ELLIPJ

Jacobi elliptic functions generalize circular and hyperbolic functions and are parameterized by an argument u and parameter m. This function returns the tuple \operatorname{sn}(u|m), \operatorname{cn}(u|m), \operatorname{dn}(u|m), and amplitude \phi.

The amplitude satisfies:

u = F(\phi,m),\qquad \operatorname{sn}(u|m)=\sin(\phi),\qquad \operatorname{cn}(u|m)=\cos(\phi)

This wrapper returns all four outputs in a single 2D row suitable for Excel spill output.

Excel Usage

=ELLIPJ(u, m)

u (float, required): Real argument of the Jacobi elliptic functions.
m (float, required): Elliptic parameter, typically in the range from zero to one.

Returns (list[list]): One-row array with sn, cn, dn, and amplitude ph in that order.

Example 1: Jacobi outputs at zero argument

Inputs:

u	m
0	0.5

Excel formula:

=ELLIPJ(0, 0.5)

Expected output:

Result
0	1	1	0

Example 2: Jacobi outputs with zero parameter

Inputs:

u	m
1	0

Excel formula:

=ELLIPJ(1, 0)

Expected output:

Result
0.841471	0.540302	1	1

Example 3: Jacobi outputs at moderate argument and parameter

Inputs:

u	m
1	0.5

Excel formula:

=ELLIPJ(1, 0.5)

Expected output:

Result
0.803002	0.595977	0.823161	0.932315

Example 4: Jacobi outputs with parameter close to one

Inputs:

u	m
1	0.99

Excel formula:

=ELLIPJ(1, 0.99)

Expected output:

Result
0.762448	0.64705	0.651526	0.867088

Python Code

from scipy.special import ellipj as scipy_ellipj

def ellipj(u, m):
    """
    Compute Jacobi elliptic functions sn, cn, dn and amplitude for scalar input.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.ellipj.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        u (float): Real argument of the Jacobi elliptic functions.
        m (float): Elliptic parameter, typically in the range from zero to one.

    Returns:
        list[list]: One-row array with sn, cn, dn, and amplitude ph in that order.
    """
    try:
        sn, cn, dn, ph = scipy_ellipj(u, m)
        return [[float(sn), float(cn), float(dn), float(ph)]]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

u *

Real argument of the Jacobi elliptic functions.

m *

Elliptic parameter, typically in the range from zero to one.