I posed to myself two problems:

$\displaystyle 1^{\pi}=?$

and

$\displaystyle (-1)^{\pi}=?$.

For the second one I did $\displaystyle (-1)^{\pi}=(e^{\pi i})^{\pi}=\cos (\pi^2)+ i\sin(\pi^2)$ it seems good.

But for the second one

$\displaystyle 1^{\pi}=(e^{2\pi i})^{\pi}=\cos (2\pi^2)+i\sin (2\pi^2)$ yet it should be one!?!?

I am guessing the exponent for complex numbers is defined for $\displaystyle e^{\theta i}$ where $\displaystyle 0\leq \theta<2\pi$ Thus, my manipulation in problem 2 is out of the domain of the definition and it causes problems.

Am I right?