       Question: DiracDelta simplifies/integrates incorrectly?

• To: mathgroup at smc.vnet.net
• Subject: [mg65132] Question: DiracDelta simplifies/integrates incorrectly?
• From: John Harker <harker at me.rochester.edu>
• Date: Wed, 15 Mar 2006 06:29:26 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Hello,

I have a question regarding the behavior of the DiracDelta function in
Mathematica 5.2.

The following two inputs produce the following results:

In:=
Simplify[Pi*DiracDelta[Pi*x]]

Out=
DiracDelta[x]

In:=
Simplify[Pi*DiracDelta[Pi*(x-3)]]

Out=
\[Pi] DiracDelta[\[Pi] (-3+x)]

As you can see, although the Pi is correctly simplified out in the first
case, it is not simplified in the second case.  This is a problem because
of the following result:

In:=
Clear[f];

In:=
Integrate[f[x]*Pi*DiracDelta[Pi*x],{x,-Infinity,Infinity}]

Out=
f

In:=
Integrate[f[x]*Pi*DiracDelta[Pi*(x-3)],{x,-Infinity,Infinity}]

Out=
0

As you can see, the output  is correct, but the output  should
correctly be f, and instead it returns 0.

Is there a flaw in my understanding of the DiracDelta function, or is this
a bug?

All of the above poses a problem because Mathematica will return results
such as the following:

In:=
Simplify[
FourierTransform[Exp[I*2*Pi*3*x],x,f,
FourierParameters\[Rule]{0,-2*\[Pi]}]
]

Out=
\[Pi] DiracDelta[(-3+f) \[Pi]]

So you see that just by asking for a simple Fourier transform, I can get
an output result which Mathematica cannot integrate correctly.

Does anyone have any ideas about a better way to perform this math
in order to get around the problem?  Or something illuminating about how
the DiracDelta function works?

Many thanks!

John

```

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