BESSEL_HANKEL1

The Hankel function of the first kind, H_v^{(1)}(z), combines Bessel functions as H_v^{(1)} = J_v + iY_v and represents outward-propagating cylindrical waves.

Because this function is generally complex-valued, the result is returned as a 2D array containing real and imaginary parts.

H_v^{(1)}(z) = J_v(z) + iY_v(z)

Excel Usage

=BESSEL_HANKEL1(v, z)

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

Returns (list[list]): One-row array with real and imaginary parts as [[real, imag]].

Example 1: Zero order at unit argument

Inputs:

v	z
0	1

Excel formula:

=BESSEL_HANKEL1(0, 1)

Expected output:

Result
0.765198	0.088257

Example 2: First order at moderate argument

Inputs:

v	z
1	2

Excel formula:

=BESSEL_HANKEL1(1, 2)

Expected output:

Result
0.576725	-0.107032

Example 3: Half order at positive argument

Inputs:

v	z
0.5	3

Excel formula:

=BESSEL_HANKEL1(0.5, 3)

Expected output:

Result
0.0650082	0.456049

Example 4: Second order at larger argument

Inputs:

v	z
2	5

Excel formula:

=BESSEL_HANKEL1(2, 5)

Expected output:

Result
0.0465651	0.367663

Python Code

from scipy.special import hankel1 as scipy_hankel1

def bessel_hankel1(v, z):
    """
    Compute the cylindrical Hankel function of the first kind and return real and imaginary parts.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.hankel1.html

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

    Args:
        v (float): Order of the Hankel function (dimensionless).
        z (float): Argument where the function is evaluated (dimensionless).

    Returns:
        list[list]: One-row array with real and imaginary parts as [[real, imag]].
    """
    try:
        v = float(v)
        z = float(z)
        result = scipy_hankel1(v, z)
        return [[float(result.real), float(result.imag)]]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

v *

Order of the Hankel function (dimensionless).

z *

Argument where the function is evaluated (dimensionless).