ELLIPK

The complete elliptic integral of the first kind is a special function that appears in geometry, mechanics, and wave problems. It is defined over a parameter m and integrates over a quarter period.

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

This wrapper evaluates K(m) for a scalar real input using SciPy.

Excel Usage

=ELLIPK(m)

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

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

Example 1: First-kind complete integral at zero parameter

Inputs:

m
0

Excel formula:

=ELLIPK(0)

Expected output:

1.5708

Example 2: First-kind complete integral at half parameter

Inputs:

m
0.5

Excel formula:

=ELLIPK(0.5)

Expected output:

1.85407

Example 3: First-kind complete integral at quarter parameter

Inputs:

m
0.25

Excel formula:

=ELLIPK(0.25)

Expected output:

1.68575

Example 4: First-kind complete integral at negative parameter

Inputs:

m
-0.5

Excel formula:

=ELLIPK(-0.5)

Expected output:

1.41574

Python Code

from scipy.special import ellipk as scipy_ellipk

def ellipk(m):
    """
    Compute the complete elliptic integral of the first kind.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.ellipk.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 first kind at parameter m.
    """
    try:
        return float(scipy_ellipk(m))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

m *

Elliptic parameter (dimensionless).