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

• To: mathgroup at smc.vnet.net
• Subject: [mg92030] Re: [mg91997] Re: Apparent error integrating product of DiracDelta's
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Thu, 18 Sep 2008 06:10:08 -0400 (EDT)
• References: <gag2lg$39k$1@smc.vnet.net> <gal3ht$dv1$1@smc.vnet.net> <200809162324.TAA24634@smc.vnet.net>

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, Infinity}=
> ]
>>> 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 there 
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

• References:
  • Re: Apparent error integrating product of DiracDelta's
    • From: magma <maderri2@gmail.com>