What's $i^e$, and why is there an imaginary part?

Question

I am not sure if this is a straightforward question, since I am not familiar with complex analysis.

$i^e = e^{(i*π/2)*e} = (e^{iπ})^{e/2} = (-1)^{e/2}$

I think that's somewhat right, but when I put this into a calculator an imaginary part also comes out.

Is this because of $e$?

Answer 1

$i^e = e^{(i*π/2)*e} = (e^{iπ})^{e/2} = (-1)^{e/2}$

You still have $(-1)$ to a power of a multiple of $\frac 12$ and that will create an imaginary part.

$(e^{i\frac {eπ}{2}}) = \cos \frac {eπ}{2} + i\sin \frac {eπ}{2}$

Answer 2

Complex exponentiation is in general multi-valued. Since $i=e^z$ iff $z=\frac{\pi i (4n+1)}{2}$ for some integer $n$, $i^e$ can be any value of the form $\exp\frac{\pi e i (4n+1)}{2}=\cos\frac{\pi e (4n+1)}{2}+i\sin\frac{\pi e (4n+1)}{2}$.