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Topic: Delta functions.
Replies: 14   Last Post: Apr 22, 2013 8:58 AM

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Posts: 1,245
Registered: 4/26/08
Re: Delta functions.
Posted: Apr 13, 2013 4:15 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply 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

> 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.


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