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From: agd@houem.UUCP (A.DEACON)
Newsgroups: net.math
Subject: More on i**i
Message-ID: <229@houem.UUCP>
Date: Mon, 13-Feb-84 10:32:50 EST
Article-I.D.: houem.229
Posted: Mon Feb 13 10:32:50 1984
Date-Received: Tue, 14-Feb-84 01:53:02 EST
Organization: Bell Labs, Holmdel NJ
Lines: 39


In response to Seaman and Rentsch:

The value of i^i is very well defined.  I gave
the values in a previous article and they are:

        i       -(4n+1)*pi/2
       i   =   e                  for all integer n.
  

There is no problem in defining log(i) either:
for any complex number z


     ln(z) = ln|z| + i(theta + 2n*pi) for all integer n

where -pi < theta <= pi and |z| is the modulus of z.
Theta is called the principle
argument of the ln.  As you can see there are an infinite
number of values for ln(z).  Of course the ln function
cannot be extended continuously to the entire complex
plane because of the pole.  However, if you consider
the Riemann surface, it can be.

In the exp(x) sum, we need x=i*ln(i) to compute i^i,
so it does require something "funny".  Being clever
has nothing to do with it.  That's the way it is.

For Seaman:
           i       -(2n+1)*pi
       (-1)   =   e              for all integer n

and
           i       -(4n-1)*pi/2
       (-i)   =   e              for all integer n.


Art Deacon
AT&T Bell Labs