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MathGroup Archive 2006

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Re: Re: Delta function could not be got when delta function is the answer

  • To: mathgroup at smc.vnet.net
  • Subject: [mg64351] Re: [mg64342] Re: Delta function could not be got when delta function is the answer
  • From: Pratik Desai <pdesai1 at umbc.edu>
  • Date: Mon, 13 Feb 2006 03:15:12 -0500 (EST)
  • References: <dsk8cn$i6d$1@smc.vnet.net> <200602120900.EAA13299@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

albert wrote:

>PengYu.UT at gmail.com wrote:
>
>  
>
>>Sum[E^(-2*Pi*I*n*p*k), {n, -Infinity, Infinity}]
>>
>>The above summation should give an delta function. 
>>    
>>
>
>which is no function but a distribution (or generalized function)...
>
>  
>
>>However, 0 is given 
>>    
>>
>
>which is correct for almost every p and k :-)
>
>  
>
>>by Mathematica 5.0. Is it a bug. 
>>    
>>
>
>  
>
Check out the site
http://functions.wolfram.com/GeneralizedFunctions/DiracDelta/
and you can download a notebook with all kinds of info about DiracDelta, 
among which is this
DiracDelta[x] == Sum[HoldForm[E^(I*k*x)], {k, -Infinity, 
Infinity}]/(2*Pi) /; -2*Pi < x < 2*Pi
I added the holdform because mathematica automatically evaluates the sum 
to zero, I tried to use
2.5.2 Manipulating Sets of Transformation Rules of m-book but was not 
able to apply the transformation??

Hope this helps

Pratik

>depends on what you expect that mathematica can do. As far as I know, it can
>handle the delta function only to some extend in some functions like
>Integrate but does not claim to support distributions in general (see
>documentation for DiracDelta and 'Generalized Functions'). But maybe I am
>wrong here, I personally wouldn't expect mathematica to do too much useful
>stuff with DiracDelta especially if it is not in the realm of D,  Integrate
>and FourierTransform... 
>
>  
>
>>Is there any workaround to get the delta function?
>>    
>>
>
>from the sum you used as input: I don't know.
>if input doesn't matter try these :-): 
> D[UnitStep[x], x]
> FourierTransform[1/Sqrt[2 Pi], k, x]
>
>hth,
>
>albert
>
>  
>


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