Re: Defining piece-wise functions and drawing, integrating, ...

Sumant S.R. Oemrawsingh wrote:
>
> Say, I wish to define a piece-wise function,
>
> (1) -> f(x|x<0)==-x**2
>                                                                    Type: Void
> (2) -> f(x)==x**2
>                                                                    Type: Void
> (3) -> draw(f(x),x=-1..1)
>    Compiling function f with type Variable x -> Polynomial Integer 
>   
As an aside I would like to point out that there is an analytic
extension from Non-Standard analysis (specifically the Dirac Delta
function) that provides a neat uniform framework to describing,
transforming, and manipulating piecewise polynomials. It consists of
simply writing out the waveforms in terms of the Dirac Delta function of
various orders. For instance a ramp that starts at zero, continues for
1.3 units and then goes back to zero is:
int(int(Dirac(x))-int(int(Dirac(x-1.5)) -1.5*int(delta(x-1.5)) or
d^2(x)-d^2(x-1.5)-1.5*d^1(x-1.5)
I haven't seen this approach properly implemented in CAS packages, and
don't know if it's of any use. I have used it to do Laplace and Mellin
transforms/analysis of discontinuous waveforms though; and found it
simplifies the calculations a lot.

Ray