Elliptic integral and function

[Liam Healy]

>>>>> "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