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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 2:15 PM
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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?

I was a bit tooooo fast - so once again:

Maple's evaluations are consistent applications of the rule

f(x)*delta(x-a) -> f(a)*delta(x)

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.

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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