BESSEL_KV

The modified cylindrical Bessel function of the second kind, K_v(z), is a decaying solution of the modified Bessel equation and is frequently used for radial decay and boundary-layer behavior.

This function evaluates K_v(z) for real order v and positive real argument z.

z^2 y'' + z y' - (z^2 + v^2)y = 0

where one decaying solution is y = K_v(z).

Excel Usage

=BESSEL_KV(v, z)

v (float, required): Order of the modified Bessel function (dimensionless).
z (float, required): Positive argument where the function is evaluated (dimensionless).

Returns (float): Value of the modified Bessel function of the second kind at the specified order and argument.

Example 1: Zero order at unit argument

Inputs:

v	z
0	1

Excel formula:

=BESSEL_KV(0, 1)

Expected output:

0.421024

Example 2: First order at small positive argument

Inputs:

v	z
1	0.8

Excel formula:

=BESSEL_KV(1, 0.8)

Expected output:

0.861782

Example 3: Half order at moderate argument

Inputs:

v	z
0.5	2

Excel formula:

=BESSEL_KV(0.5, 2)

Expected output:

0.119938

Example 4: Second order at larger argument

Inputs:

v	z
2	4

Excel formula:

=BESSEL_KV(2, 4)

Expected output:

0.0174014

Python Code

from scipy.special import kv as scipy_kv

def bessel_kv(v, z):
    """
    Compute the modified cylindrical Bessel function of the second kind for real order.

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

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

    Args:
        v (float): Order of the modified Bessel function (dimensionless).
        z (float): Positive argument where the function is evaluated (dimensionless).

    Returns:
        float: Value of the modified Bessel function of the second kind at the specified order and argument.
    """
    try:
        v = float(v)
        z = float(z)
        if z <= 0:
            return "Error: z must be greater than 0 for real-valued output"
        return float(scipy_kv(v, z))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

v *

Order of the modified Bessel function (dimensionless).

z *

Positive argument where the function is evaluated (dimensionless).