Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10 5/3/83; site houem.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!mhuxl!houxm!houem!agd
From: agd@houem.UUCP (A.DEACON)
Newsgroups: net.math
Subject: i to the i
Message-ID: <223@houem.UUCP>
Date: Fri, 10-Feb-84 09:08:07 EST
Article-I.D.: houem.223
Posted: Fri Feb 10 09:08:07 1984
Date-Received: Sat, 11-Feb-84 08:10:15 EST
Organization: Bell Labs, Holmdel NJ
Lines: 28




I tried to send this solution to you Mike but I don't think
it made it.  Jim's answer was only partially correct.  The
complex exponential is a multi-valued function, so i^i
is also multi-valued.  The values of i^i are:



            i        -(4n+1)*pi/2
           i   =    e                   n=...,-2,-1,0,1,2,3...


                     a (set) of real numbers!

Note that for n=0, we get Jim's result.  The error in Jim's
solution is in the step where he says

        i = exp(0 +pi/2) ==> ....

The argument (arg) for a complex number is not unique.  Pi/2
is the principle arg but you must add 2n*pi for all positive and
negative n.  If you insert this factor in Jim's proof, you will
end up with the proper answer.

                  Art Deacon
                  AT&T Bell Laboratories.