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## [Axiom-developer] Integral challenge

 From: Bob McElrath Subject: [Axiom-developer] Integral challenge Date: Wed, 21 Dec 2005 23:12:28 -0800 User-agent: Mutt/1.5.11

```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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