ELLIPEINC

The incomplete elliptic integral of the second kind integrates the second-kind elliptic integrand up to amplitude \phi.

E(\phi,m)=\int_0^{\phi}\sqrt{1-m\sin^2(t)}\,dt

This wrapper evaluates E(\phi,m) for scalar real inputs.

Excel Usage

=ELLIPEINC(phi, m)

• phi (float, required): Amplitude angle in radians.
• m (float, required): Elliptic parameter (dimensionless).

Returns (float): Incomplete elliptic integral of the second kind at amplitude phi and parameter m.

Example 1: Second-kind incomplete integral with zero amplitude

Inputs:

phi	m
0	0.5

Excel formula:

=ELLIPEINC(0, 0.5)

Expected output:

0

Example 2: Second-kind incomplete integral at moderate amplitude and parameter

Inputs:

phi	m
1	0.5

Excel formula:

=ELLIPEINC(1, 0.5)

Expected output:

0.92733

Example 3: Second-kind incomplete integral with negative amplitude

Inputs:

phi	m
-0.8	0.3

Excel formula:

=ELLIPEINC(-0.8, 0.3)

Expected output:

-0.776908

Example 4: Second-kind incomplete integral with negative parameter

Inputs:

phi	m
0.7	-0.4

Excel formula:

=ELLIPEINC(0.7, -0.4)

Expected output:

0.720225

Python Code

from scipy.special import ellipeinc as scipy_ellipeinc

def ellipeinc(phi, m):
    """
    Compute the incomplete elliptic integral of the second kind.

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

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

    Args:
        phi (float): Amplitude angle in radians.
        m (float): Elliptic parameter (dimensionless).

    Returns:
        float: Incomplete elliptic integral of the second kind at amplitude phi and parameter m.
    """
    try:
        return float(scipy_ellipeinc(phi, m))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

phi *

Amplitude angle in radians.

m *

Elliptic parameter (dimensionless).