Re: Question: DiracDelta simplifies/integrates incorrectly?

• To: mathgroup at smc.vnet.net
• Subject: [mg65161] Re: [mg65132] Question: DiracDelta simplifies/integrates incorrectly?
• From: Bruce Miller <brucem at wolfram.com>
• Date: Wed, 15 Mar 2006 23:59:36 -0500 (EST)
• References: <200603151129.GAA23260@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Mr. Harker,

The problem is when DiracDelta has a complicated argument (i.e.,
more than one operation).  Examples are below.
A correction for this is in the works for a future release.

In[1]:= \$Version

Out[1]= 5.2 for Mac OS X (June 20, 2005)

In[2]:= Integrate[f[x]*Pi*DiracDelta[Pi*x],{x,-Infinity,Infinity}]

Out[2]= f[0]

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

Out[3]= 0

In[4]:= Integrate[f[x]*Pi*DiracDelta[(x-3)],{x,-Infinity,Infinity}]

Out[4]= Pi f[3]

In[5]:= Integrate[f[x]*Pi*DiracDelta[Pi*x-Pi*3],{x,-Infinity,Infinity}]

Out[5]= 0

Bruce Miller
Wolfram Research

On Mar 15, 2006, at 5:29 AM, 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

```

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