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MathGroup Archive 2006

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Re: Please help overcome problems with integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72448] Re: Please help overcome problems with integration
  • From: "David W. Cantrell" <DWCantrell at sigmaxi.net>
  • Date: Thu, 28 Dec 2006 05:15:55 -0500 (EST)
  • Organization: NewsReader.Com Subscriber
  • References: <200612151206.HAA00084@smc.vnet.net> <em39l4$7bb$1@smc.vnet.net> <emb59i$9mi$1@smc.vnet.net>

"dimitris" <dimmechan at yahoo.com> wrote:
> Maybe the following link will help you a little
>
> http://www.cs.cmu.edu/~adamchik/articles/mier.htm

The first part of that article addresses the problem, yes. But some
comments about that part of the article seem in order.

Adamchik begins by mentioning

Integrate[(x^2 + 2x + 4)/(x^4 - 7x^2 + 2x + 17), {x, 0, 4}]

and then notes that the corresponding indefinite integral

int = Integrate[(x^2 + 2x + 4)/(x^4 - 7x^2 + 2x + 17), x]

can be expressed in terms of elementary functions. Simplified (which was
not done in his article), we get

int = ArcTan[(1 + x)/(4 - x^2)].

That antiderivative has jump discontinuities at x = +/- 2 and so, of
course, we cannot _naively_ use it with the Fundamental Theorem to evaluate
the definite integral previously mentioned.

Adamchik then shows how int can be used with the Fundamental Theorem to
evaluate the definite integral, namely,

Limit[int, x -> 4, Direction ->  1] -
Limit[int, x -> 2, Direction -> -1] +
Limit[int, x -> 2, Direction ->  1] -
Limit[int, x -> 0, Direction -> -1]

and states that "Mathematica evaluates definite integrals in precisely that
way." [I was quite surprised by that statement. Is that always the case?]

A little later, he says "The origin of discontinuities along the path of
integration is not in the method of indefinite integration but rather in
the integrand." If that were true, since the integrand cannot be changed,
it would seem that there would be absolutely no way to avoid
discontinuities along the path of integration. But that is simply not the
case! Here, for example, is an antiderivative which is _continuous along
the whole real line_:

In[37]:= contint =
ArcTan[(4x + Sqrt[2(15 + Sqrt[241])])/(2 - Sqrt[2(-15 + Sqrt[241])])] +
ArcTan[(4x - Sqrt[2(15 + Sqrt[241])])/(2 + Sqrt[2(-15 + Sqrt[241])])];

While it might be difficult to use Mathematica to find contint, it is at
least easy to use Mathematica to confirm that it is an antiderivative:

In[38]:= Simplify[D[contint, x]]
Out[38]= (4 + 2*x + x^2)/(17 + 2*x - 7*x^2 + x^4)

Adamchik mentions next that the four zeros of the integrand's denominator
are two complex-conjugate pairs having real parts +/- 1.95334. It then
seems that he is saying that, connecting these conjugate pairs by vertical
line segments in the complex plane, we get two branch cuts... But didn't
the relevant branch cuts for his int cross the real axis at x = +/- 2,
rather than at x = +/- 1.95334? (Note: The difference between 1.95334 and 2
is not due to numerical error.) Exactly what's going on here?

Later on, he says "Thinking hard, we can build an antiderivative that does
not have a branch cut crossing a given interval of integration. However, we
can never get rid of branch cuts!" I agree. However, integration along the
real line is often so useful that it would be nice to be able to ask
Mathematica, as an option, to give an antiderivative (such as my contint
above) which is continuous along the real line.

In the last part of that section, Adamchik notes that "...Integrate may not
be able to detect all singular points on the interval of integration, which
will result in a warning message..." But it's also worth noting that such a
failure may occur _without_ any warning message. A recent example was
posted to this newsgroup by Dimitris in the thread "integrate"; the problem
with his last item seems to be caused by branch cuts of EllipticF[z, m].

David W. Cantrell


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