Re: Interaction of Sum/Plus and KroneckerDelta

*To*: mathgroup at smc.vnet.net*Subject*: [mg55244] Re: [mg55178] Interaction of Sum/Plus and KroneckerDelta*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Thu, 17 Mar 2005 03:30:15 -0500 (EST)*References*: <200503161035.FAA23756@smc.vnet.net> <a7454fce3ae1c6ad98c491a9b18cb3ab@mimuw.edu.pl>*Sender*: owner-wri-mathgroup at wolfram.com

On 16 Mar 2005, at 19:11, Andrzej Kozlowski wrote: > 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 Ã¢Ë?Ë? 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/ > Actually, I was not quite correct. Let again KD[i_, j_] := Piecewise[{{1, i == j}}] This does not work as one might have hoped: Simplify[KD[1, j] + KD[2, j] + KD[3, j] + KD[4, j] + KD[5, j], j â?? Integers && 1 <= j <= 5] Piecewise[{{1, j == 1 || j == 2 || j == 3 || j == 4 || j == 5}}] but this: Simplify[KD[1, j] + KD[2, j] + KD[3, j] + KD[4, j] + KD[5, j], Or @@ Thread[j == Range[5]]] 1 does and to me seems the best solution to your problem. Andrzej Kozlowski

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