# 1 = e^2π. Where did I make a mistake? [duplicate]

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

## marked as duplicate by user228113, egreg, Did, Parcly Taxel, Eric StuckyOct 23 '16 at 10:41

• $(ab)^n)=a^n b^n$ is not true in general for complex numbers, for example $1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=i^2=-1$ is clearly false. – Sophie Oct 22 '16 at 20:46
• If you define, as you seem to do $$x^{\alpha}:=\exp\left(\alpha\ln \lvert x\rvert+i\alpha\arccos\frac{\Re( x)}{\lvert x\rvert}\right)$$ then the identity $x^{\alpha\beta}=\left(x^\alpha\right)^\beta$ no longer holds. – user228113 Oct 22 '16 at 20:51
• Almost exactly the same question was asked yesterday, see also For which complex $a$, $b$, $c$ does $(a^b)^c = a^{bc}$ hold? – Andrew D. Hwang Oct 22 '16 at 20:54
$i^{2i} \neq (-1)^i$; 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.