1 = e^2π. Where did I make a mistake?

Question

I seem to have proven that $e^{2\pi} = 1$. What is my mistake?
See here for proof.

Answer 1

$i^{2i} \neq (-1)^i$; this behavior with powers works over the reals, but because of branching, you can't do this over arbitrary complex numbers. You always want to go back to the exp function when computing powers, just to be safe.