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**TD!** You have to be careful with complex numbers, not all the properties you know from the real numbes (about square roots, powers, logarithms, ...) still hold for complex numbers.

In particular, the rule $\displaystyle (z^a)^b = z^{ab}$ no longer holds.

The correct version is $\displaystyle (z^a)^b = z^{ab}e^{2bk\pi i}$.

So $\displaystyle (e^{2\pi i})^{\pi} = e^{2\pi^2 i}e^{2k\pi^2 i} = e^{2\pi^2 i+2k\pi^2 i} = e^{(k+1)2\pi^2 i}$

And this again yields $\displaystyle e^0$ for $\displaystyle k = -1$.

This has to do with the analytical continuation of the natural logarithm as a complex function and the fact that for complex values of z, $\displaystyle e^z$ is a periodic function with period $\displaystyle 2\pi i$.