Hi,
I have always thought that the main use of pwdelta() was to introduce a discontinuity in the ant-derivative of a function, then I found out about the derivatives of the pwdelta() function and the associated formulas and thought that their main use was to introduce discontinuities in the higher order anti-derivatives. Now I find out that this comes at a price. If you have a simple polynomial like x^2+x+1 and you want to put a discontinuity into its anti-derivative, then you can write (x^2+x+1)*pwdelta(x-2), but now the derivatives of the function do not stop at zero. Simple polynomials have a finite number of anti-derivatives and then you stop at zero. Not true for the modified polynomial with the pwdelta(). It has an infinite number of anti-derivatives and they are not trivial. The represent real curves on the Cartesian Coordinate system. I don?t pretend to understand why but this formula predicts this fact. If it is a fact, or is there a flaw in my reasoning?
Rich
> ------------------------------
> 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).
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