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MathGroup Archive 2006

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Re: Question: DiracDelta simplifies/integrates incorrectly?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65162] Re: [mg65132] Question: DiracDelta simplifies/integrates incorrectly?
  • From: John Harker <harker at me.rochester.edu>
  • Date: Wed, 15 Mar 2006 23:59:38 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

On Wed, 15 Mar 2006, Daniel Lichtblau wrote:

>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
>

Thanks!  It's good to know it's a bug and not my imagination.  :-)

In the meantime, for my specific purposes I did figure out an acceptable
workaround.  My mathematical problem was written in the form

FourierTransform[Exp[I*2*Pi*3*x],x,f,FourierParameters->{0,-2*\[Pi]}]

And so I would get the unusable output

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

I decided that I could simplify by rewriting my problem so it could be
written in the form

FourierTransform[Exp[I*3*x],x,f,FourierParameters->{-1,-1}]

which produces the "correct" and useful result

DiracDelta[-f + n]


Thanks again!

John


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