ELLIPE

The complete elliptic integral of the second kind is a special function that appears in arc-length and energy-style expressions involving elliptic geometry.

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

This wrapper evaluates E(m) for a scalar real parameter.

Excel Usage

=ELLIPE(m)

• m (float, required): Elliptic parameter (dimensionless).

Returns (float): Complete elliptic integral of the second kind at parameter m.

Example 1: Second-kind complete integral at zero parameter

Inputs:

m
0

Excel formula:

=ELLIPE(0)

Expected output:

1.5708

Example 2: Second-kind complete integral at half parameter

Inputs:

m
0.5

Excel formula:

=ELLIPE(0.5)

Expected output:

1.35064

Example 3: Second-kind complete integral near one

Inputs:

m
0.99

Excel formula:

=ELLIPE(0.99)

Expected output:

1.01599

Example 4: Second-kind complete integral at negative parameter

Inputs:

m
-0.5

Excel formula:

=ELLIPE(-0.5)

Expected output:

1.75177

Python Code

from scipy.special import ellipe as scipy_ellipe

def ellipe(m):
    """
    Compute the complete elliptic integral of the second kind.

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

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

    Args:
        m (float): Elliptic parameter (dimensionless).

    Returns:
        float: Complete elliptic integral of the second kind at parameter m.
    """
    try:
        return float(scipy_ellipe(m))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

m *

Elliptic parameter (dimensionless).