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MathGroup Archive 2006

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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[61]:=
Simplify[Pi*DiracDelta[Pi*x]]

Out[61]=
DiracDelta[x]

In[60]:=
Simplify[Pi*DiracDelta[Pi*(x-3)]]

Out[60]=
\[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[68]:=
Clear[f];

In[69]:=
Integrate[f[x]*Pi*DiracDelta[Pi*x],{x,-Infinity,Infinity}]

Out[69]=
f[0]

In[70]:=
Integrate[f[x]*Pi*DiracDelta[Pi*(x-3)],{x,-Infinity,Infinity}]

Out[70]=
0

As you can see, the output [69] is correct, but the output [70] should
correctly be f[3], 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[78]:=
Simplify[
  FourierTransform[Exp[I*2*Pi*3*x],x,f,
    FourierParameters\[Rule]{0,-2*\[Pi]}]
  ]

Out[78]=
\[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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