Re: Mathematica and Maple disagree on this integral

• To: mathgroup at smc.vnet.net
• Subject: [mg33252] Re: Mathematica and Maple disagree on this integral
• From: "John Doty" <jpd at w-d.org>
• Date: Tue, 12 Mar 2002 05:08:38 -0500 (EST)
• Organization: Wampler-Doty Family
• References: <200203011152.GAA27858@smc.vnet.net> <a61vd9\$gqb\$1@smc.vnet.net> <a6cia0\$eio\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <a6cia0\$eio\$1 at smc.vnet.net>, "Ben Crain"
<bcrain at bellatlantic.net> wrote:

> Thanks for the clarification.  I have learned about the "principal
> value" integral, but did not know that one must distinguish between
> various definitions of the integral in Mathematica (or in Maple).  I
> think the "default" integral should be the most general, or most useful,
> definition of "integral".  Clearly
> (in my opinion) the most useful definition of "integral" would be (at
> least) the
> "principal value" integral;  so if you ask for Mathematica's integral of
> this function, what it returns is correct (i.e., most useful).  My
> standard for judgment is very practical:  suppose you had a model for a
> real-word process
> (e.g., designing an engine for a rocket), which required the evaluation
> of this integral.  Would you reject the design because the integral is,
> according to some definition, "divergent", or would you simply set the
> integral to zero, and accept the model. I think almost all engineers
> would accept the model  (assuming that is the only possible problem),
> and I am certain that the engine would work
> (assuming everthing else in the engine's design is correct).  That means
> that the "principal value" integral really is the "correct" answer.
> Other integrals are simply wrong!
>

The principal value is not always the correct solution to a physics or
engineering problem. It depends on what mathematical idealization produces
the singularity and why you find yourself attempting to integrate through
it. Sometimes the correct solution is to take a path around the
singularity in the complex plane.

--
| John Doty		"You can't confuse me, that's my job."
| Home: jpd at w-d.org
| Work: jpd at space.mit.edu

```

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