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MathGroup Archive 2005

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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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Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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