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MathGroup Archive 2003

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Re: Problem with Sum

  • To: mathgroup at smc.vnet.net
  • Subject: [mg44037] Re: [mg44004] Problem with Sum
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sat, 18 Oct 2003 03:12:45 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

On Friday, October 17, 2003, at 06:14 PM, Mariusz Jankowski wrote:

> Hello, can someone explain the following result:
>
> Sum[(-1)^m * (DiscreteDelta[1 - 2*m] - 2*DiscreteDelta[1 + 3*m]), {m, 
> -Infinity, Infinity}]
>
> gives I + 2 (-1)^(2/3)
>
> Shouldn't the sum evaluate to zero, since for any value of m the 
> summand is identically zero. Note that
>
> Sum[(-1)^m * (DiscreteDelta[1 - 2*m] - 2*DiscreteDelta[1 + 3*m]), {m, 
> -10, 10}] returns 0, as expected.
>
> This looks like a bug, is it?
>
> Thanks, Mariusz
>
>
>
>
Hello Mariusz,

It is difficult to say if such things should be considered bugs or not, 
since the answer does have a sensible interpretation. It is possible to 
interpret

In[1]:=
Sum[(-1)^m*(DiscreteDelta[1 - 2*m] -
     2*DiscreteDelta[3*m + 1]), {m, -Infinity, Infinity}]

Out[1]=
I + 2*(-1)^(2/3)


as meaning the same as:

In[2]:=
Integrate[(-1)^x*(DiracDelta[x - 1/2] -
     2*DiracDelta[x + 1/3]), {x, -Infinity, Infinity}]

Out[2]=
I + 2*(-1)^(2/3)

Of course if this is indeed what is done one would expect it to be 
documented somewhere!


Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/


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