I read somewhere that:

$\displaystyle i^i = e^{-\frac{\pi}{2}} $

Where i is the complex unit number

There wasn't any further explanation on this. Only a short developpment.

I don't have any problem with the developpment. But what puzzles me is the fact the the complex power of a complex number is in fact a real number.

Does anybody have a logical explanation for this. I think it's quite a unique phenomenon.

If you take a number out of R and take its power, it will be a number in R again and not Q.

$\displaystyle \pi^\pi \text{\; is in R for example} $