos θ + i \sin θ) + (\cos θ - i \sin θ)$
$\sin θ = \frac{1}{2 i} (e^{i θ} - e^{- i θ})$

important note

$x = \sqrt[3]{1}$ has roots $1, ω, ω^{2}$
- solve $x^{3} - 1$ for values of $ω$
if $z = (r, θ)$ :

z^{- 1} = (- r, - θ)

note: added during tutorial