ELLIPKM1

This function evaluates the complete elliptic integral of the first kind using the complementary parameter p=1-m, which improves numerical behavior when m is close to 1.

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

This wrapper computes the same K value by accepting p directly.

Excel Usage

=ELLIPKM1(p)

• p (float, required): Complementary elliptic parameter where m equals one minus p (dimensionless).

Returns (float): Complete elliptic integral of the first kind for m equals one minus p.

Example 1: Stable first-kind complete integral with p one-half

Inputs:

p
0.5

Excel formula:

=ELLIPKM1(0.5)

Expected output:

1.85407

Example 2: Stable first-kind complete integral with small p

Inputs:

p
0.01

Excel formula:

=ELLIPKM1(0.01)

Expected output:

3.69564

Example 3: Stable first-kind complete integral with p equal to one

Inputs:

p
1

Excel formula:

=ELLIPKM1(1)

Expected output:

1.5708

Example 4: Stable first-kind complete integral with p greater than one

Inputs:

p
2

Excel formula:

=ELLIPKM1(2)

Expected output:

1.31103

Python Code

from scipy.special import ellipkm1 as scipy_ellipkm1

def ellipkm1(p):
    """
    Compute the complete elliptic integral of the first kind near m equals one.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.ellipkm1.html

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

    Args:
        p (float): Complementary elliptic parameter where m equals one minus p (dimensionless).

    Returns:
        float: Complete elliptic integral of the first kind for m equals one minus p.
    """
    try:
        return float(scipy_ellipkm1(p))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

p *

Complementary elliptic parameter where m equals one minus p (dimensionless).