Re: Jacobi Elliptic Function definition

*To*: mathgroup at smc.vnet.net*Subject*: [mg68360] Re: Jacobi Elliptic Function definition*From*: "did" <didier.oslo at hotmail.com>*Date*: Wed, 2 Aug 2006 05:24:25 -0400 (EDT)*References*: <eanenb$aae$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Roland Franzius skrev: > 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 As far as I know, there are two traditional ways of defining the Jacobian elliptic functions. One involving the modulus k, the other one involving the parameter m=k^2. The first one is more common in, e.g., Russian, German and French literature, the second one in the Anglo-Saxon literature (e.g. Abramowitz & Stegun). A good reference is the book by Byrd & Friedman.