firstname.lastname@example.org 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)/(x-4); > > 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(x-a) -> f(a)*delta(x)
arrrgh: f(x)*delta(x-a) -> f(a)*delta(x-a)
> > 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 (x-4) * 1/(x-4) -> 1; the second factor is not differentiable at x = 4, whereas the Maple-simplified product is. The validity (in Maple, as in most or all CAS) of this simplification is the source of the inconsistency: a non-differentiable 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 so-called "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.