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

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