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Re: Delta functions.
Posted:
Apr 13, 2013 4:15 PM


clicliclic@freenet.de schrieb: > > Waldek Hebisch schrieb: > > > > Using delta functions can lead to problems with zero divisors. > > For example in Maple I get: > > > > > (x  4)*Dirac(x  4); > > (x  4) Dirac(x  4) > > > > > simplify((x  4)*Dirac(x  4)); > > 0 > > > > > (x  4)*Dirac(x  4)/(x4); > > Dirac(x  4) > > > > which is clearly inconsistent. I wonder I there is any theory > > how to avoid such problems? I mean, what CAS can do to > > protect users from wrong results? > > > > Maple's evaluations are consistent applications of the rule > > f(x)*delta(xa) > f(a)*delta(x)
arrrgh: f(x)*delta(xa) > f(a)*delta(xa)
> > which defines the meaning of a Dirac delta times a function that is > differentiable infinitely often. So the cofactor must not be split > into subfactors here.
In a few more words: Maple simplifies (x4) * 1/(x4) > 1; the second factor is not differentiable at x = 4, whereas the Maplesimplified product is. The validity (in Maple, as in most or all CAS) of this simplification is the source of the inconsistency: a nondifferentiable cofactor of a Dirac delta can vanish, and a meaningless product involving a delta thereby become meaningful. Note that in a meaningless product associativity doesn't hold! Strictly speaking, a product involving a Dirac delta must be declared meaningless if any one cofactor cannot be differentiated an infinite number of times. I suppose this can relaxed by checking the product of all cofactors for differentiability after its simplification (in Maple)  provided that associativity involving the Dirac delta is never assumed to hold until such test is passed.
> > The Dirac delta is a socalled "distribution"; these object are > defined via their action on certain spaces of "test functions". You > may for example refer to Constantinescu (1974), "Distributionen und > ihre Anwendungen in der Physik", but any other text on "distributions" > should do as well.
Martin.



