de Moivre identity

Primary tabs

Synonym:
de Moivre's theorem, de Moivre's formula
Type of Math Object:
Theorem
Major Section:
Reference

Mathematics Subject Classification

This entry is classified as unproven, yet the
entry itself already contains the short proof, in my opinion, for it follows easily from Euler's identity. Is there a way to change the classification of this object?

It's not really proven. Ok, so we have shown that if (exp(ix))^n = exp(nix) then the result follows, but we need to show that (exp(ix))^n does actually equal exp(nix). Recall that exp(ix) is DEFINED as isin(x) + cos(x). So we need to show the above identity; the proof for real numbers does not carry across to here.

It's like saying "dy/dx = dy/du * du/dx does not need proof, its just obvious". We are dealing with different objects than usual fractions there, and we are dealing here with different objects than real numbers.

Proof for integer n is trivial, by the way. Simply show that it is true for all n >= 0 by induction on n, and then use exp(-ix) = 1/exp(ix) (which IS provable from the definition) to show that it extends to negative numbers.