       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
School of Physics, M013                         Fax: +61 8 6488 1014
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```

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