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Re: Elliptic integral and function
- From: Liam Healy <Liam dot Healy at nrl dot navy dot mil>
- To: Brian Gough <bjg at network-theory dot co dot uk>
- Cc: Liam Healy <Liam dot Healy at nrl dot navy dot mil>,gsl-discuss at sources dot redhat dot com
- Date: Wed, 19 Dec 2001 15:12:29 -0500 (EST)
- Subject: Re: Elliptic integral and function
- References: <15390.8891.712402.651008@shadow.nrl.navy.mil><15391.46588.180743.817113@debian>
- Reply-to: Liam Healy <Liam dot Healy at nrl dot navy dot mil>
>>>>> "Brian" == Brian Gough <bjg@network-theory.co.uk> writes:
Brian> Liam Healy writes:
>> My understanding is that the Jacobi elliptic function is the inverse
>> of the elliptic function. That is,
>> sn(K(k),k) = 1
>> cn(K(k),k) = 0
>> dn(K(k),k) = sqrt(1-k^2)
>> see http://mathworld.wolfram.com/JacobiEllipticFunctions.html
>>
Brian> Hi,
Brian> Using the conventions in the GSL manual the relation is,
Brian> sn(K(k),k^2) = 1
Brian> cn(K(k),k^2) = 0
Brian> dn(K(k),k^2) = sqrt(1-k^2)
Brian> which should work correctly. I think there is a note about the
Brian> different notations used by Carlson and Abramowitz&Stegun somewhere in
Brian> the chapter there.
You're absolutely right, I had overlooked the m (where m=k^2). And it
is documented, if a bit obscurely, "The Jacobian Elliptic functions are
defined in Abramowitz & Stegun, Chapter 16." so one has to hunt down
A&S for the definition and see how they've defined the arguments.
Thank you for solving this mystery.
Liam