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

• To: mathgroup at smc.vnet.net
• Subject: [mg56233] Re: [mg56198] (x-y) DiracDelta[x-y] does not simplify to 0
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 20 Apr 2005 05:30:06 -0400 (EDT)
• References: <200504190854.EAA02509@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

On 19 Apr 2005, at 17:54, Alain Cochard wrote:

>
> Mathematica 4.0 for Linux
> Copyright 1988-1999 Wolfram Research, Inc.
>  -- Motif graphics initialized --
>
> Considering that
>
>      In[1]:= FullSimplify[x DiracDelta[x]]
>
>      Out[1]= 0
>
> I was surprised about this one:
>
>      In[2]:= FullSimplify[(x-y) DiracDelta[x-y]]
>
>      Out[2]= (x - y) DiracDelta[x - y]
>
> whereas:
>
>      In[3]:= FullSimplify[(x-y) DiracDelta[x-y]/.x-y->z]
>
>      Out[3]= 0
>
> Is it simply that Mathematica is a little weak on Out[2], or does it have a
> good reason for not simplifying, i.e., am I missing something at the
> mathematics level?
>
> Thanks in advance,
> Alain
>
>
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.




Andrzej Kozlowski
Chiba, Japan
http://www.akikoz.net/andrzej/index.html
http://www.mimuw.edu.pl/~akoz/

• References:
  • (x-y) DiracDelta[x-y] does not simplify to 0
    • From: Alain Cochard <alain@geophysik.uni-muenchen.de>