"Richard Hennessy" <rich.hennessy at verizon.net> writes:
> It seems consistent.
> If x=0 then
>
> (x*delta(x))/x = 0/x = 0/0=?und.
> x*(delta(x)/x) = 0*(?inf/0) = 0*?inf=?und.
> x/x*delta(x) = ?und*?inf = ?und.
>
> if x#0 then
> (x*delta(x))/x = (x*0)/x = 0/x=0.
> x*(delta(x)/x) = x*(0/x) = 0.
> x/x*delta(x) = 1*0 = 0.
>
> I thought delta(x) was only defined inside of an integral? I am
> confused how it is possible to make statements about delta(x) outside
> an integral.
Well, Maxima can't (yet). Mathematicians can. delta is properly defined
as a functional from (for example) the space of bounded smooth functions
on R. That's just a linear map and it's perfectly fine to talk about
such things.
Of course, you can also define delta to be the limit of a sequence of
functions that get narrower and higher. For example, I could take
delta[n] to be a box of height 2n with support [-1/n, 1/n]. Then what
should (x*delta(x))/x be? I'd better make sure that the result is
independent of the approximating functions, of course.
Well, I need to be a bit more careful still. You see, this is supposed
to be a distribution. I know how to multiply distributions by functions,
but 1/x does not describe a function R \to R, since it isn't defined at
zero. That's fine though. Let's say that
r(x) = { 1/x x ? 0
{ a x = 0
for some number a. So now I can ask what x*delta(x)*r(x) is.
Let's look at the approximating functions. But this is easy! You see,
integral(x*delta[n](x)*r(x), x) = integral(delta[n](x), x)
because the two functions are actually equal at every point except
possibly x=0 (when x*delta[n](x)*r(x) = 0 and delta[n](x) probably
doesn't equal zero).
Ahah! So as long as I make sense of x*delta(x)/x as a functional
carefully, the question of what "x*delta(x)/x" means is easy to
answer. Notice
(1) It was important to remember that this was supposed to be an
expression describing a functional. So dividing by x doesn't make
sense. The only thing that does make sense is multiplying by a
function, which is why we had to talk about r(x).
(2) It does *not* make sense to ask what the "value" of x*delta(x)/x
is at a given point. Indeed, the only way to do that for an
arbitrary distribution is to try to integrate it against a
delta. This may or may not be defined. So your table trying to
work out what it is for varying x is doomed to failure.
(3) How to deal with such questions programmatically is "left as an
exercise"... Definitely, one will have to store more information
about a distribution than just a bare expression in x.
Rupert
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