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
> 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

--
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School of Physics, M013                         Fax: +61 8 9380 1014
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Crawley WA 6009                      mailto:paul at physics.uwa.edu.au
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```

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