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