polar coordinates and Euler’s formula

Polar coordinates:
let r and ? be polar coordinates of the point(x,y) that corresponds to anon-zero complex number z=x+iy , since x=rcos? & y=rsin? ,z can be written in polar form as
z=r(cos?+isin?)……………(1)
if z=0 ,the coordinate ? is undefined
note that R=r=?z?
the real number ? represents the angle ,measured in radians ,that z makes with the positive real axis when z is interpreted as radius vector
? has an infinite number of possible values ,including negative ones,each value of ? is called an argument of z .
and the set of all such values is denoted argz ,the principle value of argz ,denoted by Argz ,is that unique value ? such that -?>???
and value of ? that have differ by integral multiples of 2? its means (0???2?)
argz=Argz+2k? (k=0,?1,?2,………)
also, when z is a negative real number ,Argz has value ? not -?
so that
r?the number which is the non-negative square root of x2+y2 is called the modulus
of z and is denoted by ?z?
??is called an amplitude or argument of z and we write ?=argz or ampz
note: the real value of ? are given by the equation ?=tan-1(y/x)
Euler’s formula:
Using the symbol e^i? or exp(i?) which is defined by Euler’s formula for any real values of ? as
e^i?=cos?+isin? ………………(2)
? Z=re^i?=r(cos?+isin?)