SPHERICAL_IN

The modified spherical Bessel function of the first kind, i_n(z), is used in spherical problems with exponentially growing or decaying radial behavior.

It is related to the modified cylindrical Bessel function by:

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

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

Excel Usage

=SPHERICAL_IN(n, z, derivative)

n (int, required): Nonnegative integer order of the modified spherical Bessel function (dimensionless).
z (float, required): 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 first kind (or derivative).

Example 1: Zero order function value

Inputs:

n	z	derivative
0	1	false

Excel formula:

=SPHERICAL_IN(0, 1, FALSE)

Expected output:

1.1752

Example 2: First order function value

Inputs:

n	z	derivative
1	2	false

Excel formula:

=SPHERICAL_IN(1, 2, FALSE)

Expected output:

0.974383

Example 3: Second order derivative value

Inputs:

n	z	derivative
2	1.5	true

Excel formula:

=SPHERICAL_IN(2, 1.5, TRUE)

Expected output:

0.270594

Example 4: Third order at moderate argument

Inputs:

n	z	derivative
3	2.5	false

Excel formula:

=SPHERICAL_IN(3, 2.5, FALSE)

Expected output:

0.208439

Python Code

from scipy.special import spherical_in as scipy_spherical_in

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

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.spherical_in.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): 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 first 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"
        return float(scipy_spherical_in(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 *

Argument where the function is evaluated (dimensionless).

derivative

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