Re: Elliptic integral
- To: mathgroup at smc.vnet.net
- Subject: [mg67888] Re: Elliptic integral
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Tue, 11 Jul 2006 05:59:19 -0400 (EDT)
- Organization: The University of Western Australia
- References: <e8d1a7$6mu$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <e8d1a7$6mu$1 at smc.vnet.net>,
Ian Linington <i.e.linington at sussex.ac.uk> wrote:
> 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]
Sqrt() is not the correct syntax. Instead, you need
f[a_, b_, c_, d_][x_] = 1 / 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]
> but Mathematica won't do it in one go, even if I choose specific values
> for A, B, C, D and q.
I don't have the patience to see.
> I believe that an analytic solution does exist,
> but to decompose the integral in terms of Jacobi ellpitic functions
> looks like a painful process.
The result will be truly horrendous, and will be expressed in terms of
explicit roots of a 4-th order polynomial. If you change variables,
x -> Log[y]/I
Exp[q I x] f[a,b,c,d][x] Dt[x]/Dt[y] /. x -> Log[y]/I // Simplify
the integrand becomes
(-I) y^(q - 1) /
Sqrt[-(a (1 + y^4)) + y ((c + I d) y - I b (-1 + y^2))]
and it is clear that this will lead to elliptic integrals of the first
kind (EllipticF), as can be seen by computing the explicitly factored
form,
Integrate[1/Sqrt[(y - r[1]) (y - r[2]) (y - r[3]) (y - r[4])], y]
where the roots, r[i], correspond to the four roots of
-(a (1 + y^4)) + y ((c + I d) y - I b (-1 + y^2)) == 0
But will this be useful? You will still need to compute very complicated
EllipticF with complex coefficients involving Root objects.
So, what is your goal? I should mention that, essentially, you are
computing Fourier coefficients:
c[q_] := Integrate[ Exp[q I x] f[a,b,c,d][x], {x, 0, 2 Pi}]
A good approximation to the c[q] can be computed quite efficiently using
Fourier:
With[{n = 200}, (1/Sqrt[n])
Fourier[ Table[ f[1, 2, 3, 4][x], {x, 0, 2Pi - Pi/n, 2 Pi/n}]]]
Increasing n improves the quality of the approximation.
Cheers,
Paul
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
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