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From: csc@watmath.UUCP (Computer Sci Club)
Newsgroups: net.math
Subject: Re: References on i ** i, "principal logs"
Message-ID: <6964@watmath.UUCP>
Date: Tue, 21-Feb-84 12:18:14 EST
Article-I.D.: watmath.6964
Posted: Tue Feb 21 12:18:14 1984
Date-Received: Wed, 22-Feb-84 01:59:04 EST
References: <205@pucc-i>, <1960@mcnc.UUCP>, <1964@mcnc.UUCP>
Organization: U of Waterloo, Ontario
Lines: 28

    The complex log fuction can be defined in a number of ways.
Perhaps the most intuitive is as the inverse of the exponential
function (which can be defined as the sum of an infinite series).
Hence we define log(z) to be a complex number such that exp(log(z))=z.
Now any non zero complex number z can be written as exp(r + it),
(r and t real numbers).  r is uniquely determined but t isn't.
If t is valid so is t+2n(pi) with n an integer.  Therefore there
are an infinitely many complex numbers g such that exp(g)=z.
(if z is non zero).  Hence log is multi valued.  There is no
complex number g such that exp(g)=0, therefore log(0) is undefined.
Hence log is a multi valued function defined on the complex plane
minus zero.
      It does not make sense to define log(0)=0 as then
exp(log(0))=1.  The article which argued that log(0)=0 contained
a division by zero which implied i*2n(pi)=i*2n(pi)exp(i*(pi)/2)
or as exp(i*pi/2)=i this implies i*2n(pi)=-2n(pi).  A contradiction.
It is because dividing by zero leads to such contradictions that
such division is not defined.
    One can define a single valued log function by choosing one
value of t for each z, usually done by restricting t to some half
open interval of length 2(pi).  However one cannnot do this in such
a way as to have the resulting function continuous on the complex
plane (minus zero).  Also such equations as log(ab)=log(a) + log(b)
and log(exp(z))=z cannot hold for all a,b,z. The usual practice is
to use whichever "branch" (ie. choice of an interval for t) that
is most convenient for the task at hand.
                                           William Hughes