Question

What is the shortest way to prove:\[
e^{ix}=\cos(x)+i\sin(x)
\]

Answer 1

I know one way is to use the series expansion for cos(x) and sin(x) and show that it matches the series expansion for e^(ix) - but is there a shorter proof?

Answer 2

Do you know what is the polar form of a complex number ?

Answer 3

yes

Answer 4

I read once a short proof by using differential equations.

Answer 5

do you recall that proof @Mr.Math?

Answer 6

Look here http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs

Answer 7

@FoolForMath - do you mean this: