(adsbygoogle = window.adsbygoogle || []).push({}); i^i means WHAT?!?!?!?!?!

My friend gave me the brain-teaser "i^i = what?", and with a little bit of coaching I finally discovered that

i^i = e^(-pi/2)

Which is cool, I suppose. But the more I think about it, the more I wonder:

what the heck does it mean???

For that matter, what does anything raised to an imaginary power mean?

For instance:

e^(ix) = cos(x)+i*sin(x)

It's so counter-intuitive, is there some way to make sense of it by analogy with real numbers? Or will it just remain abstract until the end of time?

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# I^i means WHAT?

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