I'm having a huge lapsus. Let $\alpha \in [0,1]$. From one point of view we have
$e^{2 \pi i \alpha} = (e^{2 \pi i})^{\alpha} = 1$
But I could also write
$e^{2 \pi i \alpha} = \cos(2 \pi \alpha) + i\sin(2 \pi \alpha) \neq 1$
What's wrong? I feel like i'm missing something REALLY obvius.
$\endgroup$ 31 Answer
$\begingroup$There is nothing wrong with your argument.
According to the Euler's formula, we have $$e^{i \theta}=\cos(\theta) + i \sin(\theta)$$
Thus $$e^{2\pi i\alpha } = \cos(2\pi\alpha) + i \sin(2\pi\alpha)$$
The power rule $$e^{xy}=(e^x)^y$$ does not hold for complex numbers.
For example $$e^{2\pi i\alpha}\neq 1$$.
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