Re: [Axiom-mail] Defining piece-wise functions and drawing, integrating, ...

[Raymond E. Rogers]

Ralf Hemmecke wrote:
> On 06/03/2007 07:30 PM, Raymond E. Rogers wrote:
>> Ralf Hemmecke wrote:
>>> Agreed, but what about telling Axiom the following:
>>>
>>> (1) -> g(x)==if x>0 then x else -x
>>>                                 Type: Void
>>>
>>> (
>> I am sorry for being a nuisance but I would like to point out:
>> This function can be expressed as
>> g(x)=  -x +2*DD^2
>> Where DD is the second order of the Dirac Delta function;  and DD^2 =
>> int(int(DD*1)) , a ramp starting at 0.  The point is that piecewise
>> polynomials  can be handled as generalized functions.
>> The integral int(g(x),a,b) would be [-(x^2/2)+DD^3]^b_a
>>  I am retiring soon and perhaps will implement the useful parts of using
>> the Dirac Delta function in Axiom.
>> Ray Rogers
>
> I don't know whether one can easily explain to a not so well trained
> user what a "generalised function" is and that instead of
Roughly generalized functionals can be taken as the limit of the
sequence of functions that are constrained.  For instance a sequence of
Gaussian curves that shrink in width a have an area of one.  It turns
out the constraints, smaller and smaller footprints, define the limit
more than the particular functional sequence.  This is similar to
considering the real numbers being the limit of sequences of rational
numbers.  You could consider a particular real number as the limit of a
lot rational series; for instance subsequent approximations in different
base systems.  Generalized functions have properties that are  useful in
a lot of ways.  For instance a piecewise polynomial can be represented
like I showed.  In the case of Electrical Engineering the representation
can be directly Fourier transformed and written down by inspection.
>
> g(x)==if x>0 then x else -x
>
> he/she should rather write
>
> g(x)=  -x +2*DD^2,
Because it's a "true" function that you can integrate and differentiate
by standard rules; independent of the number of terms.
The only problem is in multiplication, D*D, and such.  But there is a
mathematical interpretation that apparently also allows this, but so far
I would say "don't try this at home".   I hope to understand this soon. 
Why use it?  Well in EE land it's hard to start and stop things
symbolically without step functions.  Differentiating and integrating
step functions drag in the rest of the standard generalized functions. 
As is typical there are two methods of definition.  Extrinsic and
intrinsic definitions.   That is to say direct construction or by
properties.  I like and I think symbolic algebra programs are more
comfortable with intrinsic definitions.  That is to lay down a set of
operational rules and the using them like a machine.   As I said, I and
not 100% clear on how to do this, but am sure it can be done.

Ray Rogers