I am assuming that you are familiar with basic Trignometry and concepts of vectors .I will also assume that you are familiar with basics of complex no.
so we are going to start with euler representation of complex no. using complex exponentials.
We know that complex no. exist in a plane called the complex plane that is made of a real axis and an imaginary axis which are orthogonal to each other. just like a unit vector for some direction like the i(hat) being the basis vector for x direction we are going to introduce a similar ides for representing complex no.
lets consider a unit circle in the complex plane with centre as the origin. lets consider a point on that circle , let the line connecting the point and centre make an angle of ψ with positive direction of real axis, now the x and y coordinates of the point can be given by cos(ψ) and sin(ψ) respectively.
complex no. z is written in cartesian coordinates as x+iy. in polar coordinates z can be written as cos(ψ)+isin(ψ) for any z on the unit circle. this cos(ψ)+isin(ψ) can be considered as a basis for any complex no. on the straight line with slope tan(ψ). so just like for vectors which ae written as mangnitude times the basis , any general complex no. z can be written as |z| times cos(ψ)+isin(ψ) with ψ being in the range of [0,2*pi).
We are going to use the Taylor series for e^x ,cos(x),sin(x) . The proof of these expansions is not within the scope of this article , nor is it important for the progress of this course, we may cover this in a miscellaneous article later on.
the expansions are :cos x = 1 — x²/2! + x⁴/4! — x⁶/6! +….
sin x= x—x³/3! + x⁵/5! — x⁷/7! + ….
isinx=i(x — x³/3! + x⁵/5! — x⁷/7! + ….)
e^x = 1+ x + x²/2! + x³/3! + x⁴/4! + ….
e^ix=1+(ix)+(ix)²/2!+(ix)³/3!+(ix)⁴/4!+(ix)⁵/5!+(ix)⁶/6!+(ix)⁷/7!+….
=(1—x²/2!+x⁴/4! — x⁶/6!+….)+i(x — x³/3! + x⁵/5! — x⁷/7! + ….)
=cosx+isinx.
now that the euler formula is proved ,euler identity would follow.
Alright I will see you in the next lecture.