ELLIPRD

Carlson’s symmetric integral R_D is a three-argument elliptic integral that emphasizes the third argument and is used in transformations of Legendre second-kind integrals.

R_D(x,y,z)=\frac{3}{2}\int_0^{\infty}[(t+x)(t+y)]^{-1/2}(t+z)^{-3/2}\,dt

This wrapper evaluates R_D for scalar real inputs with nonzero z.

Excel Usage

=ELLIPRD(x, y, z)

x (float, required): First nonnegative real parameter.
y (float, required): Second nonnegative real parameter.
z (float, required): Third real parameter and weighting pole location; must be nonzero.

Returns (float): Value of Carlson symmetric elliptic integral RD.

Example 1: Carlson RD with positive parameters

Inputs:

x	y	z
1	2	3

Excel formula:

=ELLIPRD(1, 2, 3)

Expected output:

0.29046

Example 2: Carlson RD with zero first parameter

Inputs:

x	y	z
0	2	1

Excel formula:

=ELLIPRD(0, 2, 1)

Expected output:

1.79721

Example 3: Carlson RD when all parameters are equal and nonzero

Inputs:

x	y	z
2	2	2

Excel formula:

=ELLIPRD(2, 2, 2)

Expected output:

0.353553

Example 4: Carlson RD with fractional parameters

Inputs:

x	y	z
0.5	1.5	2.5

Excel formula:

=ELLIPRD(0.5, 1.5, 2.5)

Expected output:

0.439501

Python Code

from scipy.special import elliprd as scipy_elliprd

def elliprd(x, y, z):
    """
    Compute Carlson's symmetric elliptic integral RD.

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

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

    Args:
        x (float): First nonnegative real parameter.
        y (float): Second nonnegative real parameter.
        z (float): Third real parameter and weighting pole location; must be nonzero.

    Returns:
        float: Value of Carlson symmetric elliptic integral RD.
    """
    try:
        if z == 0:
            return "Error: z must be nonzero"
        return float(scipy_elliprd(x, y, z))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

x *

First nonnegative real parameter.

y *

Second nonnegative real parameter.

z *

Third real parameter and weighting pole location; must be nonzero.