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Re: Question: DiracDelta simplifies/integrates incorrectly?

• To: mathgroup at smc.vnet.net
• Subject: [mg65152] Re: [mg65132] Question: DiracDelta simplifies/integrates incorrectly?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Wed, 15 Mar 2006 23:59:26 -0500 (EST)
• References: <200603151129.GAA23260@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

John Harker wrote:
> 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

Your second integral example

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

should certainly return f[3], and the failure to do so is a bug. This 
will be fixed in a future release (the next one).

Daniel Lichtblau
Wolfram Research

• References:
  • Question: DiracDelta simplifies/integrates incorrectly?
    • From: John Harker <harker@me.rochester.edu>