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

marray_like

Parameter.

uarray_like

Argument.

Returns

sn, cn, dn, phndarrays

The returned functions:

sn(u|m), cn(u|m), dn(u|m)

The value ph is such that if u = ellipkinc(ph, m), then sn(u|m) = sin(ph) and cn(u|m) = cos(ph).

See also

ellipk

Complete elliptic integral of the first kind

ellipkinc

Incomplete elliptic integral of the first kind

Notes

Wrapper for the Cephes [1] routine ellpj.

These functions are periodic, with quarter-period on the real axis equal to the complete elliptic integral ellipk(m).

Relation to incomplete elliptic integral: If u = ellipkinc(phi,m), then sn(u|m) = sin(phi), and cn(u|m) = cos(phi). The phi is called the amplitude of u.

Computation is by means of the arithmetic-geometric mean algorithm, except when m is within 1e-9 of 0 or 1. In the latter case with m close to 1, the approximation applies only for phi < pi/2.

References

1

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/