Re: Re: Help -- Weird integration behavior

• To: mathgroup at smc.vnet.net
• Subject: [mg49407] Re: [mg49390] Re: Help -- Weird integration behavior
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 18 Jul 2004 08:09:18 -0400 (EDT)
• References: <200407151100.HAA11167@smc.vnet.net> <cd8a3f\$og0\$1@smc.vnet.net> <200407171038.GAA10339@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```On 17 Jul 2004, at 19:38, Mike D wrote:

> *This message was transferred with a trial version of CommuniGate(tm)
> Pro*
> Thanks so much!
>
> I had never heard of the "GenerateConditions" option on definite
> integrals before. After looking at the help, it makes a little sense
> that this is necessary for an integral with parameters. But was this
> option present in earlier versions of Mathematica?

On the surface the idea is very sensible, when you try to evaluate a
definite integral Mathematica tries to give you conditions on the
limits under which its answer is valid. In very simple cases it works
well, here is an example form the Documentation"

Integrate[Exp[a*x], {x, 0, Infinity}]

If[Re[a] < 0, -(1/a), Integrate[E^(a*x),
{x, 0, Infinity}, Assumptions -> Re[a] >= 0]]

This is correct and nice, but unfortunately the attempt to give this
sort of answer is more complex cases is much too ambitious and tends to
give answers which so extremely as to be unusable complicated but still
usually not universally valid. In previous versions Mathematica only
sometimes tried to return conditional answers and often was satisfied
with returning answers which were valid only with certain assumptions
on the parameters, without trying to explicitly state these conditions.

> I'm still a little bothered that I couldn't simply run an old notebook
> "as is" in a newer version of Mathematica -- more so that the program
> suggested that it was giving a valid answer ("0"), with no errors --
> if I had not had an old printout of the notebook, I would have thought
> I was going crazy.
>

This is just a consequence of trying to do something a lot more
ambitious in Version 5 compared with the previous versions. Integrate
was changed in various ways: it tries to return more "correct answers"
for definite integrals, it performs more tests to check for
singularities, it tries to simplify the integrand etc. This leads in
some cases to different answers than those returned by older versions.
In many cases the new answer are better but in quite a few cases they
are worse. It is difficult to give an objective judgement on whether
the new Integrate in version 5 is, on the whole, better or worse
without performing some sort of controlled test. But on the whole I am
pretty sure that the many changes made in version 5 are a foundation
for further improvements that will make future versions much better
than the previous ones. At the same time I remain unconvinced that the
ambitious approach to definite integration (trying to return an answer
together with full conditions under which it holds) will ever work well
enough to justify the much increased complexity of answers and the time
taken to reach them.

Andrzej Kozlowski
Chiba, Japan
http://www.mimuw.edu.pl/~akoz/

```

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