scipy.special.ellipj¶

scipy.special.
ellipj
(u, m) = <ufunc 'ellipj'>¶ Jacobian elliptic functions
Calculates the Jacobian elliptic functions of parameter m between 0 and 1, and real argument u.
Parameters: m : array_like
Parameter.
u : array_like
Argument.
Returns: sn, cn, dn, ph : ndarrays
The returned functions:
sn(um), cn(um), dn(um)
The value ph is such that if u = ellipk(ph, m), then sn(um) = sin(ph) and cn(um) = cos(ph).
See also
ellipk
 Complete elliptic integral of the first kind.
Notes
Wrapper for the Cephes [R451] routine ellpj.
These functions are periodic, with quarterperiod on the real axis equal to the complete elliptic integral ellipk(m).
Relation to incomplete elliptic integral: If u = ellipk(phi,m), then sn(um) = sin(phi), and cn(um) = cos(phi). The phi is called the amplitude of u.
Computation is by means of the arithmeticgeometric mean algorithm, except when m is within 1e9 of 0 or 1. In the latter case with m close to 1, the approximation applies only for phi < pi/2.
References
[R451] (1, 2) Cephes Mathematical Functions Library, http://www.netlib.org/cephes/index.html