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Elliptic integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67679] Elliptic integral
  • From: Ian Linington <i.e.linington at sussex.ac.uk>
  • Date: Tue, 4 Jul 2006 01:58:03 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Hello, does anybody know of a Mathematica program which converts an
elliptic integral of the form int z^n/sqrt(P(z)) dz, where P is a
fourth-order polynomial, into one of the standard forms?

The reason that I would like this is to try and evaluate

Integrate[
  Exp[q*I*x]/(Sqrt(-A*Exp[4*I*x] - I*B*Exp[3*I*x] + (C+I*D)*Exp[2*I*x]
  + I*B*Exp[I*x] - A)), {x, 0, 2*Pi},
  Assumptions -> {Element[q, Integers], A > 0, B > 0, C > 0, D > 0]

but Mathematica won't do it in one go, even if I choose specific values
for A, B, C, D and q. I believe that an analytic solution does exist,
but to decompose the integral in terms of Jacobi ellpitic functions
looks like a painful process.

Any help would be greatly appreciated!

Thanks,
Ian


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