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Jacobi Elliptic Function definition

• To: mathgroup at smc.vnet.net
• Subject: [mg68340] Jacobi Elliptic Function definition
• From: Roland Franzius <roland.franzius at uos.de>
• Date: Tue, 1 Aug 2006 07:00:18 -0400 (EDT)
• Organization: Universitaet Hannover
• Sender: owner-wri-mathgroup at wolfram.com

The JacobiAmplitude[x,k], JacobiSN[x,k] and related functions are 
defined in Mathematica by what is not the normal use of the modulus k

http://mathworld.wolfram.com/JacobiEllipticFunctions.html

Generally ist seems to be the case

cn(x,k) = JacobiCN[x,k^2]

and so on.

E.g the Talor series is

Series[JacobiSN[x, k^2], {x, 0, 5}] ->

SeriesData[x, 0, {1, 0, -1/6 - k^2/6, 0,
    1/120 + (7*k^2)/60 + k^4/120}, 1, 6, 1]

and the ODE for sn

sn'^2 =(1-sn^2)(1-k^2 sn^2)

which is now

D[JacobiSN[x, k], x]^2 /.
   {JacobiCN[x, k]^2 -> 1 - JacobiSN[x, k]^2,
    JacobiDN[x, k]^2 -> 1 - k*JacobiSN[x, k]^2}

results in

(1 - JacobiSN[x, k]^2)*(1 - k*JacobiSN[x, k]^2)



Does anybody know if this is a feature or a mistake? I don't find any 
literature using the Mathematica convention.

-- 

Roland Franzius