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 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul