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

• To: mathgroup at smc.vnet.net
• Subject: [mg56297] Re: (x-y) DiracDelta[x-y] does not simplify to 0
• From: Alain Cochard <alain at geophysik.uni-muenchen.de>
• Date: Thu, 21 Apr 2005 05:36:16 -0400 (EDT)
• References: <d42kg5\$39t\$1@smc.vnet.net> <d45agf\$ieu\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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

```

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