ELLIPRJ

Carlson’s symmetric integral R_J is a four-argument elliptic integral associated with third-kind behavior and a simple pole at p.

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

This wrapper evaluates R_J for scalar real inputs with nonzero p.

Excel Usage

=ELLIPRJ(x, y, z, p)

x (float, required): First nonnegative real parameter.
y (float, required): Second nonnegative real parameter.
z (float, required): Third nonnegative real parameter.
p (float, required): Pole parameter and fourth argument; must be nonzero.

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

Example 1: Carlson RJ with positive parameters

Inputs:

x	y	z	p
1	2	3	4

Excel formula:

=ELLIPRJ(1, 2, 3, 4)

Expected output:

0.239848

Example 2: Carlson RJ with p equal to z

Inputs:

x	y	z	p
1	2	3	3

Excel formula:

=ELLIPRJ(1, 2, 3, 3)

Expected output:

0.29046

Example 3: Carlson RJ with one zero among x y z

Inputs:

x	y	z	p
0	2	3	4

Excel formula:

=ELLIPRJ(0, 2, 3, 4)

Expected output:

0.421143

Example 4: Carlson RJ with fractional parameters

Inputs:

x	y	z	p
0.5	1.5	2.5	3.5

Excel formula:

=ELLIPRJ(0.5, 1.5, 2.5, 3.5)

Expected output:

0.348748

Python Code

from scipy.special import elliprj as scipy_elliprj

def elliprj(x, y, z, p):
    """
    Compute Carlson's symmetric elliptic integral RJ.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.elliprj.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 nonnegative real parameter.
        p (float): Pole parameter and fourth argument; must be nonzero.

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

Online Calculator

x *

First nonnegative real parameter.

y *

Second nonnegative real parameter.

z *

Third nonnegative real parameter.

p *

Pole parameter and fourth argument; must be nonzero.