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MathGroup Archive 2006

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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg64393] Re: [mg64322] Delta function could not be got when delta function is the answer
  • From: Devendra Kapadia <dkapadia at wolfram.com>
  • Date: Wed, 15 Feb 2006 03:32:10 -0500 (EST)
  • References: <200602110832.DAA18329@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On Sat, 11 Feb 2006, 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. However, 0 is given
> by Mathematica 5.0. Is it a bug. Is there any workaround to get the
> delta function?
>
> Thanks,
> Peng
>
Hello Peng,

The answer 0 given by Sum for your example is incorrect.

The Sum function gives classical (non-distributional)
results for convergent sums. In the example sent by
you, the sum is computed by dividing the interval
(-Infinity, Infinity) into the sub-intervals
(-Infinity, 0) and (1, Infinity). However, there is
a failure to check convergence before applying this
procedure, leading to the incorrect result.

The convergence failure itself appears to be due
to the presence of the symbolic parameters 'p'
and 'k' in the input. As seen below, if we
give specific numerical values to these parameters
then Sum returns unevaluated.

===================================
In[1]:= $Version

Out[1]= 5.2 for Linux (June 27, 2005)

In[2]:= Sum[E^(-2*Pi*I*n*p*k), {n, -Infinity, Infinity}]

Out[2]= 0

In[3]:= p = 1;

In[4]:= k = 1;

In[5]:= Sum[E^(-2*Pi*I*n*p*k), {n, -Infinity, Infinity}]

Sum::div: Sum does not converge.

Sum::div: Sum does not converge.

              (-2 I) n Pi
Out[5]= Sum[E           , {n, -Infinity, Infinity}]

===========================================

I apologize for the confusion caused by this problem.

Sincerely,

Devendra Kapadia.
Wolfram Research, Inc.


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