The following is an example of this kind of trouble that one can get
into in Maple. Very likely similar things appear in Mathematica,
Maxima, Sage, Sympy and other systems.
> diff(conjugate(x), x);
_
x 2 abs(1, x)
- - + ---------------
x signum(x)
I was curious how Maple could justify this result but so far haven't
been able to locate any appropriate documentation or literature.
Clearly conjugate(x) is not analytic (non-holomorphic) but they must
have had some good reason to define the derivative in that way.
Perhaps for the solution of certain differential equations involving
generalized functions (distributions)?
With no additional assumptions for some reason Maple does not attempt
to differentiate abs which might be reasonabe given that abs is also
non-analytic. But consider:
> diff(sqrt(x*conjugate(x)), x);
/ _ \
_ | x 2 abs(1, x) |
x + x | - - + ---------------- |
\ x signum(x) /
---------------------------------
(1/2)
/ _\
2 \x x/
(You need fixed width fonts to view this ASCII art properly.)
This is actually OK, given the *assumption*:
> diff(conjugate(x), x) assuming x, real
1
But from my point of view (based on Wirtinger
http://en.wikipedia.org/wiki/Wirtinger_derivatives ), by definition we
should have
diff(conjugate(x),x) = 0
then for x Hermitean, i.e. real-valued
real(x) == x/2 + conjugate(x)/2
diff(real(x),x) = 1
One point I want to make is that we can do also this algebraically
without making any assumptions about x!
Similarly I expect to be able to compute:
diff(sqrt(x*conjugate(x)),x) = conjugate(x)/sqrt(x*conjugate(x))
Both of these results necessarily require Wirtinger's chain rule.
Without this I think it might be very hard to "do something
interesting" but avoid the consequences as seen in Maple. If you
simply leave derivative of conjugate unevaluated (implicitly non-zero)
then none of this is possible.
--
You received this message because you are subscribed to the Google Groups
"FriCAS - computer algebra system" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to fricas-devel+unsubscr...@googlegroups.com.
To post to this group, send email to fricas-devel@googlegroups.com.
Visit this group at https://groups.google.com/group/fricas-devel.
For more options, visit https://groups.google.com/d/optout.
