Subject: "Unexpected behavior of log() in complex plane"
From: Michel Talon
Date: Fri, 28 Oct 2011 21:57:38 +0000 (UTC)
Karl-Dieter Crisman wrote:
> On Fri, Oct 28, 2011 at 2:25 PM, Richard Fateman
> <fateman at eecs.berkeley.edu> wrote:
>> The behavior you report below is not the behavior of the Maxima that I just
>> tried it on, 5.23.2.
>>
>> It reports for an answer, log(w).
>
> Interesting. In that version I get
>
> Maxima 5.23.2 http://maxima.sourceforge.net
> using Lisp ECL 11.1.1
> Distributed under the GNU Public License. See the file COPYING.
> Dedicated to the memory of William Schelter.
> The function bug_report() provides bug reporting information.
> (%i1) declare(w,real);
> (%o1) done
> (%i2) assume(w<0);
> (%o2) [w < 0]
> (%i3) limit(log(w+%i*eps),eps,0);
> (%o3) und
> (%i4) limit(log(w+%i*eps),eps,0,plus);
> (%o4) log(w) + %i %pi
>
I get exactly the same on maxima 5.25.1 here, as a matter of fact.
>> So complex log -- eh, not necessarily supported. ?But log has multiple
>> values, ?and to distinguish between
>> log(w)+n*i*pi ?for odd integer n, ? ? ?and log(-w), is a kind of subtle
>> point, not one that is easily handled by
>>
>> ... just tell me the limit, don't confuse me with, uh, ?the mathematics...
>
> Well, the point is that there is supposed to be a well-defined branch
> cut for such functions,
This is a point where i disagree (and probably prof Fateman also). This branch
cut is completely conventional. The only intrinsic datum is that there is a
singularity at 0. Same as for the sqrt(x), which people say has a branch cut
for x<0. But then what about sqrt((x+1)*(x-1)*(x+2)*(x-2)) for example
(related to elliptic functions) or more complicated. What if i replace the
roots 1, 2, etc. by 3+4*I etc. In what way are you going to say there are
"canonical cuts"? For a sufficiently complex expression you will have hard
time to understand things in terms of cuts, and you will be confronted to the
Riemann surface which is the true object here.
--
Michel Talon