[25] in mathematical software users group
RE: A formalism for discontinuities of fucntions ?
daemon@ATHENA.MIT.EDU (daemon@ATHENA.MIT.EDU)
Sat Mar 28 09:30:22 1992
Date: Sat, 28 Mar 1992 09:11:44 EST
From: jim@fpr.com (James E. O'Dell)
Reply-To: jim@fpr.com
To: msug@Athena.MIT.EDU
On Fri, 27 Mar 92, Reid M. Pinchback <reidmp@Athena.MIT.EDU> wrote:
>
>>Please tell me the result for the following integration by Maple and Mma:
>>
>>inte(abs(x^3), x from -2 to 2)
>>
>>inte(sgn(x)*x^3, x from -2 to 2)
>
>I don't have Mma handy on this machine, but I'll tell you the results
>for Maple.
>
>I guess that your "sgn" function extracts the sign of a number.
>To the best of my knowledge, there is no *exact* equivalent of this in
>Maple unless you "make it yourself". There is a "sign" function, but
>what it does is extract the sign of the leading coefficient of an
>expression, so "sign(x)" is always 1, since the leading coefficient (1)
>of the polynomial is non-negative.
>
>Maple can numerically integrate the "abs(x^3)" case (with a result of 4),
>but it cannot symbolically integrate it. This is because there isn't a
>*single* Maple mathematical function or expression that can be used to
>express a closed-form solution. What is needed is to split the integral
>into 2 at the point x=0, and symbolically integrate the left and right
halves.
>
>You don't run into this problem when you try to integrate expressions like
>abs(x^n) where n is "even" for the obvious reasons, ie: abs(x^(2*k)) =
x^(2*k).
>For non-even exponents, Maple doesn't really examine the limits of
integration
>until it first generates the symbolic solution.
>
>Actually, to be accurate, I think the process that Maple goes through for
>symbolic integration is a mathematically strict one:
>
> 1. Test whether the function is continuous or not over the interval
defined
> by the limits of integration.
>
> 2. If continuous, it then generates the general symbolic solution.
>
> 3. It then substitutes the endpoints into the general solution to
> evaluate the integral.
>
>Now, if a function is found continuous in step 2, Maple will look for a
>***general*** closed-form solution, without consideration of the limits
>of integration. This is probably why Maple couldn't evaluate abs(x^3),
>since it knows of no such ***general*** solution. (1/4)sgn(x)x^4 would
>probably count as such a solution, if Maple knew about the sgn function,
>but alas it does not.
>
>If a function is *not* found to be continuous, then I think Maple chooses
the
>mathematically conservative approach and doesn't perform the integration.
>This would be because a discontinuity is often the result of a limit
>not existing near some point, and so the "proper" thing to do
mathematically
>is to go back to first principles and express the integral as a limit and
>try and find the limit. In calculus courses we sometimes get into the bad
>habit of abusing notation and saying something like
"int(1/x,x=1..infinity)"
>when we should be saying "limit as c -> infinity of int(1/x,x=1..c)".
>When a function contains a discontinuity, then sometimes this latter
>expression suggests to us how we need to go about finding the integral
>(if it exists).
=============
This is pretty much the approach that ALJABR takes. I suppose that Macsyma
still does also since the intanalysis switch mentioned by Jeff controls
whether this analysis is performed.
The one trick that ALJABR can also do is to find "Principal Value"
integrals.
This is doen by looking for poles, dividing the integral in to regions
around the poles and examining the limiting behavior of the sum of the
integrals of the segments.
Jim
