TITLE: 1 to the power i
QUESTION [9 upvotes]: Possible Duplicate:
What is the value of $1^i$? 

I was thinking, what would 1^i be? Then I did:
$e^{i\pi}=-1\rightarrow e^{i\pi}\cdot e^{i\pi}=e^{2i\pi}=-1\cdot -1=1$
Now raise to the power i:
$1^i=(e^{2i\pi})^i=e^{2i^2\pi}=e^{-2\pi}=\frac{1}{e^{2\pi}}$
Is this correct?

REPLY [9 votes]: The power function $(x,y)\mapsto x^y$ is indeed defined via the exponential $x\mapsto e^x$, as
$x^y:=e^{\log x\cdot y}$,  where $\log$ is the inverse of $\exp$. But, $\exp$ is not injective: it is periodic by $2\pi i$, hence the $\log$ is not unique, only up to $+k2\pi i$ for some $k\in\mathbb Z$. So, $1^i$ has infinite many values:
$$1^i=e^{(2k\pi i)i} = e^{-2k\pi} $$
Similarly for $i^i$. Can you find all of its values?