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Re: Assuming and Integrate

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  • Subject: [mg128362] Re: Assuming and Integrate
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Tue, 9 Oct 2012 00:41:08 -0400 (EDT)
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On Oct 8, 2012, at 2:29 AM, Roland Franzius <roland.franzius at uos.de> wrote:

> Am 07.10.2012 07:37, schrieb Murray Eisenberg:
>> How to treat exceptional cases, here m = n an integer, is always an issue with Integrate (and other operations). Here one would hope to see at least a ConditionalExpression for the general situation Element[{n,m}, Integers]. After all, such integrals are so common in Fourier analysis.
>>
>> Perhaps this should even be reported as a bug to Wolfram Research.
>>
>> On Oct 6, 2012, at 1:53 AM, hamiltoncycle at gmail.com wrote:
>>
>>> When I try the line below in Mathematica 8 I get the answer 0 which is what I expect when m and n are different but not when m=n. Can anyone explain how to do this correctly?
>>>
>>> Assuming[Element[{n, m}, Integers], Integrate[Sin[n*x]*Sin[m*x], {x, 0, Pi}]]
>>>
>>> When m=n we should get Pi/2, as in this case:
>>>
>>> Assuming[Element[n, Integers], Integrate[Sin[n*x]*Sin[n*x], {x, 0, Pi}]]
>>>
>
>
>
> For integration you have to use the specialized Assumptions option of
> Integrate.
>
> Integrate[Sin[n x] Sin[m x],{x,0,Pi},Assumptions->{n,m} \in Integers]
> is giving the correct Kronecker symbol 0 or +-pi/2 for for n=+-m.
>
> The reason will be that - with the exception of linear real
> substitutions - external assumptions cannot be translated to domain
> conditions in substitutions in the complex domains or through
> transcendental replacements of variables.
>
> Instead the integration machine has to plan ist table lookup or
> simplification and transformation process with the special domain
> assumptions to be designed by hand for an successful attempt.
>
>
> The half period sines are a complete basis of the Hilbert space on
> (0,Pi) but in Fourier analysis they are used only for zero boundary
> conditions. So the orthogonality relations are not so evident contrary
> to the every day formulas for standard periodic boundary conditions.
>

I don't know which version of Mathematica you're using, but in 8.0.4, one does NOT get any kind of conditional (hence correct) result from:

  Integrate[Sin[n x] Sin[m x],{x,0,Pi},Assumptions->{n,m}\[Element] Integers]

(Note the syntax correction from "\in" to "\[Element]" there.)

Rather, one gets the same "general" result as from the OP's Assuming expression, namely:

 (n Cos[n \[Pi]] Sin[m \[Pi]]-m Cos[m \[Pi]] Sin[n \[Pi]])/(m^2-n^2)

And as originally said, this is wrong when m = n.

---
Murray Eisenberg                                     
murray at math.umass.edu
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University of Massachusetts                  413 545-2859 (W)
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