Complex Numbers

z = x + iy with (x,y) as point coordinates.

z = r(cosθ + isinθ), where x = r cosθ & y = r sinθ
$r = √{x^2+y^2}$
θ = $tan^{-1}(y/x)$

amp(z) / arg(z) = θ

As arg(z) is multivalued function & cos, sin have 2π radians period:

arg(z) = θ+2nπ, n = 0,1,2,…

θ+2nπ is general value and value of θ in between -π & π is principle value

Principle value of arguement

(1)$I^{st}$ quadrant (x>0,y>0):
θ = arg(z) = $tan^{-1}(y/x)$


(2)$II^{nd}$ quadrant (x<0,y>0):
θ = arg(z) = π – $tan^{-1}(y/x)$

(3)$III^{rd}$ quadrant (x<0,y<0):
θ = arg(z) = $tan^{-1}(y/x)$ ± π

(4)$IV^{rd}$ quadrant (x>0,y<0):
θ = arg(z) = -$tan^{-1}(y/x)$

Exponential form of complex numbers

$e^x = 1 + x + x^2/{2!} + x^3/{3!} + …$

$cosx = 1 – x^2/{2!} + x^4/{4!} -…$
$sinx = x – x^3/{3!} + x^5/{5!} – …$

Note: $i^2$= -1, $i^3$= -i, $i^4$= 1, $i^5$= i

$e^{ix}$ = cosx + isinx
$e^{-ix}$ = cosx – isinx

z = r $e^{iθ}$ (exponential form)

Some Useful Results:
(1) (cosα + isinα) (cosβ + isinβ) = cos(α+β) + isin(α+β)

(2) ${cosα + isinα}/{cosβ+isinβ}$ = cos(α-β) + isin(α-β)
(3) ${1}/{cosθ+isinθ}$ = cosθ – isinθ
(4) ${1}/{cosθ-isinθ}$ = cosθ + isinθ

(5) $1/i = -i$

Demoivre’s theorem:
$(cosθ +isinθ)^n$ = cos nθ + isin nθ