Introduction to fractional calculus: problem with identity

Question

I can't see the next step:

$D^\alpha e^{ix} = i^{\alpha}e^{ix} = e^{i\alpha \frac \pi2}e^{ix}$

Answer 1

Using Euler's identity $$e^{ix} = \cos x + i \sin x$$ A complex number must be in the form: $z = x+iy$, then $e^z = e^{x+iy} = e^x+e^{iy} = e^x(\cos y + i \sin y)$ and I want that this last equality be $i$, then, $e^x(\cos y + i \sin y) = i$ only if $x = 0$ and $y = \frac \pi2$.