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MathGroup Archive 2005

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Re: Re: (x-y) DiracDelta[x-y] does not simplify to 0

  • To: mathgroup at smc.vnet.net
  • Subject: [mg56369] Re: [mg56297] Re: (x-y) DiracDelta[x-y] does not simplify to 0
  • From: "Wolf, Hartmut" <Hartmut.Wolf at t-systems.com>
  • Date: Fri, 22 Apr 2005 06:25:52 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

>-----Original Message-----
>From: Alain Cochard [mailto:alain at geophysik.uni-muenchen.de] 
To: mathgroup at smc.vnet.net
>Sent: Thursday, April 21, 2005 11:36 AM
>Subject: [mg56369] [mg56297] Re: (x-y) DiracDelta[x-y] does not simplify to 0
>
>yehuda ben-shimol writes:
>
> > As I remember, DiracDelta is singular and has a meaning only under
> > integration.  Anyway the properties of the DiracDelta are kept by
> > Mathematica i.e., Integrate[(x - y)DiracDelta[x - y], {x, -1, 1},
> > {y, -1, 1}] returns 0 as expected
>
>Andrzej Kozlowski writes pretty much the same:
>
> > On the one hand I think the Mathematica implementation of DiracDelta

> > (and KroneckerDelta) leaves a lot to be desired... and that is
putting 
> > it mildly. (That means I have plenty of much worse examples...).
> > 
> > On the other hand, I am not convinced that Mathematica ought to
perform 
> > this sort of simplification at all.  DiracDelta is a generalised 
> > function.  The statement x DiracDelta[x] == 0 needs a lot of 
> > interpreting to make sense of (I prefer to think of it as nonsense).

> > However
> > 
> > 
> > Integrate[(x-y) DiracDelta[x-y], {x,-Infinity,Infinity}]
> > 
> > 0
> > 
> > is correct.
>
>I don't understand these reservations. I learned the statement x delta
>= 0 in my lectures on distributions at university, and I checked today
>that it also appears in one of Laurent Schwartz's ("father" of
>distribution theory) books.  Plus I find it very intuitive and it's
>straightforward to demonstrate.
>
>But anyway, I find in your responses the answer to my initial
>questions, and I thank you for your time.
>
>AC
>
>
>

A bit hesitatingly: The meaning of the distributions is defined by their
operation on a space of test functions. Sometimes the answers differ
according to these.  

Mathematica doesn't specify what is that test space for DiracDelta (or
any other dstribution), in fact it doesn't want to specify this, as to
either leave that up to the user (who is aware of it), or just proceed
in a hands-on, pragmatic way (with possible pitfalls).

Laurent Schwartz's test space are the C-infinite functions, zero outside
a compact region. Such clearly the Distribution x DiracDelta[x] is
identical to the regular distribution 0& (so to speak).


Perhaps someone liked to have (whether this makes any sense with a
closed theory or not)

In[14]:= f[x_,y_] := (1 + x + y + x^2)/(x - y)

In[15]:= Integrate[f[x,y]*(x - y)DiracDelta[x - y], {x, -Infinity,
Infinity}]
Out[15]= (1 + y)^2


Certainly it is wise not to mix up distributions with functions with
numbers. 

--
Hartmut Wolf


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