Re: Dirac-Delta function

From: David McAnally (D.McAnally_at_i'm_a_gnu.uq.net.au)
Date: 10/03/04 

Date: 3 Oct 2004 04:22:38 GMT

"Adam" <addam@rogers.com> writes:

>"A N Niel" <anniel@nym.alias.net.invalid> wrote in message
>news:021020042211149692%anniel@nym.alias.net.invalid...
>> 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.
>>
> I'm not sure what you mean.

Try reading the books on Generalized Functions by Gel'fand et al.

>It has been expressed multiple ways. In each
>course, it is taught slightly differently. I searched online and there are
>so many forms of it that it tends to make me think not so nice things about
>"it." I say "it" because it seems that many of the things that are said to
>be the same dirac delta function are not the same at all. Unless the
>"function" is pathological, lol.

The Dirac delta function is *defined* by Integral(-infinity,infinity)
delta(x) f(x) dx = f(0) for all test functions f(x). No other definition
is necessary, since the only significance that a generalized function has
is as a continuous linear functional on a space of test functions.

> There is also the kronecker delta function taught in relation to tensor
>notation.
> delta_a_b = { 1 if a = b, 0 if a != b } where a and b are einstein index
>notation indices.

> That seems to me to be the same object but for discrete cases. When
>continuous cases occur, the object becomes the dirac delta function.

> The dirac delta is said to have a limit over infinity of 1 but be
>infinite at 1 and 0 everywhere else, or some other nonsense.

That particular "definition" is not legitimate. The definition that I
gave above is the correct definition to take, i.e. (delta,f) = f(0) for
all test functions f. Various statements that are made about the delta
function should be treated more as hints as to its behaviour.

The statement that delta(x) = 0 for all nonzero x states that for any
nonzero number x_0, for any y > 0 such that y < |x_0|, and for any test
function f(x) such that f(x) = 0 outside the interval (x_0 - y,x_0 + y),
(delta,f) = 0.

The statement that the integral of delta(x) is 1, along with the previous
observation that delta(x) = 0 for all nonzero x, tells us that
(delta,f) = f(0). In other words, f(x) delta(x) = f(0) delta(x) for all
nonzero x since delta(x) = 0 at such x, and f(x) delta(x) = f(0) delta(x)
at x = 0, so that f(x) delta(x) = f(0) delta(x) for all x, and so
Integral f(x) delta(x) dx = Integral f(0) delta(x) dx = f(0). Of course,
such an argument is nonsense, and is only useful for *informally* thinking
about evaluating Integral f(x) delta(x) dx. For one thing, delta(x) is
not defined as an ordinary function, and delta(0) is certainly not
defined. For another, Integral delta(x) dx is not defined, since 1 is not
a test function.

The thing to remember is that delta(x) is not a function delta : R -> R,
but delta(x) is a function mapping from the space of test functions to R
(specifically, it evaluates the test function at x = 0, so
delta(f) = f(0)).

> I'm starting to think that this is some sort of object that when used in
>specific ways and specific situations, just happens to result in desired
>outcomes. It doesn't seem logical at all.

IF you regard the delta function as a linear function mapping the space of
test functions to R, then maybe the examples of its uses would start to
make more sense.

>> True. Perhaps you can convince the theoretical physicists to stop using
>> it.
>> But I wouldn't count on it.
>>
> More and more areas of physics are using it as time goes on. Dirac gave
>it to us for good. I'd rather take longer to do mathematics, but have an
>actual understanding of it and have it use common logic, than to use some
>strange thing in strange ways to save time or simplify notation. However,
>that's just me.

Try reading the books by Gel'fand et al on Generalized Functions. They
give all the rigourous basis that you need.

You will be alright with generalized functions so long as you always bear
in mind that they are actually linear functions from a space of test
functions to R.

David

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