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Re: Apparent error integrating product of DiracDelta's

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92080] Re: Apparent error integrating product of DiracDelta's
  • From: Michael Mandelberg <mmandelberg at comcast.net>
  • Date: Fri, 19 Sep 2008 05:17:50 -0400 (EDT)
  • References: <gag2lg$39k$1@smc.vnet.net> <gal3ht$dv1$1@smc.vnet.net>

Hmmm.

I appreciate (mostly) the niceties of this argument, but what about
the practical matter that as an intermediate step in some problem I
may want to have some DiracDelta's floating around prior to a final
integration?

Michael

On Sep 18, 6:10 am, 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
> >> I think you have synatx error in the limit part. I assume you mean to =
wri=
> > te
> >> {z, -Infinity,Infinity}
>
> >> Given that, I think zero is the correct answer.  When you multiply 2=
 de=
> > ltas
> >> at different positions, you get zero. Integral of zero is zero.
>
> >> Nasser
>
> > No Nasser, the correct value of the integral should be DiracDelta[x-
> > y], as Michael said.
> > This value is indeed 0 if x != y but it is not 0 if x==y.
>
> It is not 0 at x==y, but neither is it DiracDelta[x-y]. The value the=
re
> is undefined.
>
> > Mathematica correctly calculates:
>
> > Integrate[f[z - x] DiracDelta[z - y], {z, -Infinity, Infinity},
> >  Assumptions -> y \[Element] Reals]
>
> > as
>
> > f[-x + y]
>
> This is making a tacit assumption that f is a "nice" function. Nice, in
> this context, means it is an element of Schwartz space S: C^infinity and
> vanishing faster than any polynomial at +-infinity. DiracDelta, suffice
> it to say, is not an element of S (it's not even a function).
>
> > However it fails to recognize that if f[z-x] is replaced by
> > DiracDelta[z-x], the result should be
>
> > DiracDelta[-x + y]
>
> > or the equivalent
>
> > DiracDelta[x - y]
>
> This is not a failure but rather an active intervention.
>
> > In the help file, under "possible issues" it is mentioned that
> > "Products of distributions with coinciding singular support cannot be
> > defined:"
>
> This is a statement of mathematics and not specific to Mathematica.
>
> > So perhaps at the moment the only way to do the integral is:
>
> > Integrate[f[z - x] DiracDelta[z - y], {z, -Infinity, Infinity},
> >   Assumptions -> y \[Element] Reals] /. f -> DiracDelta
>
> > hth
>
> Here is a general rule of thumb. If you are working with DiracDelta
> function(al)s, instead approximate them as ordinary functions. If
> different methods of approximation will lead to different results, then
> what you have cannot be defined. One can use this notion to see that,
> for example, DiracDelta[x]^2 is not defined.
>
> Daniel Lichtblau
> Wolfram Research- Hide quoted text -
>
> - Show quoted text -



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