A question in Complex Analysis

This is for those math people who love complex space:

i think it is very difficult to conceptualize it, but i^i is actually a real number and is very easy to find using Euler's Formula:

exp(i*Pi) = -1 = i^2
=> i^i = exp(-Pi/2) = 0.207879......

i think this is fascinating and would like to be able to understand why this is so....can any of you give me a geometric or any other reason why i^i should be a real number?

thanks

Originally posted by davegage
This is for those math people who love complex space:

i think it is very difficult to conceptualize it, but i^i is actually a real number and is very easy to find using Euler's Formula:

exp(i*Pi) = -1 = i^2
=> i^i = exp(-Pi/2) = 0.207879......

i knew this also, but haven't seen the proof as to why.
it is logical, though, since i is such an unusual quantity.
hope soemone else can help us here.

It's something I've always just taken for granted. Never eally thought about it, but now I'm curious. Anybody able to elucidate?

Originally posted by davegage
This is for those math people who love complex space:

i think it is very difficult to conceptualize it, but i^i is actually a real number and is very easy to find using Euler's Formula:

exp(i*Pi) = -1 = i^2
=> i^i = exp(-Pi/2) = 0.207879......

i think this is fascinating and would like to be able to understand why this is so....can any of you give me a geometric or any other reason why i^i should be a real number?

thanks

Is it because raising a number to an imaginary power rotates it in the complex plane? Raising i to an imaginary power looks like it rotates i exactly onto the real plane.