[Date Index]
[Thread Index]
[Author Index]
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.
Prev by Date:
**webMathematica 2.3 now available**
Next by Date:
**NMinimize**
Previous by thread:
**Jacobi Elliptic Function definition**
Next by thread:
**Re: Using implicit information about row indices**
| |