A Way to Express i to the i

Date: 17 Apr 1995 22:42:49 -0400
From: Denise L. Kunz 813 757-9370
Subject: Curious

Dr. Math,

        I know e to the iA = cosA + i sinA, where iA is the exponent. Is 
there another way to express i to the i?

        Thank you.
        Denise Kunz (kunzd@mail.firn.edu)

Date: 18 Apr 1995 13:25:24 -0400
From: Dr. Ken
Subject: Re: Curious

Hello there!

Stuff like this is really neat.  One standard trick in evaluating weird
powers is to make everything be a power with base e.  So we can do that by
writing i^i (which means i to the i) = e^Log[i^i] = e^(i*Log[i]).  Then we
use the fact that Log[i] = i*Pi/2 to get i^i = e^(i*i*Pi/2) = e^(-Pi/2).
You can then use a calculator or the Taylor expansion for e^x to see that
e^(-Pi/2) is about 0.20788.  Nice question!!

-Ken "Dr." Math