Re: Dirac-Delta function

From: A N Niel (anniel_at_nym.alias.net.invalid)
Date: 10/03/04 

Date: Sat, 02 Oct 2004 22:11:14 -0400

In article <nbKdnQmCfvd5xcLcRVn-uQ@rogers.com>, Adam <addam@rogers.com> wrote:

> Hi,
>
> I would like to know what mathematicians think of the Dirac-Delta function.
> Also, what the pure mathematics way of writing and defining it would be.

It is a "generalized function" or a "distribution". Mathematicians
cover these in a course known as "Functional Analysis". Maybe you
will take that course one day.

>
> This is how the delta function has been "defined."
> Let h denote the delta function.
> Integral(-infinity, +infinity)h(x)f(x)dx = f(0).
> Integral(-infinity, +infinity)h(x)dx = 1.
>
> I was used to reading things like: "Let f: R -> R denote a function defined
> by f(x) = x^2 for all x in R."
>
> The dirac-delta function doesn't really make any sense.

True. Perhaps you can convince the theoretical physicists to stop using it.
But I wouldn't count on it.

> I understand that
> can be thought of as the limit of ever increasing functions centered at the
> origin, but how do pure mathematicians define and describe it?
>
> Please provide a description like "h: R -> R" etc, if possible. The function
> seems very strange.
>
> Thanks, Adam.
>
>

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