explain where the error on the statment?

Question

can someone say what wrong with the bellow stament?

$$ e^{\imath\pi}+1=0\\ e^{\imath\pi}=-1\\ (e^{\imath\pi})^2=(-1)^2\\ e^{2\imath\pi}=1\\ (e^{2\imath\pi})^\imath=1^\imath\\ e^{2\imath^2\pi}=1^\imath\\ e^{-2\pi}=1^\imath $$

Answer 1

Nothing is wrong yet. You didn't write $1^i=1$ at the end of the proof, you left it for the reader to assume. In the complex plane, $a^b=\exp(b \log a)$ and the logarithm is multivalued unless you are careful with branch cuts. We have $\log 1=0+2k\pi i$ for $k \in \Bbb Z$. If we take $k=-1, 1^i=\exp(-2\pi)$ and there is no problem.