On 18 November 2014 12:29, Ondřej Čertík <ondrej.cer...@gmail.com> wrote:
> On Tue, Nov 18, 2014 at 9:28 AM, David Roe <roed.m...@gmail.com> wrote:
>> ...
>> Because derivative is not just used in the context of functions of a
>> complex variable (whether they are analytic or not). Probably more
>> than 90% of Sage users don't know any complex analysis (as frequently
>> lamented by rtf). I will certainly acknowledge that people get things
>> wrong with regard to sqrt and log by not knowing about branch cuts.
>> But when it comes to the definition of derivative, we need to stay
>> consistent with the standard definition of lim_{h -> 0} (f(x + h) -
>> f(x))/h for functions of a real variable (or functions that many
>> people interpret as just functions of a real variable). Any other
>> decision will cause frustration for the vast majority of our users.
>
> Well, I think it doesn't matter if you know complex analysis or not.
I agree, but apparently for a different reason.
> The point is rather that there is a real derivative and a complex
> derivative. The complex derivative being a generalization of the
> real one (http://en.wikipedia.org/wiki/Derivative#Generalizations,
> http://en.wikipedia.org/wiki/Generalizations_of_the_derivative#Complex_analysis).
> As such, it must reduce to the real derivative as a special case when
> all variables are real, otherwise you get inconsistencies.
>
As I said in another email, I think this is highly dependent on one's
education and experience. Although I admit that it is very common
(almost ubiquitous) to teach calculus starting from the notion of
continuity and limits, I my opinion these references on wikipedia are
especially biased. To me from the point of view of computer algebra,
the algebraic properties of derivatives are more important.
For the sake of continuing the argument, from the point of view of
algebra why should we consider derivatives of complex functions as a
generalization of the real one rather than the real derivative as a
defined in terms of something more general? In particular notice that
the so called Wirtinger derivatives also make sense in the case of
quaternion analysis, so should we be expecting to view quaternion
calculus also as a "generalization' of real derivatives?
OK, maybe I am pushing this a little too far. I admit that the
argument from the point of view of limits and without any reference to
conjugate is quite convincing.
> ...
> Bill, you wrote "I think rather that one should strive for the most
> general solution
> consistent with the mathematics.". Well, the above (i.e.
> x.conjugate()/(2*abs(x)) + x/(2*abs(x)) * exp(-2*I*theta)) is the
> most general solution consistent with mathematics.
>
> Of these options, only theta=0 gives the real derivative as a special
> case, that's what the GiNaC proposal does.
>
Have you had a chance to consider the issue of the chain-rule yet?
Bill.
--
You received this message because you are subscribed to the Google Groups
"sage-devel" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to sage-devel+unsubscr...@googlegroups.com.
To post to this group, send email to sage-devel@googlegroups.com.
Visit this group at http://groups.google.com/group/sage-devel.
For more options, visit https://groups.google.com/d/optout.
