A question in Complex Analysis

davegage

Posers and Puzzles

13 Mar 04

davegage

Joined: 04 Aug 01
Moves: 2408

13 Mar 04

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

BarefootChessPlayer

Barefoot Chessplayer

central usa

Joined: 22 Jul 03
Moves: 63080

14 Mar 04

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.

qwiksylver

I'm in Checkmate

Joined: 11 Mar 04
Moves: 842

14 Mar 04

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

iamatiger

Joined: 26 Apr 03
Moves: 26771

14 Mar 04

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.