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Re: Another strange bug in Mathematica 4.0's Integrate
*To*: mathgroup at smc.vnet.net
*Subject*: [mg25406] Re: Another strange bug in Mathematica 4.0's Integrate
*From*: Richard Fateman <fateman at cs.berkeley.edu>
*Date*: Fri, 29 Sep 2000 01:06:51 -0400 (EDT)
*Organization*: University of California, Berkeley
*References*: <8q21lp$hoh@smc.vnet.net> <ztgz5.89556$Zh6.152771@ralph.vnet.net> <8qlo62$q42@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Hendrik van Hees wrote:
>
> It's funny what the CA-fans argue and don't admit that a bug is a bug.
> If I put in
>
> Integrate[Sqrt[r^2-x^2],{x,-r,r},Assumptions->{r>0}]
>
> this is a well defined expression in terms of a real integral. The
> square root is positive because the principle value is realised
> (according to the online documentation). Further as long as I do not
> tell the path of integration it must be a real integral, otherwise
> Integrate is not well defined!
>
> --
> Hendrik van Hees Phone: ++49 6159 71-2751
> c/o GSI-Darmstadt SB3 3.183 Fax: ++49 6159 71-2990
> Planckstr. 1 mailto:h.vanhees at gsi.de
> D-64291 Darmstadt http://theory.gsi.de/~vanhees/index.html
Documenting a bug in the on-line documentation does not
make it a feature. If Mathematica makes the "wrong" choice
of a value for square root (or in general any function
with a non-unique value) becaue it makes some "standard"
choice of a principal value or some "standard" location
for a branch cut in the complex plane, then the user
can get surprising answers. OR answers that are
not surprising, but subtlely wrong.
A good mathematician will do manipulations being
attentive to such matters as locations of branch cuts,
and make the appropriate choices. When Mathematica
makes a choice (and this is the same for other
computer algebra systems, most of the time), it
(a) doesn't tell you there are other possibilities
and
(b) doesn't allow you to influence the choice.
The point I've tried to make in several notes to this
newsgroup recently is that if the CAS is just too
naive to understand these issues, then inevitably
it will make blunders. The fact that these blunders
often show up in definite integrals does not mean
that the bug is in the integration program. One
can try to patch up the integration program but that
won't cure the underlying difficulty. If the kernel
program that does simplification cannot deal with
multivalued functions (like square root), then there
will always be problems.
These are research problems that are not resolved
by assertions like "we choose the principal value".
For example in another system, Paul Wang's PhD (1972)
definite integration program made explicit use of
notations of Principal and other values. For
example, he used Log and log to distinguish them.
And he made some effort to manipulate them correctly as
he computed path integrals in the complex plane.
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