SPHERICAL_YN

The spherical Bessel function of the second kind, y_n(z), is the second independent spherical solution paired with j_n(z) in radial problems.

It is related to the cylindrical Bessel function by:

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

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

Excel Usage

=SPHERICAL_YN(n, z, derivative)

n (int, required): Nonnegative integer order of the 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 spherical Bessel function of the second kind (or derivative) at order n and argument z.

Example 1: Zero order function value

Inputs:

n	z	derivative
0	1	false

Excel formula:

=SPHERICAL_YN(0, 1, FALSE)

Expected output:

-0.540302

Example 2: First order function value

Inputs:

n	z	derivative
1	2	false

Excel formula:

=SPHERICAL_YN(1, 2, FALSE)

Expected output:

-0.350612

Example 3: Second order derivative value

Inputs:

n	z	derivative
2	1.5	true

Excel formula:

=SPHERICAL_YN(2, 1.5, TRUE)

Expected output:

1.99499

Example 4: Third order at moderate argument

Inputs:

n	z	derivative
3	4	false

Excel formula:

=SPHERICAL_YN(3, 4, FALSE)

Expected output:

-0.218642

Python Code

from scipy.special import spherical_yn as scipy_spherical_yn

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

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

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

    Args:
        n (int): Nonnegative integer order of the 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 spherical Bessel function of the second kind (or derivative) at order n and argument z.
    """
    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_yn(n, z, derivative=derivative))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

n *

Nonnegative integer order of the 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).