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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