Re: Interaction of Sum/Plus and KroneckerDelta
- To: mathgroup at smc.vnet.net
- Subject: [mg55259] Re: Interaction of Sum/Plus and KroneckerDelta
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Thu, 17 Mar 2005 03:31:21 -0500 (EST)
- Organization: The University of Western Australia
- References: <d192un$ngm$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <d192un$ngm$1 at smc.vnet.net>, "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. But, as stated, Mathematica's answer is perfectly correct, whereas returning 1 would not be! > The direct approach: > Simplify[%, Assumptions -> {Element[j, Integers], 0 < j < 3}] > > still gives: > KroneckerDelta[1, j] + KroneckerDelta[2, j] Again, without a definite value for j, this is the correct answer. It should _not_ be transformed to one. Why? Because if you multiply this expression by a function depending on the parameter j, say (KroneckerDelta[1, j] + KroneckerDelta[2, j]) f[j] then which one of KroneckerDelta[1, j] and KroneckerDelta[2, j] evaluates to 1 effects the result. > Can Mathematica somehow automatically transform this to 1? > modification of the original sum are welcome too, of course. Explicit summation such as Sum[KroneckerDelta[i, j] KroneckerDelta[j, k], {j, 1, Infinity}] is essentially just a notational device: the Einstein summation convention, where summation over repeated indices is implied rather than explicitly stated, i.e. writing KroneckerDelta[i, j] KroneckerDelta[j, k] instead of the sum can often be used to simplify the algebra. The easiest way to reduce the above sum to KroneckerDelta[i, k] is to use pattern-matching. For example, KroneckerDelta[i, j] KroneckerDelta[j, k] /. KroneckerDelta[a_, b_] KroneckerDelta[b_, c_] :> KroneckerDelta[a,c] Interestingly, this convention translates into a general principle for analyzing and evaluating integrals and products of sums in Mathematica -- just focus on the summand (i.e., the general term of the sum) and drop the summation symbol. Cheers, Paul -- Paul Abbott Phone: +61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul
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