Re: Simplifying Jacobian elliptic functions
- To: mathgroup at smc.vnet.net
- Subject: [mg55654] Re: Simplifying Jacobian elliptic functions
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Fri, 1 Apr 2005 05:36:31 -0500 (EST)
- Organization: The University of Western Australia
- References: <d2g5qo$f3h$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <d2g5qo$f3h$1 at smc.vnet.net>,
John Billingham <John.Billingham at Nottingham.ac.uk> wrote:
> I am doing a problem involving Jacobian elliptic functions, and am trying to
> use Mathematica to help. However, I find that I am unable to persuade
> Mathematica to simplify the expression
>
> JacobiDN[p, k^2]^2 + k^2 JacobiSN[p, k^2]^2
>
> which is equal to 1.
One approach is to use replacement rules.
EllipticRules = {
JacobiCN[u_, m_]^(p_ /; EvenQ[p]) -> (1 - JacobiSN[u, m]^2)^(p/2),
JacobiDN[u_, m_]^(p_ /; EvenQ[p]) -> (1 - m JacobiSN[u, m]^2)^(p/2)};
JacobiDN[p, k^2]^2 + k^2 JacobiSN[p, k^2]^2 /. EllipticRules
> It is also unable to integrate powers of Jacobian
> elliptic functions higher than 2, which are given by Byrd and Friedman in
> terms of elliptic functions and integrals.
It can integrate _some_ powers of Jacobian elliptic functions higher
than 2. For example,
Integrate[JacobiSN[u, m]^3, u]
Cheers,
Paul
--
Paul Abbott Phone: +61 8 6488 2734
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