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MathGroup Archive 2004

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Re: how to explain this weird effect? Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg46315] Re: how to explain this weird effect? Integrate
  • From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
  • Date: Fri, 13 Feb 2004 21:58:13 -0500 (EST)
  • References: <c0ftbt$c7p$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

nma124 at hotmail.com (steve_H) wrote:

You have gotten several responses showing how, rather than merely replacing
m or n by specific values, the results you desire can be obtained by taking
limits as m or n approach those values. However, your expectation that mere
replacement should give the desired results is _not unjustified_. First,
let's look at

In[1]:= Integrate[Sin[m*x]*Sin[n*x], {x, 0, 2*Pi}]

Out[1]= (n*Cos[2*n*Pi]*Sin[2*m*Pi] - m*Cos[2*m*Pi]*Sin[2*n*Pi])/(m^2 - n^2)

This expression may be rewritten as

  Sin[2*Pi*(m - n)]/(2(m - n)) - Sin[2*Pi*(m + n)]/(2(m + n))

{Perhaps a clever person could show how to get Mathematica to do that
rewriting, but I did it by hand.}

To avoid problems when we substitute using n = m or n = -m, we can again
rewrite the expression in terms of a function which is well known, but
(unfortunately?) not implemented in Mathematica. It's the sine cardinal
function, often abbreviated "sinc".

In[2]:= SinC[x_] := If[x == 0, 1, Sin[x]/x]

Using it, the result of the integration may then be written nicely as

#    Pi*(SinC[2*Pi(m - n)] - SinC[2*Pi(m + n)])

The above form is valid for all m and n; in other words, merely replacing
m and n by specific values, we always get the correct result for the
integral. In an ideal world, I think that SinC would be implemented in
Mathematica, which would then give #, rather than what it currently gives,
for Out[1].

David Cantrell

> I type:
>
> r = Integrate[Sin[m  x] Sin[n x], {x, 0, 2 Pi}]
>
> then I type
>
> r /. {n -> 2, m -> 2}
>
> I get error (1/0 expression encountered) and no result.
>
> but when I let m=2 and n=2 right into the integral first, it works:
>
> r = Integrate[Sin[2 x] Sin[2 x], {x, 0, 2 Pi}]
>
> and I get Pi as expected.
>
> I wanted to integrate this once, and try the output for different n,m.
>
> I did not think it will make a difference as to when I replace m and n
> by their numerical values, but Mathematica disagrees.
>
> I know Mathematica is correct in this, since it is clear from the result
> of the integration why I get 1/0. But it seems to me I should
> get the same result if I replace m,n inside the integral before
> the integration operation starts, or replace them afterwords.
>
> For example, when I type
>
> Integrate[Sin[m x], {x, 0, Pi}]
> % /. m -> 4
>
> I get zero.
>
> and when I replace m with 4 inside the integral first, I get the same
> result as above:
>
> Integrate[Sin[4 x], {x, 0, Pi}]
> 0
>
> So, what do you think? is there something I am missing here?
>
> thanks
> Steve


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