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Re: Elliptic Integral problem with 5.0
*To*: mathgroup at smc.vnet.net
*Subject*: [mg46742] Re: Elliptic Integral problem with 5.0
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Fri, 5 Mar 2004 01:46:41 -0500 (EST)
*Organization*: The University of Western Australia
*References*: <c1onol$3cc$1@smc.vnet.net> <c238g6$igt$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Andreas Stahel <sha at hta-bi.bfh.ch> wrote:
> we just upgraded from version 4.2 to 5.0 (5.0.1)
> and it seems that an some integrals are not solvabla any more
> Here the input on the system
> $Version
> "5.0 for Sun Solaris (UltraSPARC) (November 26, 2003)"
>
> Integrate[Cos[phi]/(x^2 - 2*x*r*Cos[phi] + r^2 + z^2)^(3/2),
> {phi, 0, 2*Pi}]
>
> and the kernel just happily uses up CPU cycles with no result. I have
> not found any hint on similar problems on the wri site and google
> did not turn up information either.
I do not know why this integral is taking so long to compute. I don't
have the patience to wait for it to return, even if I set
SetOptions[Integrate, GenerateConditions -> False]
which should help.
However, observe that you can get the result for
int = Integrate[1/Sqrt[a - b Cos[p]], p]
quickly, and the associated definite integral is given by
defint = Simplify[Subtract @@ (int /. {{p -> 2Pi}, {p -> 0}})]
4 EllipticK[2 b/(b - a)]/Sqrt[a - b]
Since
2 D[1/Sqrt[a - b Cos[p]], b]
is just
Cos[p]/(a - b Cos[p])^(3/2)
the integral you are after can be computed by parametric differentiation
of defint:
FullSimplify[2 defint, b]
(4a EllipticE[2b/(b - a)] - 4(a + b) EllipticK[2b/(b - a)])/
(Sqrt[a - b] b (a + b))
where a = x^2 + r^2 + z^2 and b = 2 x r.
Clearly a > b since a - b = (x-r)^2 + z^2 > 0.
Cheers,
Paul
--
Paul Abbott Phone: +61 8 9380 2734
School of Physics, M013 Fax: +61 8 9380 1014
The University of Western Australia (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://physics.uwa.edu.au/~paul
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