BESSEL_IV

The modified cylindrical Bessel function of the first kind, I_v(z), solves the modified Bessel equation and grows exponentially for large positive arguments. It is used in diffusion, heat transfer, and radial problems with hyperbolic behavior.

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

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

where one standard solution is y = I_v(z).

Excel Usage

=BESSEL_IV(v, z)

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

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

Example 1: Zero order at unit argument

Inputs:

v	z
0	1

Excel formula:

=BESSEL_IV(0, 1)

Expected output:

1.26607

Example 2: First order at positive argument

Inputs:

v	z
1	2

Excel formula:

=BESSEL_IV(1, 2)

Expected output:

1.59064

Example 3: Half order at moderate argument

Inputs:

v	z
0.5	1.5

Excel formula:

=BESSEL_IV(0.5, 1.5)

Expected output:

1.38716

Example 4: Second order at negative argument

Inputs:

v	z
2	-1

Excel formula:

=BESSEL_IV(2, -1)

Expected output:

0.135748

Python Code

from scipy.special import iv as scipy_iv

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

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.iv.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): Argument where the function is evaluated (dimensionless).

    Returns:
        float: Value of the modified Bessel function of the first kind at the specified order and argument.
    """
    try:
        v = float(v)
        z = float(z)
        return float(scipy_iv(v, z))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

v *

Order of the modified Bessel function (dimensionless).

z *

Argument where the function is evaluated (dimensionless).