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Re: case differentiation problem with "Assumptions"

  • To: mathgroup at smc.vnet.net
  • Subject: [mg83681] Re: case differentiation problem with "Assumptions"
  • From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
  • Date: Wed, 28 Nov 2007 05:32:23 -0500 (EST)
  • References: <figumr$fnp$1@smc.vnet.net>

Paul.Wenk at googlemail.com wrote:
> I found the following problem using $Assumptions:
> Let us calculate the integral: Integrate[Cos[(n \[Pi] (-W + y))/(2
> W)]^2/W,{y,-W,W}]
> for n \[Element] Integers and n>0 you receive 1, for n=0 you receive 2.
> If you make the assumption
> $Assumptions = n \[Element] Integers
> then you get for the above integral the output 1, although for n=0
> (also integer) it should be 2!
> Why mathematica is not using a case differentiation in this case? How
> is the information about the integer element inserted?
>
> p.s.
> I'm using Mathematica 6.0.0 on Linux x86

I'm using version 5.2, but that may not make a difference.

Your observation seems to be that

In[7]:= Integrate[Cos[n Pi (-W + y)/(2 W)]^2/W, {y, -W, W},
Assumptions -> Element[n, Integers]]

Out[7]= 1

despite the fact that, if n == 0, the correct result is 2 instead.

Let's examine the source of this problem and a way in which this problem
could perhaps be eliminated in version 6.

Note that, without any assumption on n, we have

In[8]:= Integrate[Cos[n Pi(-W + y)/(2 W)]^2/W, {y, -W, W}]

Out[8]= 1 + Sin[2*n*Pi]/(2*n*Pi)

That result is valid for nonzero n, integer or not. But it fails if we
literally substitute 0 for n, giving Indeterminate. OTOH, if we take its
limit as n approaches 0, we do get the desired result, 2 .

Now, for In[8], there is a result which is _literally_ valid for all n:

1 + Sinc[2*n*Pi]

I have mentioned in this newgroup and others several times that it would be
useful to implement the sine cardinal function in computer algebra systems.
One use is illustrated above. And I see that version 6 now does implement
the sine cardinal function as Sinc. But perhaps, in version 6, In[8] still
produces Out[8]. If so, that would hardly be surprising, since Sinc was
such a recent addition. But I would hope that, eventually, Mathematica will
take full advantage of having Sinc implemented and that then the result of
In[8] would be 1 + Sinc[2*n*Pi].

Because it's probably related to why Out[7] was simply 1, also note that

In[9]:= Assuming[Element[n, Integers], Simplify[Sin[2 n Pi]/(2 n Pi)]]

Out[9]= 0

despite the fact that Sin[2 n Pi]/(2 n Pi) is literally Indeterminate
if n == 0.

And the above is probably related to

In[10]:= 0/x

Out[10]= 0

despite the fact that 0/x is literally Indeterminate if x == 0.

David W. Cantrell


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