e^(iΘ) = cos (Θ) + i sin (Θ)
Typically the proof involves using the summation e^x = Sum (x^n / n!) from 1 to infinity. Then you set x = iΘ, and separate the summation into two halves: cosine and sine.
The part I never understood is, how do you establish that the summation e^x = Sum (x^n / n!) holds for complex values of x? The proofs I've seen just assume that it does. Is there a way to develop summation definition for C?