Re: Re: Integrate question
- To: mathgroup at smc.vnet.net
- Subject: [mg82480] Re: [mg82474] Re: Integrate question
- From: DrMajorBob <drmajorbob at bigfoot.com>
- Date: Sun, 21 Oct 2007 05:11:10 -0400 (EDT)
- References: <20071020082209.763$0x_-_@newsreader.com> <13065124.1192936090864.JavaMail.root@m35>
- Reply-to: drmajorbob at bigfoot.com
> principal value sense. But this actually confirms the other point I
> made: that one cannot exclude the possibility that there is a sense
> in which the answer 1/16 is "acceptable". However, and this is the
> key point, the fact that such a sense exists is only a curiosity
> unless this "sense" fits the intended purpose of this "integration".
I gotta agree with Andrzej, and I'll repeat my earlier comment that you
can make the integral anything you want by approaching the pole
(in)appropriately from left and right.
If you want the result to be Pi, choose epsilon1 so that the right
integral is Pi/2 + n and epsilon2 so that the left integral is Pi/2 - n.
e1 and e2 are not equal, but they both tend to zero as n -> Infinity, and
the partial integral is Pi for every n.
A version of this is precisely what happened in the change of variables:
unequal epsilons (original problem) mapped to equal epsilons (new
problem), so the new problem has a finite Cauchy principal value when the
original did not. I suspect there's a change of variables that
accomplishes the same trick for ANY integrand whose antiderivative tends
to +Infinity on one side of the pole and -Infinity on the other.
Integrals that aren't invariant (even as to the question of convergence)
under a change of variables?
Yikes. Have fun with that.
Bobby
On Sat, 20 Oct 2007 21:21:08 -0500, Andrzej Kozlowski <akoz at mimuw.edu.pl>
wrote:
> You are right of course, I should have checked it "by hand" (as I did
> in my original response) before writing it was not true in the Cauchy
> principal value sense. But this actually confirms the other point I
> made: that one cannot exclude the possibility that there is a sense
> in which the answer 1/16 is "acceptable". However, and this is the
> key point, the fact that such a sense exists is only a curiosity
> unless this "sense" fits the intended purpose of this "integration".
> This is why I am usually not very impressed when someone asks a
> question to which, in the intended context, the answer is "no" and
> then someone writes to point out that there is some other context in
> which the answer is "yes". There is a countless number of situations
> of this kind and it is easy to multiply examples. You can take almost
> any theorem of elementary analysis and find that in some more
> abstract context it is not valid. The first example that comes to my
> mind (just as I am writing this): the familiar and even "obvious"
> fact that a smooth function is continuous is not true in the context
> of analysis on locally convex spaces more general than Banach spaces
> (the Frolicher-Kriegl theory). But this is completely irrelevant to
> anyone who never deals with anything other than finite dimensional
> euclidean spaces. Similarly, the fact that this particular integral
> (after the change of variable) has Cauchy Principal value 1/16 is
> probably irrelevant - "probably" because the OP never told us what
> sort of "integral" he had in mind.
>
> Andrzej Kozlowski
>
>
> On 20 Oct 2007, at 21:22, David W. Cantrell wrote:
>
>> [Message also posted to: comp.soft-sys.math.mathematica]
>>
>> Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
>>> It represents the same mathematica object, so you will get the sam=
e
>>> answer:
>>>
>>> Integrate[1/6/u^3, {u, -1, 2}]
>>> Integrate::idiv:Integral of 1/u3 does not converge on {-1,2}. >>
>>> Integrate[1/(6*u^3), {u, -1, 2}]
>>>
>>> What else did you expect? It's the same and therefore 1/16 it is
>>> equally "unacceptable" or "non-acceptable" as an answer as before.
>>
>> Of course, you are correct in saying that 1/16 is unacceptable for
>> Integrate[1/6/u^3, {u, -1, 2}] since nothing other than Mathematica's=
>> standard integration was specified. But curiously, there is a sense
>> (see
>> below) in which 1/16 is now the correct answer, even though it
>> wasn't for
>> the original integral in terms of x.
>>
>>> The integral does not converge in any usual mathematical sense : as
>>> Riemann or Lebesgue integral or even in the sense of "Principal
>>> value".
>>
>> No, it _does_ converge in the sense of Cauchy principal value. Of
>> course,
>> as I noted before, PrincipalValue in the current version of
>> Mathematica
>> doesn't handle anything but _simple_ poles. But the integral in
>> terms of
>> u converges to 1/16 in the sense of Cauchy principal value,
>> although the
>> original integral in terms of x did not:
>>
>> In[11]:= Simplify[Assuming[0 < e < 1,
>> Integrate[1/(6*u^3), {u, -1, -e}] + Integrate[1/(6*u^3), {u, e, 2}]]]=
>>
>> Out[11]= 1/16
>>
>> David
>>
>>
>>> On 19 Oct 2007, at 18:09, Oskar Itzinger wrote:
>>>
>>>> Would setting
>>>>
>>>> u=3 x^2 - 1
>>>>
>>>> and integrating
>>>>
>>>> (1/6)/u^3 on [-1,2]
>>>>
>>>> be acceptable?
>>>>
>>>> /oskar
>>>>
>>>> "Oskar Itzinger" <oskar at opec.org> wrote in message
>>>> news:ff77b9$ncr$1 at smc.vnet.net...
>>>>> Hmm, from Mathematica 2.1 Help:
>>>>>
>>>>> Integrate can evaluate definite integrals whenever the correct
>>>>> result can be found by taking limits
>>>>> of the indefinite form at the endpoints.
>>>>>
>>>>> ?
>>>>>
>>>>> /oskar
>>>>>
>>>>> "Andrzej Kozlowski" <akoz at mimuw.edu.pl> wrote in message
>>>>> news:ff4hpp$k0e$1 at smc.vnet.net...
>>>>>>
>>>>>> On 16 Oct 2007, at 16:28, Oskar Itzinger wrote:
>>>>>>
>>>>>>> Mathematica 5.2 under IRIX complains that
>>>>>>>
>>>>>>> Integrate[x/(3 x^2 - 1)^3,{x,0,1}]
>>>>>>>
>>>>>>> doesn't converge on [0,1].
>>>>>>>
>>>>>>> However, Mathematica 2.1 under Windows gives the corrrect answer=
,
>>>>>>> (1/16).
>>>>>>>
>>>>>>> When did Mathematica lose the ability to do said integral?
>>>>>>>
>>>>>>> Thanks.
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>
>>>>>>
>>>>>> The reason is that Mathematica 2.1 was wrong and Mathematica
>>>>>> 5.2 is
>>>>>> much more careful and right. What Mathematica 2.1 did here was
>>>>>> simply:
>>>>>>
>>>>>> Subtract @@ (Integrate[x/(3 x^2 - 1)^3, x] /. {{x -> 1}, {x ->
>>>>>> 0}})
>>>>>> 1/16
>>>>>>
>>>>>> in other words, it applied the Newton-Leibnitz rule in a mindless=
>>>>>> way. Later versions are more intelligent and see that the
>>>>>> singularity at
>>>>>> =CE=B1 = Last[x /. Solve[3*x^2 - 1 == 0, x]]
>>>>>> 1/Sqrt[3]
>>>>>>
>>>>>> One can also see this graphically (of course!):
>>>>>>
>>>>>> Plot[x/(3 x^2 - 1)^3, {x, 0, 1}]
>>>>>>
>>>>>>
>>>>>> the integral still might exist in the sense of Cauchy
>>>>>> PrincipalValue
>>>>>> but we see that it does not:
>>>>>>
>>>>>> Integrate[x/(3*x^2 - 1)^3, {x, 0, 1}, PrincipalValue -> True]
>>>>>> Integrate::idiv:Integral of x/(3 x^2-1)3 does not converge on
>>>>>> {0,1}. >>
>>>>>> Integrate[x/(3*x^2 - 1)^3, {x, 0, 1}, PrincipalValue -> True]
>>>>>>
>>>>>> If you still don't beleive it, you can do it "by hand":
>>>>>>
>>>>>> int = FullSimplify[Integrate[x/(3*x^2 - 1)^3,
>>>>>> {x, 0, 1/Sqrt[3] - =CE=B5}] +
>>>>>> Integrate[x/(3*x^2 - 1)^3,
>>>>>> {x, 0, 1/Sqrt[3] + =CE=B5}], =CE=B5 > 0]
>>>>>> (9*=CE=B5^2*(3*=CE=B5^4 - 8*=CE=B5^2 + 5) - 4)/
>>>>>> (18*(3*=CE=B5^3 - 4*=CE=B5)^2)
>>>>>>
>>>>>> Limit[int, =CE=B5 -> 0]
>>>>>> -=E2=88=9E
>>>>>>
>>>>>>
>>>>>> Andrzej Kozlowski
>
>
>
-- =
DrMajorBob at bigfoot.com