SPHERICAL_KN

The modified spherical Bessel function of the second kind, k_n(z), is a decaying spherical solution used in screened or evanescent radial fields.

It is related to the modified cylindrical Bessel function by:

k_n(z) = \sqrt{\frac{\pi}{2z}}\,K_{n+1/2}(z)

This function evaluates either k_n(z) or its derivative depending on the derivative flag.

Excel Usage

=SPHERICAL_KN(n, z, derivative)

n (int, required): Nonnegative integer order of the modified spherical Bessel function (dimensionless).
z (float, required): Positive argument where the function is evaluated (dimensionless).
derivative (bool, optional, default: false): Whether to return the derivative instead of the function value (true or false).

Returns (float): Value of the modified spherical Bessel function of the second kind (or derivative).

Example 1: Zero order function value

Inputs:

n	z	derivative
0	1	false

Excel formula:

=SPHERICAL_KN(0, 1, FALSE)

Expected output:

0.577864

Example 2: First order function value

Inputs:

n	z	derivative
1	2	false

Excel formula:

=SPHERICAL_KN(1, 2, FALSE)

Expected output:

0.159438

Example 3: Second order derivative value

Inputs:

n	z	derivative
2	1.5	true

Excel formula:

=SPHERICAL_KN(2, 1.5, TRUE)

Expected output:

-2.4145

Example 4: Third order at moderate argument

Inputs:

n	z	derivative
3	2.5	false

Excel formula:

=SPHERICAL_KN(3, 2.5, FALSE)

Expected output:

0.348651

Python Code

from scipy.special import spherical_kn as scipy_spherical_kn

def spherical_kn(n, z, derivative=False):
    """
    Compute the modified spherical Bessel function of the second kind or its derivative.

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

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

    Args:
        n (int): Nonnegative integer order of the modified spherical Bessel function (dimensionless).
        z (float): Positive argument where the function is evaluated (dimensionless).
        derivative (bool, optional): Whether to return the derivative instead of the function value (true or false). Default is False.

    Returns:
        float: Value of the modified spherical Bessel function of the second kind (or derivative).
    """
    try:
        n = int(n)
        z = float(z)
        derivative = bool(derivative)
        if n < 0:
            return "Error: n must be greater than or equal to 0"
        if z <= 0:
            return "Error: z must be greater than 0 for real-valued output"
        return float(scipy_spherical_kn(n, z, derivative=derivative))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

n *

Nonnegative integer order of the modified spherical Bessel function (dimensionless).

z *

Positive argument where the function is evaluated (dimensionless).

derivative

Whether to return the derivative instead of the function value (true or false).