Inline comment.
From: Ren? Grognard
Sent: Thursday, July 18, 2013 4:14 AM
To: maxima at math.utexas.edu
Cc: Rene Grognard
Subject: Re: [Maxima] pw.mac problem
IF I correctly understand the code
> integrate(f(x) * pwdelta(x-a, n), x);
> sum((-1)^k*binomial(n,k)*at(diff(f(x),x,k),x=a) * pwdelta(x-a,n-k-1),k,0,n)
I think that it is not entirely correct.
Given the space D of indefinitely continuously differentiable functions
with bounded support, the Dirac delta is a functional from D to R, the space of real numbers.
For all phi in D with support including x=0,
the Dirac delta can be represented by the set of formal integrals
int_R phi(x) delta(x) dx = phi(0).
Using formal integration by parts the derivative delta' can be defined as
int_R phi(x) delta'(x) dx = - int_R phi'(x) delta(x) dx = - phi'(0),
and by repeating the operation n times:
int_R phi(x) delta^(n) (x) dx = (-1)^n \phi^(n) (0). (*)
OK. As long as the function f is such that the product f phi remains in D (thus the Heaviside
function H(x) := 1 if x >0 else 0, is excluded; a fortiori 1/x) then
Why are you explicitly excluding all discontinuous functions? Obviously if you do that, you have to lose the ?1 in the pwdelta(x-a, n ? k ? 1) function and change the range of k. One of the uses of pwdelta() is to introduce a discontinuity in the anti-derivative, thus you get the heaviside function or one of its relatives.
int_R phi(x) f(x) delta^(n) (x) dx = (-1)^n [(d/dx)^n (f phi)]_{x=0}
and by the Leibniz rule
(d/dx)^n (f phi) = sum_{k=0}^n C(n,k) f^(k) phi^{n-k},
where C(n,k) is the binomial coefficient.
Thus, splitting the factor (-1)^n,
int_R phi(x) f(x) delta^(n) (x) dx =
sum_{k=0}^n C(n,k) (-1)^k f^(k)(0) (-1)^{n-k) phi^{n-k}(0),
or, returning to integrals using (*) above
= int_R phi(x) [sum_{k=0}^n C(n,k) (-1)^k f^(k)(0) delta^{n-k}(x)] dx,
which shows that as generalised function
f(x) delta^(n) (x) = sum_{k=0}^n C(n,k) (-1)^k f^(k)(0) delta^{n-k}(x).
Thus the code above AS I understand it should be:
> integrate(f(x) * pwdelta(x-a, n), x);
> sum((-1)^k*binomial(n,k)*at(diff(f(x),x,k),x=a) * pwdelta(x-a,n-k),k,0,n)
For n=0 this formula is trivial: f(x) delta(x) = f(0) delta(x)
The next non trivial case states
f(x) delta'(x) = f(0) delta'(x) - f'(0) delta(x).
To check directly a formula assumed valid for such generalised functions
just follow the fundamental rule. On the left side:
\int_R phi(x) f(x) delta'(x) dx =
- (d/dx)(phi f)_{x=0} = -phi'(0)f(0) - phi(0) f'(0),
On the right side:
f(0) \int_R phi(x) delta'(x) dx = f(0) [-\phi'(0)]
and
- f'(0) \int_R phi(x) delta(x) dx = - f'(0) phi(0).
All problems with Dirac delta and other generalised functions
arising for instance from singular integrals is that one take
unwarranted short cuts.
Reference: Gel'fand I.M. & Shilov G. E.
Generalized Functions, Vol. I: Properties and Operations,
Academic Press: 1964.
(Note to the list manager:
I tried as much as possible to format the reply to make
it more or less legible only using "plain text" for the
emails (avoiding "rich text" --- or GmailLaTeX !)
but for some reason which escapes me even this formatting,
using line returns and blank lines for legibility, is completely
removed from the reply as in comes back to me in maxima digests.
My apologies if that is delivered that way but I seem to
have no control whatever on the format.)
Rene' Grognard
> ------------------------------
> Message: 7
> Date: Wed, 17 Jul 2013 17:03:53 -0400
> From: "Richard Hennessy" <rich.hennessy at verizon.net>
> Subject: Re: [Maxima] pw.mac problem
> Hi List,
>
> Someone pointed out that there is an error in pw.mac, for integration of delta functions and their derivatives.
>
> Currently,
>
> integrate(f(x)*diff_pwdelta(m, x-a), x);
> gives the wrong answer.
>
> This person says it should be.
>
> integrate(f(x) * pwdelta(x-a, n), x);
> sum((-1)^k*binomial(n,k)*at(diff(f(x),x,k),x=a) * pwdelta(x-a,n-k-1),k,0,n)
>
> where the two argument form of pwdelta(x,n) = diff_pwdelta(n,x) and pwdelta(x,0) = pwdelta(x) and pwdelta(x,-1)=unit_step(x).
--------------------------------------------------------------------------------
_______________________________________________
Maxima mailing list
Maxima at math.utexas.edu
http://www.math.utexas.edu/mailman/listinfo/maxima