scipy.special.ellipj#

scipy.special.ellipj(u, m, out=None) = <ufunc 'ellipj'>#

Jacobi elliptic functions.

Calculates the Jacobi elliptic functions of parameter m less than or equal to 1, and real argument u.

Parameters:

uarray_like: Argument.
marray_like: Parameter.
outtuple of ndarray, optional: Optional output arrays for the function values

Returns:

sn, cn, dn, ph4-tuple of scalar or ndarray

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/

Examples

The elliptic sine sn(u|m) interpolates between the sine function and the hyperbolic tangent when m changes from 0 to 1.

>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> from scipy.special import ellipj
>>> u = np.linspace(0, 2*np.pi, 100)
>>> fig, ax = plt.subplots()
>>> ax.plot(u, np.sin(u), '--', label='sin(u)')
>>> for m in (0.2, 0.8, 0.99):
...     ax.plot(u, ellipj(u, m)[0], label=f'sn(u|{m})')
>>> ax.plot(u, np.tanh(u), '--', label='tanh(u)')
>>> ax.set_xlabel('u')
>>> ax.legend(loc='lower left')
>>> plt.show()

../../_images/scipy-special-ellipj-1_00_00.png

Like for sine and cosine, the squares of elliptic sine and elliptic cosine add up to one.

>>> u = np.linspace(0, 5, 11)
>>> sn, cn, _, _ = ellipj(u, 0.7)
>>> sn**2 + cn**2
array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.])