> From: Richard Hennessy
> Sent: Monday, April 15, 2013 12:16 PM
> To: Rupert Swarbrick ; maxima at math.utexas.edu
> Subject: Re: [Maxima] DiracDelta>
> ... I could assume phi(x) can be any
> function but by way of approximates it is possible to show my answer is
> wrong for phi(x) = unit_step(x).
>
> "As far as I understand it, the whole point of these "idealised functions"
> (ie distributions) is to avoid the icky business of computing with the
> approximants in the first place."
>
phi(x) must be a "test" function in general but in phi(x) delta(x)the minimum requirement is mere continuity at x=0 as long as nofurther differentiation is going to be required.
At the risk of introducing even more confusion in the matter of the "function" H(x) delta(x) which is in fact the product of TWO generalized functions, may I draw the attention to the paper:
http://arxiv.org/ftp/math-ph/papers/0612/0612077.pdf
"Generalized functions as a tool for nonsmooth nonlinear problems in mathematics and physics" by J.F.Colombeau, script of a talk given to the American Math. Soc. (Jan. 2007).
In brief the Heaviside function H(x) is also a generalized function, fraught with rather well known paradoxes.
The simplest one is the following.
Let H(x) = 1 if x>0 and 0 if x<0 with H(0) undefined (if not just "multi-valued" or in view of some of its approximants: H(0)=1/2) and assume that its "derivative" with respect to x is delta(x). This assumption might be justified by considering the limits of operations made on regular functions idealized by H(x) and delta(x).
Now clearly H(x)^p = H(x) since 0^p=0 and 1^p=1 for any power p whatever, real or complex. But then by "differentiation" for any p=/=1:
p H(x)^(p-1) delta(x) = p H(x) delta(x) = delta(x) ,
a paradox!
The only way out is to postulate that somehow H(x)^p =/= H(x) for p=/=1 which means that even the Heaviside "function" does not behave at all as a regular function.
More generally in the theory of generalised functions it is found that forconsistency products are only possible with a single generalized functionas factor. To go further one needs some form of non-standard analysis.(H(x)^p=/=H(x) for p=/=1 and an "infinitesimal" x.)
Thanks to Daniel Lakeland to draw the attention to a less demanding form of non-standard analysis than the "classical" Robinson's one.
Beware of the formulae used in physics and engineering as they are restricted to idealizations of specific problems, a kind of (rather dangerous) abuse oflanguage and notation for (one hopes!) a correct limiting process which can seldom be applied more generally.
Indeed it leads to "garden varieties" of H(x), delta(x),etc... !
The short-hand notations used in these cases mask a variety of definitions which might surprise anyone not already familiar with these (risky) usages.
No way then "to avoid the icky business of computing with the approximants in the first place" otherwise this becomes like the mathematical equivalentof the biblical confusion of tongues (;-)
With CAS of course it is even more dangerous since different rules shouldapply to different contexts and applications.
Sorry if the above does not seem to be of any help and might appear pedantic.
Rene' J-M Grognard.