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Re: Re: Apparent error integrating product of DiracDelta's

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92098] Re: [mg92073] Re: Apparent error integrating product of DiracDelta's
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Sat, 20 Sep 2008 04:57:18 -0400 (EDT)
  • References: <gag2lg$39k$1@smc.vnet.net> <gal3ht$dv1$1@smc.vnet.net> <200809190916.FAA15073@smc.vnet.net>

magma wrote:
> On Sep 18, 12:10 pm, Daniel Lichtblau <d... at wolfram.com> wrote:
>> magma wrote:
>>> On Sep 15, 9:40 am, "Nasser Abbasi" <n... at 12000.org> wrote:
>>>> "Michael Mandelberg" <mmandelb... at comcast.net> wrote in message
>>>> news:gag2lg$39k$1 at smc.vnet.net...
>>>>> How do I get:
>>>>> Integrate[DiracDelta[z- x]   DiracDelta[z- y], {z-Infinity, Infinit=
> y}=
>>> ]
>>>>> to give DiracDelta[x-y] as the result?  Currently it gives 0.  I =
> ha=
>>> ve
>>>>> all three variable assumed to be Reals.  I am using 6.0.0.
>>>>> Thanks,
>>>>> Michael Mandelberg

[...snip my claim to the effect that the result is not defined at x==y]


> Very interesting answer!

Yes. I have to hope it was not so interesting as to have gone to the 
point of fiction.


> Yet, taking a naive (physicist's ?) point of view, the result seems
> intuitively correct.
> Mr. Delta himself (P.A.M. Dirac) in his "The principles of quantum
> mechanics" published in 1930 confirms the result (eq. 15.9 in the
> Italian 4th edition).
> He even "proves" it, a little bit later. The "proof" is a bit
> heuristics because he assumes the integral is defined in the first
> place.
> Yet the whole Delta function story was at the time just heuristics.
> Apparently the delta was described for the first time in this book
> (Wikipedia).
> I am also pretty sure he uses the result somewhere else later in the
> book (probably where double integrals appear).
> So, could he have been so wrong after having been so right before?

If it's a question of Dirac being wrong, or me, I'd put the money on me 
being wrong. That said, I do not know precisely what it was he claimed, 
or did, in this particular setting. It might be he used a shaky 
justification for an end result that was correct.


> More to the point: I found very intriguing your claim that " If
> different methods of approximation will lead to different results,
> then what you have cannot be defined".
> I agree with you of course, but could you "show" with Mathematica - maybe with
> a Manipulate program - that this is the case with this integral?

Illustrating pathologies with singular integrals is not one of my 
stronger points (which may be why I seem to get tangled in them). I need 
to give this some thought.


> Could you find 2 different approximation methods which give 2
> different results?
> I look forward to seeing it.
> 
> Concerning DiracDelta[x]^2, I would agree it is undefined at least
> because its integral, if it existed, would be DeltaDirac[0], which is
> undefined.

Daniel Lichtblau
Wolfram Research


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