[Maxima] Howto simplify Integral with jacobi polynomials

andre maute andre.maute at gmx.de
Thu Jul 10 12:22:45 CDT 2008


On Thursday 10 July 2008, Raymond Toy (RT/EUS) wrote:
> andre maute wrote:
> > For the documentation, especially the orthopoly section,
> > Abramowitz & Stegun (A&S) is freely available
> > but Gradshteyn & Ryzhik and Merzbacher are not.
> >
> > Couldn't the documentation be a litlle bit more specific
> > at least for the nonfree citations?
> > Perhaps implementing the tables of A&S would help here also.


>
> I don't follow you.  What exactly are you suggesting here?
One point for a better documentation could also be to
have more functionality,

e.g.

1. L2-Norm (h_n in A&S),
2. leading coefficient (k_n in A&S)
3. rodriguez coeffcient (a_n in A&S) 
4. coefficients in a special linearcombination (d_n, c_m in A&S)
	these are tabulated on page 775
5. with the assumptions and declarations from my previous post i get
-----------------------------------------------------------------
(%i12) jacobi_p(k,a,b,-1)
(%o12) pochhammer(a+1,k)*('sum(pochhammer(-k,i)*pochhammer(k+b+a+1,i)
                                               *pochhammer(a+1,i)^-1*i!^-1
                                               *1^i,i,0,k))
------------------------------------------------------------------
there is a really nicer form for this one see page 777 

6. the coefficients of the second order ODE are missing (page 781 in A&S) 
7. the discrete orthogonal polynomials from A&S are missing

> I, for one, would like it if the documentation
> gave some definition for each of the orthogonal polynomials.
Exactly.
E.g.
If I want to verify a hand calculation with maxima, but don't know
what maxima uses as definitions for the orthogonal polynomial
maxima is not really helpful.

One could inspect the lisp code for orthopoly
but there a hypergeometric identity is used,
which I doubt is useful for the average user.
I also doubt that the average user is used to lisp.

But if you have the above coefficients you could guess
what definition maxima uses.

> The reference is nice, but, sometimes, I have 
> neither net access nor book access.
There is a single reference to Merzbacher,
which refers to the definition of the spherical harmonics.
nothing more.
A&S can be downloaded for offline access

Andre


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