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Re: Interaction of Sum/Plus and KroneckerDelta

• To: mathgroup at smc.vnet.net
• Subject: [mg55237] Re: [mg55178] Interaction of Sum/Plus and KroneckerDelta
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Thu, 17 Mar 2005 03:29:39 -0500 (EST)
• References: <200503161035.FAA23756@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

On 16 Mar 2005, at 11:35, Ofek Shilon wrote:

> I'm fighting Mathematica 5.1.0 to perform a (seemingly) elementary
> simplification, and Mathematica - so far - wins, so i thought i'd
> consult some veterans.
> Here's a simplified example of the problem.
>
> type:
> Sum[KroneckerDelta[i, j], {i, 1, 5}]
>
> and you get:
> KroneckerDelta[1, j] + KroneckerDelta[2, j] +
>   KroneckerDelta[3, j] + KroneckerDelta[4, j] + KroneckerDelta[5, j]
>
> which i want to simplify to 1.  The direct approach:
> Simplify[%, Assumptions -> {j â?? Integers, 0 < j < 3}]
>
> still gives:
> KroneckerDelta[1, j] + KroneckerDelta[2, j]
>
> Can Mathematica somehow automatically transform this to 1?
> modification of the original sum are welcome too, of course.
>
>   thanks for any ideas,
>
>     Ofek Shilon
>
>
>

I think the closest you can get to the answer you want is by using 
Matheamtica 5.1's new function  Piecewise. You have to define your own 
KroneckerDelta:

KD[i_, j_] := Piecewise[{{1, i == j}}]

and then use Simplify:

In[2]:=
Simplify[KD[1, j] + KD[2, j] +
    KD[3, j] + KD[4, j] + KD[5, j]]

Out[2]=
Piecewise[{{1, j == 1 ||
      j == 2 || j == 3 ||
      j == 4 || j == 5}}]


Unfortunately the answer is a Piecewise object rather than one and 
including assumption sin Simplify such as 1<=j<=5 does not help.


Andrzej Kozlowski
Chiba, Japan
http://www.akikoz.net/andrzej/index.html
http://www.mimuw.edu.pl/~akoz/

• References:
  • Interaction of Sum/Plus and KroneckerDelta
    • From: "Ofek Shilon" <ofek@simbionix.com>