Re: Interaction of Sum/Plus and KroneckerDelta

*To*: mathgroup at smc.vnet.net*Subject*: [mg55261] Re: Interaction of Sum/Plus and KroneckerDelta*From*: Maxim <ab_def at prontomail.com>*Date*: Thu, 17 Mar 2005 03:31:27 -0500 (EST)*References*: <d192un$ngm$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

On Wed, 16 Mar 2005 10:48:55 +0000 (UTC), Ofek Shilon <ofek at simbionix.com> 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 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 > You might want to try PiecewiseSum from http://library.wolfram.com/infocenter/MathSource/5117/ . As mentioned in piecewise.nb, sometimes the best way is to evaluate a sum with symbolic limits and substitute numerical values later: In[51]:= s = PiecewiseSum[KroneckerDelta[i, j], {i, n}] Out[51]= If[1 <= j && j <= n, 1 - Ceiling[j] + Floor[j], 0] In[52]:= Refine[s /. n -> 5, Element[j, Integers] && 0 < j < 3] Out[52]= 1 Or even In[53]:= Refine[s, Element[{j, n}, Integers] && 0 < j < n + 1] Out[53]= 1 A more convoluted way would be to do In[54]:= Simplify[ Refine[Sum[KroneckerDelta[i, j], {i, 5}], #]& /@ Reduce[Element[j, Integers] && 0 < j < 3]] Out[54]= 1 Finally, you can rewrite the sum as In[55]:= Sum[DiscreteDelta[i - j], {i, n}] Out[55]= UnitStep[-1 + j]*UnitStep[-j + n] but notice that the result is correct only for integer j (PiecewiseSum gives a result identical to s). Maxim Rytin m.r at inbox.ru