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Re: Question: DiracDelta simplifies/integrates incorrectly?

• To: mathgroup at smc.vnet.net
• Subject: [mg65155] Re: [mg65132] Question: DiracDelta simplifies/integrates incorrectly?
• From: gardyloo <gardyloo at mail.wsu.edu>
• Date: Wed, 15 Mar 2006 23:59:29 -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
>
>
>   
   Perhaps this is overkill (more things get evaluated symbolically when
they don't need to be?), but is it possible to use an unassigned symbol,
like "n", in place of Pi, evaluate whatever you want, and then do a
replacement with result/.{n->Pi} ?


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