ELLIPKINC

The incomplete elliptic integral of the first kind accumulates the first-kind integrand up to an amplitude \phi rather than over a full quarter period.

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

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

Excel Usage

=ELLIPKINC(phi, m)

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

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

Example 1: First-kind incomplete integral with zero amplitude

Inputs:

phi	m
0	0.5

Excel formula:

=ELLIPKINC(0, 0.5)

Expected output:

0

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

Inputs:

phi	m
1	0.5

Excel formula:

=ELLIPKINC(1, 0.5)

Expected output:

1.08322

Example 3: First-kind incomplete integral with negative amplitude

Inputs:

phi	m
-0.8	0.3

Excel formula:

=ELLIPKINC(-0.8, 0.3)

Expected output:

-0.824317

Example 4: First-kind incomplete integral with negative parameter

Inputs:

phi	m
0.7	-0.4

Excel formula:

=ELLIPKINC(0.7, -0.4)

Expected output:

0.680725

Python Code

from scipy.special import ellipkinc as scipy_ellipkinc

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

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

Online Calculator

phi *

Amplitude angle in radians.

m *

Elliptic parameter (dimensionless).