# A question in Complex Analysis

davegage
Posers and Puzzles 13 Mar '04 21:52
1. 13 Mar '04 21:52
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
=&gt; 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
2. 14 Mar '04 02:50
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.
3. 14 Mar '04 04:25
It's something I've always just taken for granted. Never eally thought about it, but now I'm curious. Anybody able to elucidate?
4. 14 Mar '04 18:15
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.