Re: [Axiom-developer] Integral challenge

[Richard Harke]

On Thu December 22 2005 05:54, M. Edward (Ed) Borasky wrote:
> There are unanswered questions about the hydrogen atom in 2005? :)
>
> Seriously, though, have you thrown this at Derive? An HP or TI handheld?
> Those pesky "student grade" tools sometimes surprise people. I've
> actually got all three, so I could do it if you haven't. Have you tried
> Reduce?
I didn't really expect this to work but I tried anyway on a
HP 28C. But as the manual says " It can return an exact inegral
(indefinite) of an expression that is a polynomial in the variable
of integration. Otherwise it can give an approximation based on
the Taylor series. But when I tried it, I just got "out of memory"
(Degree of polynomial set to 3)

Richard harke

>
> Bob McElrath wrote:
> >I don't know whether to file this as a bug or not, since it seems the
> >solution to this integral is not known.  (None of Maple, Mathematica,
> >Maxima can do it, and in addition it does not seem to be in G&R)
> >
> >So, any integral experts want to take a crack at it?  In the problem I
> >am interested in, a is real and a>0, b is real and 0 <= b <= 1, and I
> >need the integral over theta=0..2*Pi, in case the definite integral is
> >easier.
> >
> >(3) -> integrate(exp(-a*sqrt(1-b*cos(theta))), theta)
> > 3) ->
> >
> >   >> Error detected within library code:
> >
> >   Function not supported by Risch d.e.
> >
> >This has arisen in a quantum mechanical calculation involving the
> >Hydrogen atom.
> >
> >Since the integrand is well-behaved, I have chosen to do this
> >numerically for my paper, however an analytic solution would be
> >desirable.  I'm having to carry out a 3-dimensional numeric integral
> >using the Cuhre method because the above integral is not known, and the
> >second of the three integrals is over things inside 'b' above, and is in
> >general quite complicated.  The third is even more nasty, and I think
> >I'd have to do numerically anyway.
> >
> >But, I'm surprised to find that such a simple integral has no known
> >solution.
> >
> >--
> >Cheers,
> >Bob McElrath [Univ. of California at Davis, Department of Physics]
> >
> >    "In science, 'fact' can only mean 'confirmed to such a degree that it
> >    would be perverse to withhold provisional assent.' I suppose that
> >    apples might start to rise tomorrow, but the possibility does not
> > merit equal time in physics classrooms." -- Stephen Jay Gould (1941 -
> > 2002)
> >
> >
> >------------------------------------------------------------------------
> >
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