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.

```

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