Re: question of i.

From: Brian Borchers (borchers_at_nmt.edu)
Date: 07/13/04 

Date: Tue, 13 Jul 2004 23:58:52 +0000 (UTC)

israel@math.ubc.ca (Robert B. Israel) wrote:
>"Dirk Van de moortel" <dirkvandemoortel@ThankS-NO-SperM.hotmail.com> wrote in message news:<jfRIc.184083$Jc.8753205@phobos.telenet-ops.be>...
>> "G. A. Edgar" <edgar@math.ohio-state.edu.invalid> wrote in message news:130720040815319519%edgar@math.ohio-state.edu.invalid...
>>> Maple uses a consistent set of principal values in its computations,
>>> and sqrt(z) (for complex z) is just one of them. Similar is the
>>> notation log(z) for complex z. If you reject sqrt(-1) do you also
>>> reject log(-1)?
 
>> If it is made explicitly clear that both are complex functions of a
>> complex variable (and single-valued) and give the principal values,
>> then I reluctanlty accept.

>If you only have the principal-branch sqrt or log, you lose the freedom
>to put the branch cut where it is convenient for a given problem (or
>else you have to resort to some horrible circumlocution).

>>But I would at the very least explicitely
>> use another notation, like CSqrt or CLog...
>> Otherwise you have to make the students unlearn things like
>> log(xy) = log(x) + log(y).
>> I have learned that asking students to unlearn something is a
>> bad idea. In the best case it's a waste of time.

>There are a lot of things they have to unlearn in complex
>analysis. It does make the subject a bit more challenging.

An important practical problem in using symbolic computation software
with undergraduate students like Calculus and Intro to Diff EQ is that
these students have not had any exposure to complex analysis and thus
aren't in any position to understand the output from the system in
situations like these.

Personally I'm quite happy with a system that consistently does things in
terms of the principal values of complex functions. However, it would
be interesting if the system also had a mode in which it only expressed answers
in terms of real numbers and real valued functions (and presumably says
"I can't do this in the reals" in other cases.) Are there any available
packages that do this? 

-- 
Brian Borchers                          borchers@nmt.edu
Department of Mathematics               http://www.nmt.edu/~borchers/
New Mexico Tech                         Phone: 505-835-5813
Socorro, NM 87801                       FAX: 505-835-5366       
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