agd@houem.UUCP (A.DEACON) (02/10/84)
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.